Finite Group Databasebeta

← #33105232

Subgroups — #33105232

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×#17988672C211yesmaximal, characteristic
2#22612876C211yesmaximal, characteristic
3Q8×#11924007C211yesmaximal, characteristic, complemented
4#22610614C211yesmaximal, characteristic
5#22610612C211yesmaximal, characteristic
6#22610516C211yesmaximal, characteristic, complemented
7#22610514C211yesmaximal, characteristic, complemented
8#22610286C211yesmaximal, characteristic
9#22610284C211yesmaximal, characteristic
10#22610057C211yesmaximal, characteristic, complemented
11#22609934C211yesmaximal, characteristic, complemented
12#22609930C211yesmaximal, characteristic
13C8×#11924007C211yesmaximal, characteristic, complemented
14#103×Dic73C211yesmaximal, characteristic, complemented
15C4×#103×C73C211yesmaximal, characteristic, complemented
16C22×#11924015C2211yescharacteristic
17C2×Q8×Dic73C2211yescharacteristic
18C2×Q8×Dic73C2211yescharacteristic
19C2×#14556553C2211yescharacteristic
20C2×#14556548—21no—
21#17988672C418yescomplemented
22C2×#14556522C2211yescharacteristic
23C2×#14556520C2211yescharacteristic
24C2×#14556515C2211yescharacteristic
25C2×#14556513C2211yescharacteristic
26C2×#14556513C2211yescharacteristic
27C2×#14556511C2211yescharacteristic
28C2×C8×Dic73C2211yescharacteristic
29C2×C8×Dic73C2211yescharacteristic
30#17988489C2211yescharacteristic
31#17988488C2211yescharacteristic
32#17988485—21no—
33#17988485C2211yescharacteristic
34#17988483—21no—
35#17988483C2211yescharacteristic
36#17988478—21no—
37#17988476—21no—
38#17988456—21no—
39#17988456C2211yescharacteristic
40#17988456C2211yescharacteristic
41#17988446—21no—
42C4⋊C4×Dic73C2211yescharacteristic
43C4×#11924007—21no—
44C4×#11924007C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988342C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×C4×Q8×C73C2211yescharacteristic
54C2×#103×C73C2211yescharacteristic
55C2×#103×C73C2211yescharacteristic
56C2×C4×C8×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C22×D292C2311yescharacteristic
61C22×C4×D73—21no—
62C2×#11924015—22no—
63C2×#11924015—22no—
64C2×#11924015C2×C414yes—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67Q8×Dic73—22no—
68Q8×Dic73—22no—
69Q8×Dic73—41no—
70Q8×Dic73C2×C414yes—
71#14556553—22no—
72#14556553C2×C414yes—
73#14556548—44no—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×#11924007D411yescharacteristic
86C2×#11924007D411yescharacteristic
87C2×#11924005—21no—
88C2×#11924005—21no—
89C2×#11924005—21no—
90C2×#11924005—21no—
91C2×#11924005—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73—21no—
94C2×C4×Dic73—21no—
95C2×C4×Dic73C2311yescharacteristic
96C2×C4×Dic73C2311yescharacteristic
97#14556522C2×C414yes—
98#14556520—22no—
99#14556515—22no—
100#14556513—41no—
101#14556513C2×C414yes—
102#14556511—41no—
103C8×Dic73—41no—
104C8×Dic73C2×C414yes—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498D411yescharacteristic
109#14556498D411yescharacteristic
110C22×Q8×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C4×Q8×C73—22no—
114C4×Q8×C73—41no—
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C42×C73—21no—
120C2×C42×C73—21no—
121C2×C42×C73C2311yescharacteristic
122#103×C73—22no—
123#103×C73C2×C414yes—
124C4×C8×C73—41no—
125#95×C73—21no—
126C23×D73—21no—
127C2×D292—21no—
128C2×D292—21no—
129C2×D292—21no—
130C2×D292—21no—
131C2×D292—21no—
132C2×D292C22×C412yes—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73C2×D411yescharacteristic
137C22×Dic73C2×D411yescharacteristic
138C22×Dic73C2×D411yescharacteristic
139C22×Dic73C2×D411yescharacteristic
140C2×C4×D73—24no—
141C2×C4×D73—42no—
142C2×C4×D73—42no—
143#11924015—28no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831C22×C412yes—
150C4⋊Dic73—21no—
151C4⋊Dic73—24no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—42no—
157C4⋊Dic73C22×C412yes—
158#11924007—22no—
159#11924007—42no—
160#11924007—42no—
161#11924007—81no—
162#11924007—81no—
163#11924007—81no—
164#11924007—81no—
165#11924007SD1614yescomplemented
166#11924005—24no—
167#11924005—42no—
168#11924005—42no—
169#11924005—81no—
170#11924005—81no—
171C4×Dic73—21no—
172C4×Dic73—21no—
173C4×Dic73—24no—
174C4×Dic73—42no—
175C4×Dic73—42no—
176C4×Dic73C22×C412yes—
177C4×Dic73#4112yes—
178C2×Q8×C73—21no—
179C2×Q8×C73—21no—
180C2×Q8×C73—21no—
181C2×Q8×C73C22×C412yes—
182C2×Q8×C73#4112yes—
183C2×Q8×C73#4112yes—
184C22×C4×C73—21no—
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—42no—
203C4⋊C4×C73C22×C412yes—
204C42×C73—21no—
205C42×C73—21no—
206C42×C73—42no—
207C42×C73—42no—
208C23×C73C22×D411yescharacteristic
209C22×D73—22no—
210C22×D73—41no—
211C22×D73—41no—
212C22×D73—41no—
213C22×D73—41no—
214C22×D73—41no—
215C22×D73—41no—
216Q8×C73—22no—
217Q8×C73—22no—
218Q8×C73—22no—
219Q8×C73#11714yescomplemented
220C2×C4×C73—22no—
221C2×C4×C73—21no—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—21no—
225C2×C4×C73—22no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73C2×SD1612yes—
238C2×C4×C73C23×C411yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C2×C4×C73C2×#4111yescharacteristic
241C2×C4×C73C2×#4111yescharacteristic
242C2×C4×C73H(5,𝔽2)11yescharacteristic
243C2×C4×C732−1+411yescharacteristic
244C2×Dic73—21no—
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—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73C4×D412yes—
261C2×Dic73C4×D412yes—
262C2×Dic73C2×SD1612yes—
263C2×Dic73C2×SD1612yes—
264C2×Dic73#13512yes—
265D292—22no—
266D292—22no—
267D292—41no—
268D292—41no—
269C4×D73—48no—
270C8×C73—22no—
271C8×C73—41no—
272#10596831—22no—
273#10596831—41no—
274C22×C73C2×C4×D411yescharacteristic
275C22×C73#54611yescharacteristic
276C22×C73#54811yescharacteristic
277C22×C73C22×SD1611yescharacteristic
278C22×C73C2×#13511yescharacteristic
279C22×C73#57611yescharacteristic
280C22×C73#57811yescharacteristic
281C2×D73—42no—
282C2×D73—42no—
283C2×D73—42no—
284C2×D73—81no—
285C2×D73—81no—
286C2×D73—81no—
287C2×D73—81no—
288C4×C73—24no—
289C4×C73—22no—
290C4×C73—42no—
291C4×C73—41no—
292C4×C73—81no—
293C4×C73—81no—
294C4×C73C2×#11711yescharacteristic
295C4×C73#52011yescharacteristic
296C4×C73#55211yescharacteristic
297C4×C73#55311yescharacteristic
298Dic73—22no—
299Dic73—22no—
300Dic73—42no—
301Dic73—41no—
302Dic73—81no—
303Dic73—81no—
304Dic73C4×SD1614yes—
305#22767—731nomaximal, Sylow
306C2×C73#235211yescharacteristic
307C2×C73C2×C4×SD1611yescharacteristic
308C2×C73#295111yescharacteristic
309C2×C73#329811yescharacteristic
310C2×C73#330511yescharacteristic
311C2×C73#336211yescharacteristic
312C2×C73#336411yescharacteristic
313D73—82no—
314C2×C4×SD16—731no—
315#2354—731no—
316Q8×C4⋊C4—731no—
317#1934—731no—
318#1934—731no—
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×#103D7311yescharacteristic, complemented
329C73#2276711yescharacteristic, complemented, Sylow
330C22×SD16—731no—
331C2×C4×Q8—731no—
332C2×C4×Q8—731no—
333C2×C4×Q8C2×D7311yescharacteristic
334C2×C4×D4—731no—
335C2×#117—1461no—
336C4×SD16—738no—
337C2×#106—731no—
338C2×C4⋊C8—731no—
339C2×#103—731no—
340C2×#103C2×D7311yescharacteristic
341C2×#103C2×D7311yescharacteristic
342C2×#102—731no—
343C2×#102—731no—
344C2×#102—731no—
345C2×C4×C8—731no—
346C2×C4×C8—731no—
347C2×C4×C8C2×D7311yescharacteristic
348#390—731no—
349#390—731no—
350#389—731no—
351#389—731no—
352#389—1461no—
353#389—1461no—
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—1461no—
363C4×C4⋊C4C2×D7311yescharacteristic
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×Q8C22×D7311yescharacteristic
375C22×D4—731no—
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—22no—
385C4×Q8—41no—
386C4×Q8—2921no—
387C4×Q8—734no—
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—1461no—
411C2×C22⋊C4—1461no—
412C2×C22⋊C4—1461no—
413C2×C22⋊C4—1461no—
414C2×C22⋊C4—1461no—
415C2×C42—21no—
416C2×C42—21no—
417C2×C42—731no—
418C2×C42—731no—
419C2×C42—1461no—
420C2×C42—1461no—
421C2×C42—1461no—
422C2×C42C22×D7311yescharacteristic
423#106—734no—
424C4⋊C8—1462no—
425#103—22no—
426#103—1462no—
427#103C4×D7314yes—
428#102—2921no—
429#102—2921no—
430#102—734no—
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—1461no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C2×C4×D7312yes—
445C2×Q8#1192402812yes—
446C2×Q8#1192402812yes—
447C2×D4—732no—
448C2×D4—1461no—
449C2×D4—1461no—
450C2×D4—1461no—
451C2×D4—1461no—
452C2×D4—1461no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—2922no—
458C22×C4—2922no—
459C22×C4—731no—
460C22×C4—731no—
461C22×C4—731no—
462C22×C4—731no—
463C22×C4—1464no—
464C22×C4—1461no—
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—42no—
489C4⋊C4—2922no—
490C4⋊C4—2922no—
491C4⋊C4—2922no—
492C4⋊C4—5841no—
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—2922no—
507C22⋊C4—2922no—
508C22⋊C4—5841no—
509C22⋊C4—5841no—
510C22⋊C4—1464no—
511C42—21no—
512C42—21no—
513C42—42no—
514C42—42no—
515C42—2922no—
516C42—2922no—
517C42—732no—
518C42—732no—
519C42—1464no—
520C42—1461no—
521C42—1461no—
522C23—2921no—
523C23—2921no—
524C23—2921no—
525C23—2921no—
526C23—2921no—
527C23—2921no—
528C23—1462no—
529C23C2×D4×D7311yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8#1455654814yescomplemented
534D4—2921no—
535D4—2921no—
536D4—1462no—
537D4—1462no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—2928no—
556C2×C4—2921no—
557C2×C4—2921no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—2921no—
564C2×C4—732no—
565C2×C4—732no—
566C2×C4—732no—
567C2×C4—732no—
568C2×C4—732no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4—1462no—
572C2×C4—1461no—
573C2×C4—1461no—
574C2×C4—1461no—
575C2×C4—1461no—
576C2×C4—1461no—
577C2×C4SD16×D7312yes—
578C2×C4C22×C4×D7311yescharacteristic
579C2×C4#41×D7311yescharacteristic
580C2×C4C2×#1192402811yescharacteristic
581C2×C4C2×#1192402811yescharacteristic
582C2×C4#1455663611yescharacteristic
583C2×C4#1455663811yescharacteristic
584C8—22no—
585C8—41no—
586C8—2921no—
587C8—1462no—
588C22—2922no—
589C22—2922no—
590C22—2922no—
591C22—5841no—
592C22—5841no—
593C22—5841no—
594C22—5841no—
595C22C4×D4×D7311yescharacteristic
596C22#1798922711yescharacteristic
597C22#1798924611yescharacteristic
598C22C2×SD16×D7311yescharacteristic
599C22#135×D7311yescharacteristic
600C22#1798938911yescharacteristic
601C22#1798940011yescharacteristic
602C4—22no—
603C4—24no—
604C4—42no—
605C4—41no—
606C4—81no—
607C4—81no—
608C4—2921no—
609C4—2922no—
610C4—5841no—
611C4—5841no—
612C4—734no—
613C4—1462no—
614C4—1462no—
615C4C2×#1455654811yescharacteristic
616C4#1798906611yescharacteristic
617C4#1798927011yescharacteristic
618C4#1798928611yescharacteristic
619C2—5842no—
620C2#2261285411yescharacteristic
621C2C4×SD16×D7311yescharacteristic
622C2#2261880611yescharacteristic
623C2#2262179211yescharacteristic
624C2#2262187711yescharacteristic
625C2#2262260011yescharacteristic
626C2#2262261711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC4×#103×C73C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenterC4×#103D73
hypocenterC73#22767
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
3. Q8×#11924007C2semidirect
6. #22610516C2semidirect
7. #22610514C2semidirect
10. #22610057C2semidirect
11. #22609934C2semidirect
13. C8×#11924007C2semidirect
14. #103×Dic73C2semidirect
15. C4×#103×C73C2semidirect
21. #17988672C4semidirect
165. #11924007SD16semidirect
219. Q8×C73#117semidirect
328. C4×#103D73semidirect
329. C73#22767semidirect
533. Q8#14556548semidirect

Maximal subgroups

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

subgroupindex
C2×#179886722
#226128762
Q8×#119240072
#226106142
#226106122
#226105162
#226105142
#226102862
#226102842
#226100572
#226099342
#226099302
C8×#119240072
#103×Dic732
C4×#103×C732
#2276773