Finite Group Databasebeta

← #33149599

Subgroups — #33149599

Order 18688.

818 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22613627C211yesmaximal, characteristic
2#22612940C211yesmaximal, characteristic
3#22611492C211yesmaximal, characteristic, complemented
4#22611431C211yesmaximal, characteristic, complemented
5#22611310C211yesmaximal, characteristic, complemented
6#22611070C211yesmaximal, characteristic, complemented
7#22610881C211yesmaximal, characteristic, complemented
8#22610508C211yesmaximal, characteristic, complemented
9#22610355C211yesmaximal, characteristic
10#22610201C211yesmaximal, characteristic, complemented
11#22610126C211yesmaximal, characteristic
12#22610108C211yesmaximal, characteristic
13#22609934C211yesmaximal, characteristic, complemented
14C2×#17988349C211yesmaximal, characteristic
15C73×#1872C211yesmaximal, characteristic
16C23×#10596831C2211yescharacteristic
17C2×#14556590C2211yescharacteristic
18C2×#14556570—21no—
19C2×Q8×Dic73C2211yescharacteristic
20C22×C4⋊Dic73C2211yescharacteristic
21C2×#14556523C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23C2×#14556520C2211yescharacteristic
24C2×#14556520C2211yescharacteristic
25C2×#14556515C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27#17988520—21no—
28#17988507—21no—
29#17988494C2211yescharacteristic
30#17988486—21no—
31#17988486C2211yescharacteristic
32#17988486C2211yescharacteristic
33#17988483C2211yescharacteristic
34#17988473—21no—
35#17988462—21no—
36#17988460—21no—
37#17988456C2211yescharacteristic
38#17988447C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C22⋊C4×Dic73—21no—
41C4×C22⋊Dic73C2211yescharacteristic
42#17988350C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C2×C22⋊Q8×C73C2211yescharacteristic
53C22×C4⋊C4×C73C2211yescharacteristic
54C2×#103×C73C2211yescharacteristic
55C2×#103×C73C2211yescharacteristic
56C2×C22⋊C8×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59C22×Dic146—21no—
60C23×Dic73—21no—
61C22×#10596831—22no—
62C22×#10596831—41no—
63C22×#10596831—41no—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66#14556590—22no—
67#14556590—41no—
68#14556570—44no—
69Q8×Dic73—22no—
70Q8×Dic73—41no—
71C2×C4⋊Dic73—22no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73—41no—
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×C22⋊Dic73—21no—
86C2×C22⋊Dic73D411yescharacteristic
87C2×C22⋊Dic73D411yescharacteristic
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73C2311yescharacteristic
91C2×C4×Dic73C2311yescharacteristic
92#14556523—22no—
93#14556523—41no—
94#14556520—41no—
95#14556520—41no—
96#14556515—41no—
97#14556515—41no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498D411yescharacteristic
106#14556498D411yescharacteristic
107C22×Q8×C73C2311yescharacteristic
108C23×C4×C73—21no—
109C23×C4×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112C22⋊Q8×C73—22no—
113C22⋊Q8×C73Q814yescomplemented
114C2×C4⋊C4×C73—21no—
115C2×C4⋊C4×C73—22no—
116C2×C4⋊C4×C73—21no—
117C2×C4⋊C4×C73—41no—
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C4⋊C4×C73C2311yescharacteristic
120C2×C4⋊C4×C73C2311yescharacteristic
121C2×C22⋊C4×C73—21no—
122#103×C73—22no—
123#103×C73—41no—
124C22⋊C8×C73—22no—
125C2×Dic146—42no—
126C2×Dic146—42no—
127C2×Dic146—81no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73—42no—
133C22×Dic73—42no—
134C22×Dic73—81no—
135C22×Dic73C2×D411yescharacteristic
136C22×Dic73C2×D411yescharacteristic
137C22×Dic73C2×D411yescharacteristic
138C22×Dic73C2×D411yescharacteristic
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—44no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—42no—
154C4⋊Dic73—42no—
155C4⋊Dic73—42no—
156C4⋊Dic73—81no—
157C4⋊Dic73#4112yes—
158#11924007—42no—
159#11924007—42no—
160#11924007—42no—
161#11924007—81no—
162C22⋊Dic73—22no—
163C22⋊Dic73—42no—
164C22⋊Dic73SD1614yes—
165C4×Dic73—21no—
166C4×Dic73—21no—
167C4×Dic73—21no—
168C4×Dic73—21no—
169C4×Dic73—42no—
170C4×Dic73—81no—
171C24×C73C2×D411yescharacteristic
172C2×Q8×C73—21no—
173C2×Q8×C73—21no—
174C2×Q8×C73—21no—
175C2×Q8×C73—21no—
176C2×Q8×C73—21no—
177C2×Q8×C73C2×Q812yes—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—42no—
187C22×C4×C73—42no—
188C22×C4×C73—81no—
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2×Q812yes—
191C22×C4×C73#4112yes—
192C22×C4×C73C2411yescharacteristic
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C4⋊C4×C73—24no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—21no—
204C4⋊C4×C73—42no—
205C4⋊C4×C73C2×Q812yes—
206C4⋊C4×C73#4112yes—
207C22⋊C4×C73—24no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73C22⋊Q812yes—
213C23×C73C2×SD1612yes—
214C23×C73#13712yes—
215C23×C73C22×D411yescharacteristic
216Q8×C73—22no—
217Q8×C73—22no—
218Q8×C73—41no—
219Q8×C73—41no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—22no—
227C2×C4×C73—22no—
228C2×C4×C73—22no—
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×C73—41no—
238C2×C4×C73—84no—
239C2×C4×C73C22⋊Q812yes—
240C2×C4×C73C22×Q811yescharacteristic
241C2×C4×C73C2×#4111yescharacteristic
242C2×C4×C73C2×#4111yescharacteristic
243C2×C4×C73C2×#4111yescharacteristic
244C2×C4×C73H(5,𝔽2)11yescharacteristic
245C2×C4×C73H(5,𝔽2)11yescharacteristic
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—22no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—84no—
269C2×Dic73C2×SD1612yes—
270C2×Dic73C2×SD1612yes—
271Dic146—84no—
272C8×C73—41no—
273C8×C73—41no—
274#10596831—41no—
275#10596831—41no—
276C22×C73—22no—
277C22×C73—22no—
278C22×C73—22no—
279C22×C73—22no—
280C22×C73—22no—
281C22×C73—22no—
282C22×C73#47514yes—
283C22×C73C2×C22⋊Q811yescharacteristic
284C22×C73#54611yescharacteristic
285C22×C73#54611yescharacteristic
286C22×C73C22×SD1611yescharacteristic
287C22×C73C2×#13711yescharacteristic
288C22×C73#57611yescharacteristic
289C22×C73#57811yescharacteristic
290C4×C73—22no—
291C4×C73—42no—
292C4×C73—42no—
293C4×C73—42no—
294C4×C73—81no—
295C4×C73#52911yescharacteristic
296C4×C73#53811yescharacteristic
297C4×C73#55411yescharacteristic
298C4×C73#55411yescharacteristic
299Dic73—22no—
300Dic73—22no—
301Dic73—41no—
302Dic73—41no—
303Dic73—42no—
304Dic73—81no—
305Dic73—81no—
306Dic73—81no—
307#26412—731nomaximal, Sylow
308C2×C73—24no—
309C2×C73#245011yescharacteristic
310C2×C73C2×#47511yescharacteristic
311C2×C73#308311yescharacteristic
312C2×C73#329511yescharacteristic
313C2×C73#329811yescharacteristic
314C2×C73#331211yescharacteristic
315C2×C73#331411yescharacteristic
316#2440—731no—
317#2360—731no—
318#2099—731no—
319#2088—731no—
320#2066—731no—
321#2024—731no—
322#1988—731no—
323#1919—731no—
324#1892—731no—
325#1872D7311yescharacteristic
326#1865—731no—
327#1853—731no—
328#1850—731no—
329#1803—731no—
330C2×#340—731no—
331C73#2641211yescharacteristic, complemented, Sylow
332C23×C8—731no—
333C2×C4⋊Q8—731no—
334C2×C22⋊Q8—1461no—
335C2×C22⋊Q8C2×D7311yescharacteristic
336C2×C4×Q8—731no—
337C22×C4⋊C4—731no—
338C22×C4⋊C4C2×D7311yescharacteristic
339C2×#106—731no—
340C2×#106—731no—
341C2×C4⋊C8—731no—
342C2×C4⋊C8—731no—
343C2×#103—731no—
344C2×#103—731no—
345C2×#103C2×D7311yescharacteristic
346C2×#103C2×D7311yescharacteristic
347C2×C22⋊C8C2×D7311yescharacteristic
348#398—1461no—
349#395—1461no—
350#391—731no—
351#389—731no—
352#389—731no—
353#389—731no—
354#389—1461no—
355#387—1461no—
356#385—1461no—
357#385—1461no—
358#384—731no—
359#380—731no—
360C4×C4⋊C4—731no—
361C4×C22⋊C4—731no—
362C4×C22⋊C4—1461no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#340—731no—
371#340C2×D7311yescharacteristic
372#340C2×D7311yescharacteristic
373#336—731no—
374#336—731no—
375C22×Q8—1461no—
376C22×Q8C22×D7311yescharacteristic
377C23×C4—21no—
378C23×C4—1461no—
379C23×C4C22×D7311yescharacteristic
380C22×C8—2921no—
381C22×C8—2921no—
382C22×C8—731no—
383C22×C8—731no—
384C22×C8—1462no—
385C22×C8C22×D7311yescharacteristic
386C22×C8C22×D7311yescharacteristic
387C4⋊Q8—2921no—
388C4⋊Q8—1462no—
389C22⋊Q8—22no—
390C22⋊Q8—2924no—
391C22⋊Q8Dic14614yescomplemented
392C4×Q8—2921no—
393C4×Q8—1462no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—41no—
398C2×C4⋊C4—2921no—
399C2×C4⋊C4—731no—
400C2×C4⋊C4—731no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4—731no—
403C2×C4⋊C4—1461no—
404C2×C4⋊C4—1461no—
405C2×C4⋊C4—1461no—
406C2×C4⋊C4—1461no—
407C2×C4⋊C4—1461no—
408C2×C4⋊C4—1461no—
409C2×C4⋊C4—1462no—
410C2×C4⋊C4—1461no—
411C2×C4⋊C4—1461no—
412C2×C4⋊C4C22×D7311yescharacteristic
413C2×C4⋊C4C22×D7311yescharacteristic
414C2×C4⋊C4C22×D7311yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—731no—
417C2×C22⋊C4—731no—
418C2×C22⋊C4—1461no—
419C2×C42—731no—
420C2×C42—731no—
421C2×C42—1461no—
422C2×C42—1461no—
423#106—2921no—
424#106—1462no—
425C4⋊C8—2921no—
426C4⋊C8—2921no—
427#103—22no—
428#103—41no—
429#103—2921no—
430#103—2921no—
431C22⋊C8—22no—
432#95—731no—
433#95—731no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438#95—1461no—
439#95—1461no—
440#95—1461no—
441C24D4×D7311yescharacteristic
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—2922no—
448C2×Q8—2922no—
449C2×Q8—5841no—
450C2×Q8C2×Dic14612yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—81no—
462C22×C4—2922no—
463C22×C4—2922no—
464C22×C4—5841no—
465C22×C4—731no—
466C22×C4—731no—
467C22×C4—731no—
468C22×C4—731no—
469C22×C4—1461no—
470C22×C4—1461no—
471C22×C4—1461no—
472C22×C4—1461no—
473C22×C4D4×D7311yescharacteristic
474C22×C4C2×Dic14612yes—
475C22×C4#1192402712yes—
476C22×C4C23×D7311yescharacteristic
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—2924no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C4⋊C4—24no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—42no—
496C4⋊C4—2922no—
497C4⋊C4—2922no—
498C4⋊C4—2922no—
499C4⋊C4—2922no—
500C4⋊C4—2922no—
501C4⋊C4—2922no—
502C4⋊C4—5841no—
503C4⋊C4—5841no—
504C4⋊C4—732no—
505C4⋊C4—1461no—
506C4⋊C4—1461no—
507C4⋊C4—1461no—
508C4⋊C4—1461no—
509C4⋊C4—1461no—
510C4⋊C4—1461no—
511C4⋊C4—1461no—
512C4⋊C4C2×Dic14612yes—
513C4⋊C4#1192402712yes—
514C22⋊C4—24no—
515C22⋊C4—2922no—
516C22⋊C4—734no—
517C22⋊C4—1462no—
518C42—2922no—
519C42—5841no—
520C42—1461no—
521C42—1461no—
522C42—1461no—
523C42—1461no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#1455656812yes—
529C23SD16×D7312yes—
530C23#1455662312yes—
531C23C2×D4×D7311yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536Q8—5844no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—41no—
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—84no—
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—2921no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—5844no—
568C2×C4—732no—
569C2×C4—732no—
570C2×C4—1461no—
571C2×C4—1461no—
572C2×C4—1461no—
573C2×C4—1461no—
574C2×C4—1461no—
575C2×C4—1462no—
576C2×C4—1461no—
577C2×C4—1461no—
578C2×C4—1461no—
579C2×C4—1461no—
580C2×C4—1461no—
581C2×C4#1455656812yes—
582C2×C4C22×Dic14611yescharacteristic
583C2×C4#41×D7311yescharacteristic
584C2×C4C2×#1192402711yescharacteristic
585C2×C4C2×#1192402711yescharacteristic
586C2×C4#1455663611yescharacteristic
587C2×C4#1455663711yescharacteristic
588C8—41no—
589C8—41no—
590C8—2921no—
591C8—2921no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22#1798887014yes—
599C22C2×#1455656811yescharacteristic
600C22#1798922411yescharacteristic
601C22#1798922911yescharacteristic
602C22C2×SD16×D7311yescharacteristic
603C22C2×#1455662311yescharacteristic
604C22#1798939111yescharacteristic
605C22#1798940311yescharacteristic
606C4—22no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—81no—
611C4—2921no—
612C4—2922no—
613C4—2921no—
614C4—5841no—
615C4—5841no—
616C4—5841no—
617C4—1462no—
618C4—1462no—
619C4#1798911811yescharacteristic
620C4#1798917411yescharacteristic
621C4#1798929411yescharacteristic
622C4#1798929611yescharacteristic
623C2—24no—
624C2#2261375911yescharacteristic
625C2C2×#1798887011yescharacteristic
626C2#2261978611yescharacteristic
627C2#2262174811yescharacteristic
628C2#2262179411yescharacteristic
629C2#2262198611yescharacteristic
630C2#2262201011yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1872C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#1872D73
hypocenterC73#26412
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
3. #22611492C2semidirect
4. #22611431C2semidirect
5. #22611310C2semidirect
6. #22611070C2semidirect
7. #22610881C2semidirect
8. #22610508C2semidirect
10. #22610201C2semidirect
13. #22609934C2semidirect
113. C22⋊Q8×C73Q8semidirect
331. C73#26412semidirect
391. C22⋊Q8Dic146semidirect

Maximal subgroups

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

subgroupindex
#226136272
#226129402
#226114922
#226114312
#226113102
#226110702
#226108812
#226105082
#226103552
#226102012
#226101262
#226101082
#226099342
C2×#179883492
C73×#18722
#2641273