Finite Group Databasebeta

← #33108585

Subgroups — #33108585

Order 18688.

994 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988814C211yesmaximal, characteristic
2C2×#17988678C211yesmaximal, characteristic, complemented
3C2×#17988623C211yesmaximal, characteristic, complemented
4#22612973C211yesmaximal, characteristic
5#22612958C211yesmaximal, characteristic, complemented
6#22610786C211yesmaximal, characteristic, complemented
7#22610658C211yesmaximal, characteristic, complemented
8#22610570C211yesmaximal, characteristic, complemented
9#22610554C211yesmaximal, characteristic, complemented
10#22610241C211yesmaximal, characteristic, complemented
11#22610186C211yesmaximal, characteristic, complemented
12#22610176C211yesmaximal, characteristic, complemented
13#22609819C211yesmaximal, characteristic, complemented
14#22609705C211yesmaximal, characteristic, complemented
15C73×#1875C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C23×#10596831C2211yescharacteristic
18C2×#14556570C2211yescharacteristic
19C2×#14556556C2211yescharacteristic
20C2×D4×Dic73C2211yescharacteristic
21C2×#14556550C2211yescharacteristic
22#17988814C418yescomplemented
23#17988678—24no—
24#17988623—24no—
25C2×#14556524C2211yescharacteristic
26C2×#14556524C2211yescharacteristic
27C2×#14556522C2211yescharacteristic
28C2×#14556522C2211yescharacteristic
29C2×#14556522C2211yescharacteristic
30C2×#14556516C2211yescharacteristic
31C2×#14556516C2211yescharacteristic
32C2×#14556514C2211yescharacteristic
33C2×#14556512C2211yescharacteristic
34C2×C8×Dic73C2211yescharacteristic
35C2×C8×Dic73C2211yescharacteristic
36#17988492C2211yescharacteristic
37#17988490—21no—
38#17988490C2211yescharacteristic
39#17988486C2211yescharacteristic
40#17988483C2211yescharacteristic
41#17988481—21no—
42#17988480—21no—
43#17988475—21no—
44#17988471—21no—
45#17988467—21no—
46C4⋊C4×Dic73C2211yescharacteristic
47C4⋊C4×Dic73C2211yescharacteristic
48C4×#11924007—21no—
49C22⋊C4×Dic73—21no—
50C4×C22⋊Dic73C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53#17988341C2211yescharacteristic
54#17988341C2211yescharacteristic
55C2×#121×C73C2211yescharacteristic
56C2×#117×C73C2211yescharacteristic
57C2×#102×C73C2211yescharacteristic
58C2×#102×C73C2211yescharacteristic
59C2×C22⋊C8×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61#340×C73C2211yescharacteristic
62C22×Dic146C2311yescharacteristic
63C23×Dic73—21no—
64C2×#11924016—22no—
65C2×#11924016—22no—
66C2×#11924016C2×C414yes—
67C22×#10596831—22no—
68C22×#10596831—22no—
69C22×#10596831C2×C414yescomplemented
70C22×#10596831C2311yescharacteristic
71C22×#10596831C2311yescharacteristic
72#14556570—22no—
73#14556570C2×C414yescomplemented
74#14556556—22no—
75#14556556—41no—
76D4×Dic73—22no—
77D4×Dic73—41no—
78#14556550—22no—
79#14556550—41no—
80C2×C4⋊Dic73—21no—
81C2×C4⋊Dic73C2311yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×C4⋊Dic73C2311yescharacteristic
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×#11924007—21no—
87C2×C22⋊Dic73—21no—
88C2×C22⋊Dic73—21no—
89C2×C22⋊Dic73—21no—
90C2×C22⋊Dic73—21no—
91C2×C22⋊Dic73—21no—
92C2×C22⋊Dic73D411yescharacteristic
93C2×C22⋊Dic73D411yescharacteristic
94C2×C4×Dic73—21no—
95C2×C4×Dic73—21no—
96C2×C4×Dic73—21no—
97C2×C4×Dic73—21no—
98C2×C4×Dic73C2311yescharacteristic
99C2×C4×Dic73C2311yescharacteristic
100#14556524—22no—
101#14556524—22no—
102#14556522—22no—
103#14556522C2×C414yescomplemented
104#14556522D414yescomplemented
105#14556516—22no—
106#14556516—41no—
107#14556514C2×C414yescomplemented
108#14556512C2×C414yes—
109C8×Dic73—22no—
110C8×Dic73—22no—
111#14556498—21no—
112#14556498—21no—
113#14556498—21no—
114#14556498—21no—
115#14556498—21no—
116#14556498—21no—
117#14556498D411yescharacteristic
118#14556498D411yescharacteristic
119C22×D4×C73—21no—
120C22×D4×C73C2311yescharacteristic
121C23×C4×C73C2311yescharacteristic
122C22×C8×C73C2311yescharacteristic
123C22×C8×C73C2311yescharacteristic
124#121×C73—22no—
125#121×C73C2×C414yes—
126#117×C73—22no—
127#117×C73—41no—
128C2×C4⋊C4×C73C2311yescharacteristic
129C2×C4⋊C4×C73C2311yescharacteristic
130C2×C4⋊C4×C73C2311yescharacteristic
131C2×C22⋊C4×C73—21no—
132C2×C22⋊C4×C73—21no—
133C2×C42×C73—21no—
134#102×C73—22no—
135#102×C73—41no—
136C22⋊C8×C73—22no—
137C2×Dic146—21no—
138C2×Dic146—21no—
139C2×Dic146—21no—
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146C22×C412yes—
143C22×Dic73—21no—
144C22×Dic73—21no—
145C22×Dic73—21no—
146C22×Dic73—21no—
147C22×Dic73—42no—
148C22×Dic73—42no—
149C22×Dic73—81no—
150C22×Dic73C2×D411yescharacteristic
151C22×Dic73C2×D411yescharacteristic
152C22×Dic73C2×D411yescharacteristic
153C22×Dic73C2×D411yescharacteristic
154#11924016—28no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—28no—
158C2×#10596831C22×C412yes—
159C2×#10596831C22×C412yes—
160C2×#10596831C2×D412yes—
161C2×#10596831#4112yes—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—21no—
166C4⋊Dic73—81no—
167C4⋊Dic73C22×C412yes—
168C4⋊Dic73C2×D412yes—
169#11924007—24no—
170#11924007—42no—
171#11924007—42no—
172C22⋊Dic73—24no—
173C22⋊Dic73—22no—
174C22⋊Dic73—22no—
175C22⋊Dic73—42no—
176C22⋊Dic73—42no—
177C22⋊Dic73—42no—
178C22⋊Dic73—81no—
179C4×Dic73—21no—
180C4×Dic73—21no—
181C4×Dic73—21no—
182C4×Dic73—21no—
183C4×Dic73—42no—
184C4×Dic73—42no—
185C4×Dic73—42no—
186C4×Dic73—81no—
187C24×C73—21no—
188C24×C73C2×D411yescharacteristic
189C2×D4×C73—24no—
190C2×D4×C73—21no—
191C2×D4×C73—21no—
192C2×D4×C73—21no—
193C2×D4×C73—21no—
194C2×D4×C73—21no—
195C2×D4×C73—81no—
196C2×D4×C73—81no—
197C2×D4×C73C22×C412yes—
198C22×C4×C73—21no—
199C22×C4×C73—21no—
200C22×C4×C73—21no—
201C22×C4×C73—21no—
202C22×C4×C73—21no—
203C22×C4×C73—21no—
204C22×C4×C73—21no—
205C22×C4×C73C22×C412yes—
206C22×C4×C73C2×D411yescharacteristic
207C22×C4×C73#4112yes—
208C22×C4×C73C2411yescharacteristic
209C2×C8×C73—21no—
210C2×C8×C73—21no—
211C2×C8×C73—21no—
212C2×C8×C73—21no—
213C2×C8×C73—21no—
214C2×C8×C73—21no—
215C4⋊C4×C73—21no—
216C4⋊C4×C73—21no—
217C4⋊C4×C73—21no—
218C4⋊C4×C73—21no—
219C4⋊C4×C73C22×C412yes—
220C4⋊C4×C73C2×D412yes—
221C22⋊C4×C73—24no—
222C22⋊C4×C73—42no—
223C42×C73—42no—
224C23×C73—21no—
225C23×C73—21no—
226C23×C73—21no—
227C23×C73—21no—
228C23×C73—21no—
229C23×C73—21no—
230C23×C73—22no—
231C23×C73—41no—
232C23×C73—41no—
233C23×C73—41no—
234C23×C73—41no—
235C23×C73—41no—
236C23×C73—41no—
237C23×C73C4×D412yes—
238C23×C73C22×D411yescharacteristic
239D4×C73—22no—
240D4×C73—22no—
241D4×C73—41no—
242D4×C73—41no—
243D4×C73—84no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—22no—
247C2×C4×C73—22no—
248C2×C4×C73—22no—
249C2×C4×C73—22no—
250C2×C4×C73—22no—
251C2×C4×C73—22no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73C4×D412yes—
260C2×C4×C73C23×C411yescharacteristic
261C2×C4×C73C22×D411yescharacteristic
262C2×C4×C73C2×#4111yescharacteristic
263C2×C4×C73C2×#4111yescharacteristic
264C2×C4×C73H(5,𝔽2)11yescharacteristic
265C2×C4×C732−1+411yescharacteristic
266C2×Dic73—21no—
267C2×Dic73—21no—
268C2×Dic73—21no—
269C2×Dic73—21no—
270C2×Dic73—22no—
271C2×Dic73—22no—
272C2×Dic73—21no—
273C2×Dic73—22no—
274C2×Dic73—21no—
275C2×Dic73—21no—
276C2×Dic73—21no—
277C2×Dic73—21no—
278C2×Dic73—21no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—41no—
283C2×Dic73—41no—
284C2×Dic73—41no—
285C2×Dic73—41no—
286C2×Dic73—41no—
287C2×Dic73—41no—
288C2×Dic73—84no—
289C2×Dic73#13512yes—
290C2×Dic73#13512yes—
291Dic146—22no—
292Dic146—22no—
293Dic146—41no—
294Dic146—41no—
295C8×C73—41no—
296C8×C73—41no—
297#10596831—22no—
298#10596831C4×D414yes—
299C22×C73—22no—
300C22×C73—22no—
301C22×C73—22no—
302C22×C73—41no—
303C22×C73—41no—
304C22×C73—41no—
305C22×C73—41no—
306C22×C73—42no—
307C22×C73—42no—
308C22×C73—42no—
309C22×C73—81no—
310C22×C73—81no—
311C22×C73—81no—
312C22×C73—81no—
313C22×C73C2×C4×D411yescharacteristic
314C22×C73D4211yescharacteristic
315C22×C73#54711yescharacteristic
316C22×C73C2×#13511yescharacteristic
317C22×C73C2×#13511yescharacteristic
318C22×C73#57711yescharacteristic
319C22×C73#57711yescharacteristic
320C4×C73—22no—
321C4×C73—42no—
322C4×C73—42no—
323C4×C73—81no—
324C4×C73C2×C4×D411yescharacteristic
325C4×C73#52011yescharacteristic
326C4×C73#54811yescharacteristic
327C4×C73#55311yescharacteristic
328Dic73—22no—
329Dic73—22no—
330Dic73—41no—
331Dic73—42no—
332Dic73—42no—
333Dic73—42no—
334Dic73—41no—
335Dic73—81no—
336#23023—731nomaximal, Sylow
337C2×C73—42no—
338C2×C73—82no—
339C2×C73#236111yescharacteristic
340C2×C73C4×#13511yescharacteristic
341C2×C73#294911yescharacteristic
342C2×C73#328211yescharacteristic
343C2×C73#328411yescharacteristic
344C2×C73#333711yescharacteristic
345C2×C73#334011yescharacteristic
346C2×#465—731no—
347C2×C4×SD16—731no—
348C2×#427—731no—
349#2363—731no—
350#2362—731no—
351#1968—731no—
352#1941—731no—
353#1928—731no—
354#1925—731no—
355#1875D7311yescharacteristic
356#1873—731no—
357#1863—731no—
358#1862—731no—
359C4×#106—731no—
360C8×C22⋊C4—731no—
361C73#2302311yescharacteristic, complemented, Sylow
362C22×SD16—731no—
363C23×C8—731no—
364C2×C22⋊Q8—731no—
365C2×#121C2×D7311yescharacteristic
366C2×C4×Q8—731no—
367C2×C4×D4—731no—
368C2×#117—731no—
369C2×#117C2×D7311yescharacteristic
370#465—738no—
371C4×SD16—1464no—
372#427—1464no—
373C2×#107—731no—
374C2×#107—731no—
375C2×#106—731no—
376C2×#106—731no—
377C2×#106—731no—
378C2×#103—731no—
379C2×#103—731no—
380C2×#103—731no—
381C2×#102—731no—
382C2×#102C2×D7311yescharacteristic
383C2×#102C2×D7311yescharacteristic
384C2×C22⋊C8C2×D7311yescharacteristic
385C2×C4×C8—731no—
386C2×C4×C8—731no—
387#391—731no—
388#390—731no—
389#390—1461no—
390#389—731no—
391#389—731no—
392#388—1461no—
393#388—1461no—
394#388—1461no—
395#387—1461no—
396#386—1461no—
397C4×C4⋊C4—731no—
398C4×C4⋊C4—731no—
399C4×C4⋊C4—1461no—
400C4×C22⋊C4—731no—
401C4×C22⋊C4—1461no—
402#340—731no—
403#340—731no—
404#340C2×D7311yescharacteristic
405#340C2×D7311yescharacteristic
406#336—731no—
407#336—731no—
408C22×Q8—731no—
409C22×D4—21no—
410C22×D4C22×D7311yescharacteristic
411C23×C4—1461no—
412C23×C4C22×D7311yescharacteristic
413C2×SD16—734no—
414C2×SD16—1462no—
415C2×SD16—1462no—
416C22×C8—731no—
417C22×C8—734no—
418C22×C8—731no—
419C22×C8—1462no—
420C22×C8—1462no—
421C22×C8C22×D7311yescharacteristic
422C22×C8C22×D7311yescharacteristic
423C22⋊Q8—734no—
424C22⋊Q8—1462no—
425#121—22no—
426#121C4×D7314yes—
427C4×Q8—2921no—
428C4×Q8—1462no—
429C4×D4—2921no—
430C4×D4—1462no—
431#117—22no—
432#117—41no—
433#117—2921no—
434#117—1462no—
435C2×C4⋊C4—731no—
436C2×C4⋊C4—731no—
437C2×C4⋊C4—731no—
438C2×C4⋊C4—1461no—
439C2×C4⋊C4—1461no—
440C2×C4⋊C4—1461no—
441C2×C4⋊C4—1461no—
442C2×C4⋊C4C22×D7311yescharacteristic
443C2×C4⋊C4C22×D7311yescharacteristic
444C2×C4⋊C4C22×D7311yescharacteristic
445C2×C22⋊C4—21no—
446C2×C22⋊C4—21no—
447C2×C22⋊C4—731no—
448C2×C22⋊C4—731no—
449C2×C22⋊C4—1461no—
450C2×C22⋊C4—1461no—
451C2×C22⋊C4—1461no—
452C2×C22⋊C4—1461no—
453C2×C22⋊C4—1461no—
454C2×C42—21no—
455C2×C42—731no—
456C2×C42—731no—
457C2×C42—1461no—
458C2×C42—1461no—
459C2×C42—1461no—
460C2×C42—1461no—
461#107—1462no—
462#107—1462no—
463#106—734no—
464#106—734no—
465#106—1462no—
466#103—2921no—
467#103—734no—
468#103—1462no—
469#102—22no—
470#102—41no—
471#102—734no—
472C22⋊C8—22no—
473C4×C8—1462no—
474C4×C8—1462no—
475#95—731no—
476#95—731no—
477#95—1461no—
478#95—1461no—
479#95—1461no—
480#95—1461no—
481#95—1461no—
482#95—1461no—
483C24—21no—
484C24D4×D7311yescharacteristic
485C2×Q8—732no—
486C2×Q8—1461no—
487C2×Q8—1461no—
488C2×Q8—1461no—
489C2×Q8—1461no—
490C2×Q8—1461no—
491C2×D4—24no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—81no—
498C2×D4—81no—
499C2×D4C2×C4×D7312yes—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—2922no—
508C22×C4—2922no—
509C22×C4—5841no—
510C22×C4—731no—
511C22×C4—731no—
512C22×C4—731no—
513C22×C4—731no—
514C22×C4—1461no—
515C22×C4—1461no—
516C22×C4—1461no—
517C22×C4—1461no—
518C22×C4C2×C4×D7312yes—
519C22×C4D4×D7311yescharacteristic
520C22×C4#1192402712yes—
521C22×C4C23×D7311yescharacteristic
522SD16—1468no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—732no—
530C2×C8—732no—
531C2×C8—732no—
532C2×C8—732no—
533C2×C8—1468no—
534C2×C8—1461no—
535C2×C8—1461no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—2922no—
541C4⋊C4—2922no—
542C4⋊C4—5841no—
543C4⋊C4—732no—
544C4⋊C4—732no—
545C4⋊C4—1461no—
546C4⋊C4—1461no—
547C4⋊C4—1464no—
548C4⋊C4—1461no—
549C4⋊C4—1461no—
550C4⋊C4C2×C4×D7312yes—
551C4⋊C4D4×D7312yes—
552C22⋊C4—24no—
553C22⋊C4—42no—
554C22⋊C4—2922no—
555C22⋊C4—2922no—
556C22⋊C4—2922no—
557C22⋊C4—5841no—
558C22⋊C4—1462no—
559C22⋊C4—1462no—
560C22⋊C4—1464no—
561C42—42no—
562C42—2922no—
563C42—2922no—
564C42—2922no—
565C42—5841no—
566C42—1461no—
567C42—1461no—
568C42—1461no—
569C42—1461no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—22no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23#1455655112yes—
584C23C2×D4×D7311yescharacteristic
585Q8—2921no—
586Q8—2921no—
587Q8—1462no—
588Q8—1462no—
589D4—22no—
590D4—22no—
591D4—41no—
592D4—41no—
593D4—84no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—2921no—
610C2×C4—2921no—
611C2×C4—2921no—
612C2×C4—2921no—
613C2×C4—2921no—
614C2×C4—2921no—
615C2×C4—2921no—
616C2×C4—2921no—
617C2×C4—2921no—
618C2×C4—5844no—
619C2×C4—732no—
620C2×C4—732no—
621C2×C4—1461no—
622C2×C4—1461no—
623C2×C4—1462no—
624C2×C4—1461no—
625C2×C4—1461no—
626C2×C4—1462no—
627C2×C4—1462no—
628C2×C4—1461no—
629C2×C4—1461no—
630C2×C4—1461no—
631C2×C4—1461no—
632C2×C4—1461no—
633C2×C4—1461no—
634C2×C4#1455655112yes—
635C2×C4C22×C4×D7311yescharacteristic
636C2×C4C2×D4×D7311yescharacteristic
637C2×C4#41×D7311yescharacteristic
638C2×C4C2×#1192402711yescharacteristic
639C2×C4#1455663611yescharacteristic
640C2×C4#1455663811yescharacteristic
641C8—41no—
642C8—41no—
643C8—734no—
644C8—1462no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22C2×#1455655111yescharacteristic
660C22#1798921911yescharacteristic
661C22#1798923511yescharacteristic
662C22#135×D7311yescharacteristic
663C22#135×D7311yescharacteristic
664C22#1798939711yescharacteristic
665C22#1798939711yescharacteristic
666C4—22no—
667C4—42no—
668C4—42no—
669C4—81no—
670C4—2922no—
671C4—2921no—
672C4—2922no—
673C4—2922no—
674C4—2921no—
675C4—5841no—
676C4—1462no—
677C4—1462no—
678C4C4×D4×D7311yescharacteristic
679C4#1798906611yescharacteristic
680C4#1798924311yescharacteristic
681C4#1798928011yescharacteristic
682C2—42no—
683C2—82no—
684C2#2261294211yescharacteristic
685C2#2261869911yescharacteristic
686C2#2261879311yescharacteristic
687C2#2262164111yescharacteristic
688C2#2262165911yescharacteristic
689C2#2262230311yescharacteristic
690C2#2262234211yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1875C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
cosocleC22×C4×C73C24
hypercenter#1875D73
hypocenterC73#23023
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1875

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#17988678C2semidirect
3. C2×#17988623C2semidirect
5. #22612958C2semidirect
6. #22610786C2semidirect
7. #22610658C2semidirect
8. #22610570C2semidirect
9. #22610554C2semidirect
10. #22610241C2semidirect
11. #22610186C2semidirect
12. #22610176C2semidirect
13. #22609819C2semidirect
14. #22609705C2semidirect
22. #17988814C4semidirect
69. C22×#10596831C2×C4semidirect
73. #14556570C2×C4semidirect
103. #14556522C2×C4semidirect
104. #14556522D4semidirect
107. #14556514C2×C4semidirect
361. C73#23023semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179888142
C2×#179886782
C2×#179886232
#226129732
#226129582
#226107862
#226106582
#226105702
#226105542
#226102412
#226101862
#226101762
#226098192
#226097052
C73×#18752
#2302373