Finite Group Databasebeta

← #33105237

Subgroups — #33105237

Order 18688.

886 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988678C211yesmaximal, characteristic
2#22612879C211yesmaximal, characteristic
3#22612845C211yesmaximal, characteristic, complemented
4#22610616C211yesmaximal, characteristic
5#22610519C211yesmaximal, characteristic, complemented
6#22610513C211yesmaximal, characteristic, complemented
7#22610289C211yesmaximal, characteristic
8#22610283C211yesmaximal, characteristic
9#22610057C211yesmaximal, characteristic, complemented
10#22609935C211yesmaximal, characteristic, complemented
11#22609929C211yesmaximal, characteristic
12C8×#11924007C211yesmaximal, characteristic, complemented
13C4×#14556516C211yesmaximal, characteristic, complemented
14#22609750C211yesmaximal, characteristic, complemented
15C73×#1934C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18C2×#14556556C2211yescharacteristic
19C2×#14556556C2211yescharacteristic
20C2×D4×Dic73C2211yescharacteristic
21C2×#14556547—21no—
22#17988678C418yescomplemented
23C2×#14556522C2211yescharacteristic
24C2×#14556520C2211yescharacteristic
25C2×#14556516C2211yescharacteristic
26C2×#14556516C2211yescharacteristic
27C2×#14556514C2211yescharacteristic
28C2×#14556512C2211yescharacteristic
29C2×C8×Dic73C2211yescharacteristic
30C2×C8×Dic73C2211yescharacteristic
31#17988490C2211yescharacteristic
32#17988486C2211yescharacteristic
33#17988484—21no—
34#17988482—21no—
35#17988481—21no—
36#17988475—21no—
37#17988456—21no—
38#17988456C2211yescharacteristic
39#17988456C2211yescharacteristic
40#17988445—21no—
41C4⋊C4×Dic73C2211yescharacteristic
42C4×#11924007C2211yescharacteristic
43C4×C4⋊Dic73C2211yescharacteristic
44#17988438—21no—
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988342C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×#102×C73C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55C2×C4×C8×C73C2211yescharacteristic
56#390×C73C2211yescharacteristic
57#389×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C22×Dic146C2311yescharacteristic
61C23×Dic73—21no—
62C2×#11924016—22no—
63C2×#11924016—22no—
64C2×#11924016C2×C414yes—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C4×Dic146—22no—
68C4×Dic146—41no—
69#14556556—22no—
70#14556556—22no—
71#14556556—41no—
72#14556556C2×C414yes—
73D4×Dic73—22no—
74D4×Dic73C2×C414yes—
75#14556547—44no—
76C2×C4⋊Dic73—21no—
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×#11924007—21no—
87C2×#11924007D411yescharacteristic
88C2×#11924007D411yescharacteristic
89C2×C22⋊Dic73—21no—
90C2×C22⋊Dic73—21no—
91C2×C22⋊Dic73—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73—21no—
94C2×C4×Dic73—21no—
95C2×C4×Dic73—21no—
96C2×C4×Dic73—21no—
97C2×C4×Dic73C2311yescharacteristic
98C2×C4×Dic73C2311yescharacteristic
99#14556522C2×C414yes—
100#14556520—22no—
101#14556516—22no—
102#14556516C2×C414yes—
103#14556514—22no—
104#14556512—41no—
105C8×Dic73—41no—
106C8×Dic73C2×C414yes—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498D411yescharacteristic
111#14556498D411yescharacteristic
112C22×D4×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C22⋊C4×C73—21no—
120C2×C22⋊C4×C73—21no—
121C2×C42×C73C2311yescharacteristic
122#102×C73—41no—
123#102×C73C2×C414yes—
124C4×C8×C73—41no—
125#95×C73—21no—
126C2×Dic146—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146C22×C412yes—
130C2×Dic146#4112yes—
131C2×Dic146#4112yes—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—24no—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C22×Dic73—42no—
138C22×Dic73—42no—
139C22×Dic73C2×D411yescharacteristic
140C22×Dic73C2×D411yescharacteristic
141C22×Dic73C2×D411yescharacteristic
142C22×Dic73C2×D411yescharacteristic
143#11924016—28no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831C22×C412yes—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—81no—
156C4⋊Dic73C22×C412yes—
157#11924007—24no—
158#11924007—22no—
159#11924007—42no—
160#11924007—42no—
161#11924007—42no—
162#11924007—42no—
163#11924007—81no—
164#11924007—81no—
165#11924007SD1614yescomplemented
166C22⋊Dic73—24no—
167C22⋊Dic73—42no—
168C22⋊Dic73—81no—
169C4×Dic73—21no—
170C4×Dic73—21no—
171C4×Dic73—24no—
172C4×Dic73—42no—
173C4×Dic73—42no—
174C4×Dic73—42no—
175C4×Dic73—42no—
176C4×Dic73C22×C412yes—
177C4×Dic73#4112yes—
178C24×C73—21no—
179C2×D4×C73—21no—
180C2×D4×C73—21no—
181C2×D4×C73—21no—
182C2×D4×C73—21no—
183C2×D4×C73—21no—
184C2×D4×C73C22×C412yes—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2411yescharacteristic
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73C22×C412yes—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—81no—
203C4⋊C4×C73C22×C412yes—
204C22⋊C4×C73—42no—
205C22⋊C4×C73—81no—
206C42×C73—21no—
207C42×C73—21no—
208C23×C73—22no—
209C23×C73—41no—
210C23×C73—41no—
211C23×C73—41no—
212C23×C73—41no—
213C23×C73—41no—
214C23×C73—41no—
215C23×C73C22×D411yescharacteristic
216D4×C73—22no—
217D4×C73—22no—
218D4×C73—41no—
219D4×C73—41no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—22no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73C2×SD1612yes—
235C2×C4×C73C23×C411yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73H(5,𝔽2)11yescharacteristic
240C2×C4×C732−1+411yescharacteristic
241C2×Dic73—22no—
242C2×Dic73—21no—
243C2×Dic73—22no—
244C2×Dic73—22no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—22no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—48no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73C4×D412yes—
262C2×Dic73C4×D412yes—
263C2×Dic73C2×SD1612yes—
264C2×Dic73C2×SD1612yes—
265C2×Dic73#13512yes—
266Dic146—22no—
267Dic146—22no—
268Dic146—22no—
269Dic146#11714yescomplemented
270C8×C73—22no—
271C8×C73—41no—
272#10596831—22no—
273#10596831—41no—
274C22×C73—42no—
275C22×C73—42no—
276C22×C73—42no—
277C22×C73—81no—
278C22×C73—81no—
279C22×C73—81no—
280C22×C73—81no—
281C22×C73C2×C4×D411yescharacteristic
282C22×C73#54611yescharacteristic
283C22×C73#54811yescharacteristic
284C22×C73C22×SD1611yescharacteristic
285C22×C73C2×#13511yescharacteristic
286C22×C73#57611yescharacteristic
287C22×C73#57811yescharacteristic
288C4×C73—22no—
289C4×C73—42no—
290C4×C73—41no—
291C4×C73—81no—
292C4×C73—81no—
293C4×C73C2×#11711yescharacteristic
294C4×C73#52011yescharacteristic
295C4×C73#55211yescharacteristic
296C4×C73#55311yescharacteristic
297Dic73—24no—
298Dic73—22no—
299Dic73—22no—
300Dic73—42no—
301Dic73—41no—
302Dic73—81no—
303Dic73—81no—
304Dic73C4×SD1614yes—
305#22767—731nomaximal, Sylow
306C2×C73—82no—
307C2×C73#235211yescharacteristic
308C2×C73C2×C4×SD1611yescharacteristic
309C2×C73#295111yescharacteristic
310C2×C73#329811yescharacteristic
311C2×C73#330511yescharacteristic
312C2×C73#336211yescharacteristic
313C2×C73#336411yescharacteristic
314C2×C4×SD16—731no—
315#2354—731no—
316Q8×C4⋊C4—731no—
317#1934—731no—
318#1934D7311yescharacteristic
319#1920—731no—
320#1920—731no—
321#1880—731no—
322#1880—731no—
323#1834—731no—
324#1803—731no—
325#1802—731no—
326C8×C4⋊C4—731no—
327C4×#103—731no—
328C4×#103—731no—
329C73#2276711yescharacteristic, complemented, Sylow
330C22×SD16—731no—
331C2×C4×Q8—731no—
332C2×C4×Q8—731no—
333C2×C4×Q8—731no—
334C2×C4×D4—731no—
335C2×#117—1461no—
336C4×SD16—738no—
337C2×#106—731no—
338C2×C4⋊C8—731no—
339C2×#103—731no—
340C2×#103—731no—
341C2×#103—731no—
342C2×#102—731no—
343C2×#102C2×D7311yescharacteristic
344C2×#102C2×D7311yescharacteristic
345C2×C4×C8—731no—
346C2×C4×C8—731no—
347C2×C4×C8C2×D7311yescharacteristic
348#390—731no—
349#390C2×D7311yescharacteristic
350#389—731no—
351#389—1461no—
352#389—1461no—
353#389C2×D7311yescharacteristic
354#388—1461no—
355#388—1461no—
356#384—731no—
357#384—731no—
358#384—1461no—
359#380—1461no—
360C4×C4⋊C4—731no—
361C4×C4⋊C4—731no—
362C4×C4⋊C4—731no—
363C4×C4⋊C4—1461no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#340C2×D7311yescharacteristic
371#340C2×D7311yescharacteristic
372#336—731no—
373#336—731no—
374C22×Q8—731no—
375C22×D4C22×D7311yescharacteristic
376C23×C4—1461no—
377C2×SD16—734no—
378C2×SD16—1462no—
379C2×SD16—1462no—
380C22×C8—731no—
381C22×C8—731no—
382C22×C8C22×D7311yescharacteristic
383C22×C8C22×D7311yescharacteristic
384C4×Q8—2921no—
385C4×Q8—2921no—
386C4×Q8—734no—
387C4×Q8—1462no—
388C4×Q8—1462no—
389C4×Q8—1462no—
390C4×D4—734no—
391C4×D4—1462no—
392#117—2924no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—731no—
395C2×C4⋊C4—731no—
396C2×C4⋊C4—731no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—731no—
399C2×C4⋊C4—1461no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4—1461no—
402C2×C4⋊C4—1461no—
403C2×C4⋊C4—1461no—
404C2×C4⋊C4—1461no—
405C2×C4⋊C4—1461no—
406C2×C4⋊C4—1461no—
407C2×C4⋊C4C22×D7311yescharacteristic
408C2×C4⋊C4C22×D7311yescharacteristic
409C2×C4⋊C4C22×D7311yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—1461no—
413C2×C22⋊C4—1461no—
414C2×C22⋊C4—1461no—
415C2×C42—731no—
416C2×C42—731no—
417C2×C42—1461no—
418C2×C42—1461no—
419C2×C42—1461no—
420C2×C42—1461no—
421C2×C42—1461no—
422C2×C42C22×D7311yescharacteristic
423#106—734no—
424C4⋊C8—1462no—
425#103—734no—
426#103—1462no—
427#103—1462no—
428#102—41no—
429#102—2921no—
430#102C4×D7314yes—
431C4×C8—41no—
432C4×C8—2921no—
433C4×C8—734no—
434#95—21no—
435#95—731no—
436#95—731no—
437#95—1461no—
438#95—1461no—
439#95—1461no—
440C24—21no—
441C2×Q8—732no—
442C2×Q8—732no—
443C2×Q8—732no—
444C2×Q8—1461no—
445C2×Q8—1461no—
446C2×Q8—1461no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×C4×D7312yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—2922no—
457C22×C4—2922no—
458C22×C4—731no—
459C22×C4—731no—
460C22×C4—731no—
461C22×C4—731no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4—1464no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4D4×D7311yescharacteristic
468C22×C4D4×D7311yescharacteristic
469C22×C4C23×D7311yescharacteristic
470SD16—1468no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—732no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8C2×C4×D7312yes—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—81no—
489C4⋊C4—2922no—
490C4⋊C4—2922no—
491C4⋊C4—2922no—
492C4⋊C4—2922no—
493C4⋊C4—5841no—
494C4⋊C4—5841no—
495C4⋊C4—5841no—
496C4⋊C4—734no—
497C4⋊C4—732no—
498C4⋊C4—1462no—
499C4⋊C4—1464no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4—1461no—
505C4⋊C4C2×C4×D7312yes—
506C22⋊C4—42no—
507C22⋊C4—81no—
508C22⋊C4—2922no—
509C22⋊C4—5841no—
510C22⋊C4—1464no—
511C42—21no—
512C42—21no—
513C42—2922no—
514C42—2922no—
515C42—2922no—
516C42—2922no—
517C42—732no—
518C42—732no—
519C42—1464no—
520C42—1461no—
521C42—1461no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C2×D4×D7311yescharacteristic
530Q8—734no—
531Q8—1462no—
532Q8—1462no—
533Q8—1462no—
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2928no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—2921no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—732no—
562C2×C4—732no—
563C2×C4—732no—
564C2×C4—732no—
565C2×C4—732no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4—1462no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4—1461no—
572C2×C4—1461no—
573C2×C4—1461no—
574C2×C4—1462no—
575C2×C4—1462no—
576C2×C4—1462no—
577C2×C4SD16×D7312yes—
578C2×C4C22×C4×D7311yescharacteristic
579C2×C4#41×D7311yescharacteristic
580C2×C4#41×D7311yescharacteristic
581C2×C4#41×D7311yescharacteristic
582C2×C4#1455663611yescharacteristic
583C2×C4#1455663811yescharacteristic
584C8—22no—
585C8—41no—
586C8—2921no—
587C8—1462no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C4×D4×D7311yescharacteristic
596C22#1798922611yescharacteristic
597C22#1798924511yescharacteristic
598C22C2×SD16×D7311yescharacteristic
599C22#135×D7311yescharacteristic
600C22#1798938811yescharacteristic
601C22#1798940111yescharacteristic
602C4—22no—
603C4—42no—
604C4—41no—
605C4—81no—
606C4—81no—
607C4—2922no—
608C4—2921no—
609C4—5841no—
610C4—5841no—
611C4—734no—
612C4—1462no—
613C4—1464no—
614C4—1462no—
615C4#117×D7311yescharacteristic
616C4#1798906611yescharacteristic
617C4#1798927311yescharacteristic
618C4#1798928211yescharacteristic
619C2—82no—
620C2#2261285911yescharacteristic
621C2C4×SD16×D7311yescharacteristic
622C2#2261880111yescharacteristic
623C2#2262179111yescharacteristic
624C2#2262187611yescharacteristic
625C2#2262259911yescharacteristic
626C2#2262261311yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1934C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#1934D73
hypocenterC73#22767
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1934

Complements

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

NH ≅ G/Ntype
3. #22612845C2semidirect
5. #22610519C2semidirect
6. #22610513C2semidirect
9. #22610057C2semidirect
10. #22609935C2semidirect
12. C8×#11924007C2semidirect
13. C4×#14556516C2semidirect
14. #22609750C2semidirect
22. #17988678C4semidirect
165. #11924007SD16semidirect
269. Dic146#117semidirect
329. C73#22767semidirect

Maximal subgroups

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

subgroupindex
C2×#179886782
#226128792
#226128452
#226106162
#226105192
#226105132
#226102892
#226102832
#226100572
#226099352
#226099292
C8×#119240072
C4×#145565162
#226097502
C73×#19342
#2276773