Finite Group Databasebeta

← #10858708

Subgroups — #10858708

Order 768.

714 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2913C211yesmaximal, characteristic
2#82571C211yesmaximal, characteristic
3#82559C211yesmaximal, characteristic
4#79198C211yesmaximal, characteristic, complemented
5#72019C211yesmaximal, characteristic
6#72017C211yesmaximal, characteristic
7#71666C211yesmaximal, characteristic
8#71664C211yesmaximal, characteristic
9#70679C211yesmaximal, characteristic, complemented
10#70677C211yesmaximal, characteristic, complemented
11#70669C211yesmaximal, characteristic, complemented
12#70666C211yesmaximal, characteristic, complemented
13#70661C211yesmaximal, characteristic, complemented
14#70644C211yesmaximal, characteristic, complemented
15#70626C211yesmaximal, characteristic, complemented
16#26406—31nomaximal, Sylow
17C2×C3×#131C2211yescharacteristic
18C3×#520C2211yescharacteristic
19C3×#520C2211yescharacteristic
20C2×#951C2211yescharacteristic
21#5519C2211yescharacteristic
22#5516C2211yescharacteristic
23C3×#428—21no—
24C3×#428—21no—
25C3×#428C2211yescharacteristic
26C3×#428C2211yescharacteristic
27C3×#427—21no—
28C3×#427C2211yescharacteristic
29C2×C3×#107—21no—
30C2×C3×#107C2211yescharacteristic
31C2×#837C2211yescharacteristic
32#5233C2211yescharacteristic
33#5232C2211yescharacteristic
34C2×#824C2211yescharacteristic
35C2×#822C2211yescharacteristic
36C2×#816C2211yescharacteristic
37#5102C2211yescharacteristic
38#5030—21no—
39C4×#216—21no—
40#5022—21no—
41#5021—21no—
42#5018—21no—
43C4×#214—21no—
44#5009—21no—
45#5007—21no—
46#5004C2211yescharacteristic
47#4999C2211yescharacteristic
48#4997C2211yescharacteristic
49#4995C2211yescharacteristic
50#4975C2211yescharacteristic
51#4973C2211yescharacteristic
52C2×#810C2211yescharacteristic
53#4964C2211yescharacteristic
54#4964C2211yescharacteristic
55C2×#807C2211yescharacteristic
56#4877—21no—
57#4875—21no—
58#4857—21no—
59#4854—21no—
60#4555C2211yescharacteristic
61#4554C2211yescharacteristic
62#4552C2211yescharacteristic
63#4550C2211yescharacteristic
64#4532C2211yescharacteristic
65#4532C2211yescharacteristic
66#4530C2211yescharacteristic
67#4530C2211yescharacteristic
68#3083—31no—
69#3082—31no—
70#2947—31no—
71#2913D311yescharacteristic
72#2095—31no—
73#2089—31no—
74#2064—31no—
75#2060—31no—
76#1952—31no—
77#1922—31no—
78#1908—31no—
79#1882—31no—
80#1866—31no—
81#1859—31no—
82#1857—31no—
83C2×C3×#41C2311yescharacteristic
84C22×Dic6C2311yescharacteristic
85C3×#131—22no—
86C3×#131—22no—
87C2×C3×M16—21no—
88C2×C3×M16C2311yescharacteristic
89C22×C3×C8—21no—
90C22×C3×C8C2311yescharacteristic
91C3×C4×Q8—21no—
92C3×C4×Q8—21no—
93C3×C4×D4—21no—
94C3×C4×D4—21no—
95C3×C4×D4—41no—
96C3×C4×D4—41no—
97C3×#117—21no—
98C3×#117—21no—
99C3×#117C2311yescharacteristic
100C3×#117C2311yescharacteristic
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105#951—22no—
106#951Q814yescomplemented
107C2×#216—21no—
108C2×#214—21no—
109#926C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—41no—
112C2×C4×Dic3C2311yescharacteristic
113C2×#208C2311yescharacteristic
114C2×#208C2311yescharacteristic
115C2×#208C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117#890—21no—
118#888—21no—
119#887—21no—
120#887—21no—
121#878—41no—
122#877—41no—
123#870—21no—
124#869—21no—
125C4×Dic6—21no—
126C4×Dic6—21no—
127C3×#107—21no—
128C3×#107—21no—
129C3×#107—21no—
130C3×#107—21no—
131C3×#107D411yescharacteristic
132C3×#107D411yescharacteristic
133C3×#106—21no—
134C3×#106—21no—
135C3×#106D411yescharacteristic
136C3×#106D411yescharacteristic
137#837—22no—
138#835—21no—
139#835D411yescharacteristic
140#835D411yescharacteristic
141#832—21no—
142#832D411yescharacteristic
143#832D411yescharacteristic
144C3⋊#95—41no—
145#824—22no—
146#822—22no—
147#816—21no—
148#816—21no—
149Dic3⋊C8—21no—
150Dic3⋊C8—21no—
151#810—21no—
152#810—21no—
153#810—21no—
154#810—21no—
155#808—21no—
156#808—21no—
157#808—21no—
158#807—21no—
159#807—21no—
160#807—21no—
161#807—21no—
162#807—21no—
163C4⋊(C3⋊C8)—21no—
164C4⋊(C3⋊C8)—21no—
165#803—21no—
166C4×C3⋊C8—21no—
167C2×#137—31no—
168C2×#131C2×D311yescharacteristic
169#544—31no—
170#520C2×D311yescharacteristic
171#520C2×D311yescharacteristic
172#519—31no—
173#479—61no—
174#479—61no—
175#479—61no—
176#478—61no—
177#477—61no—
178#477—61no—
179#475—61no—
180#474—61no—
181#441—61no—
182#440—61no—
183C4×Q16—61no—
184C4×SD16—61no—
185C2×#108—31no—
186#428—21no—
187#428—21no—
188#428—31no—
189#428—31no—
190#428—31no—
191#428C2×D311yescharacteristic
192#428C2×D311yescharacteristic
193#427—21no—
194#427C2×D311yescharacteristic
195C2×#107—21no—
196C2×#107—31no—
197C2×#107C2×D311yescharacteristic
198#424—31no—
199#424—31no—
200C2×#104—31no—
201#419—31no—
202#417—31no—
203#416—31no—
204C2×#103—31no—
205C2×#103—31no—
206C2×#101—31no—
207#405—31no—
208#391—31no—
209#384—31no—
210#343—31no—
211#343—31no—
212#343—31no—
213#341—31no—
214#339—31no—
215#339—31no—
216#339—31no—
217#336—31no—
218C3×#41—21no—
219C3×#41—21no—
220C3×#41C2×Q812yes—
221C3×#41#4112yes—
222C2×C3×Q8C2×D411yescharacteristic
223C2×C3×D4—21no—
224C2×C3×D4C2×D411yescharacteristic
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—41no—
229C22×C3×C4—41no—
230C22×C3×C4C2411yescharacteristic
231C22×Dic3—21no—
232C22×Dic3—21no—
233C22×Dic3—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—41no—
240C2×Dic6C2×Q812yes—
241C3×M16—24no—
242C3×M16—21no—
243C3×M16#4112yes—
244C2×C3×C8—24no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8C2×D411yescharacteristic
249C2×C3×C8C2×D411yescharacteristic
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—41no—
255C3×C4⋊C4—41no—
256C3×C4⋊C4C2×D411yescharacteristic
257C3×C4⋊C4C2×D411yescharacteristic
258C3×C4⋊C4C2×D411yescharacteristic
259C3×C4⋊C4C2×D411yescharacteristic
260C3×C22⋊C4—21no—
261C3×C22⋊C4—21no—
262C3×C22⋊C4—41no—
263C3×C22⋊C4—41no—
264C3×C42—21no—
265C3×C42—21no—
266C3×C42—41no—
267C3×C42—41no—
268#217—21no—
269#216—24no—
270#214—24no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—41no—
274C4⋊Dic3C2×D411yescharacteristic
275C4⋊Dic3C2×D411yescharacteristic
276Dic3⋊C4—41no—
277Dic3⋊C4—41no—
278Dic3⋊C4—41no—
279Dic3⋊C4—41no—
280Dic3⋊C4—82no—
281C4×Dic3—21no—
282C4×Dic3—21no—
283C4×Dic3—21no—
284C4×Dic3—41no—
285C4×Dic3—41no—
286#208—21no—
287#208—21no—
288#208—21no—
289#208—21no—
290#208—21no—
291#208C2×Q812yes—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8C2×D411yescharacteristic
298C2×C3⋊C8C2×D411yescharacteristic
299C2×#41C22×D311yescharacteristic
300C22×Q8—31no—
301#137—34no—
302#137—62no—
303C2×Q16—61no—
304C2×SD16—61no—
305#131—22no—
306#131—22no—
307C2×M16—21no—
308C2×M16—31no—
309C2×M16—31no—
310C2×M16—31no—
311C2×M16C22×D311yescharacteristic
312C22×C8—21no—
313C22×C8—31no—
314C22×C8C22×D311yescharacteristic
315C4⋊Q8—61no—
316C4⋊Q8—61no—
317#125—61no—
318#125—61no—
319C22⋊Q8—121no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×Q8—61no—
323C4×Q8—61no—
324C4×Q8—61no—
325C4×Q8—61no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—41no—
329C4×D4—41no—
330#117—21no—
331#117—21no—
332#117—31no—
333#117—121no—
334#117C22×D311yescharacteristic
335#117C22×D311yescharacteristic
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—121no—
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C42—31no—
343#108—62no—
344#107—21no—
345#107—21no—
346#107—21no—
347#107—21no—
348#107—61no—
349#107—61no—
350#107—61no—
351#107—61no—
352#107—61no—
353#107#6711yescharacteristic
354#107#6711yescharacteristic
355#106—21no—
356#106—21no—
357#106—61no—
358#106—61no—
359#106—61no—
360#106#6711yescharacteristic
361#106#6711yescharacteristic
362C4⋊C8—61no—
363C4⋊C8—61no—
364C4⋊C8—61no—
365C4⋊C8—61no—
366#104—62no—
367#103—31no—
368#103—31no—
369#103—61no—
370#103—61no—
371#103—61no—
372#103—61no—
373#103—61no—
374#103—61no—
375#103—61no—
376#102—31no—
377#102—31no—
378#102—61no—
379#101—62no—
380#97—61no—
381C4×C8—61no—
382#95—121no—
383C23×C3—21no—
384C23×C3H(5,𝔽2)11yescharacteristic
385C3×Q8—21no—
386C3×Q8C22⋊Q812yes—
387C3×D4—22no—
388C3×D4—22no—
389C3×D4—21no—
390C3×D4C22⋊Q812yes—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—81no—
401C2×C3×C4—81no—
402C2×C3×C4—161no—
403C2×C3×C4—161no—
404C2×C3×C4C22×D411yescharacteristic
405C2×C3×C4C22×D411yescharacteristic
406C2×C3×C4C22×Q811yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C42−1+411yescharacteristic
410C2×Dic3—22no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—22no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—81no—
420Dic6—22no—
421Dic6—22no—
422Dic6—41no—
423Dic6—41no—
424Dic6—161no—
425C3×C8—21no—
426C3×C8—22no—
427C3×C8—21no—
428C3⋊C8—22no—
429C3⋊C8—21no—
430C3⋊C8—21no—
431C3⋊C8—41no—
432C3⋊C8—41no—
433#41—21no—
434#41—21no—
435#41#23512yes—
436#41D3×Q812yes—
437C2×Q8—32no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×Q8—61no—
442C2×Q8—61no—
443C2×Q8—121no—
444C2×Q8D3×D411yescharacteristic
445C2×D4—21no—
446C2×D4D3×D411yescharacteristic
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4C23×D311yescharacteristic
456Q16—64no—
457SD16—64no—
458M16—24no—
459M16—21no—
460M16—32no—
461M16—61no—
462M16—61no—
463M16—61no—
464M16—61no—
465M16—61no—
466M16#23512yes—
467C2×C8—24no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—31no—
472C2×C8—31no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8C2×#6711yescharacteristic
479C2×C8C2×#6711yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—31no—
485C4⋊C4—31no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—121no—
491C4⋊C4—121no—
492C4⋊C4—121no—
493C4⋊C4—121no—
494C4⋊C4—121no—
495C4⋊C4—242no—
496C4⋊C4C2×#6711yescharacteristic
497C4⋊C4C2×#6711yescharacteristic
498C4⋊C4C2×#6711yescharacteristic
499C4⋊C4C2×#6711yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—61no—
505C42—21no—
506C42—21no—
507C42—41no—
508C42—41no—
509C42—61no—
510C42—61no—
511C42—61no—
512C42—121no—
513C42—121no—
514C22×C3—21no—
515C22×C3—21no—
516C22×C3—22no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3#53911yescharacteristic
520C22×C3D4211yescharacteristic
521C22×C3#55411yescharacteristic
522C3×C4—22no—
523C3×C4—41no—
524C3×C4—41no—
525C3×C4—41no—
526C3×C4—41no—
527C3×C4—81no—
528C3×C4—81no—
529C3×C4C2×C22⋊Q811yescharacteristic
530C3×C4#52511yescharacteristic
531C3×C4#54711yescharacteristic
532C3×C4D4×Q811yescharacteristic
533Dic3—42no—
534Dic3—42no—
535Dic3—41no—
536Dic3—41no—
537Dic3—81no—
538C23—21no—
539C23#100411yescharacteristic
540Q8—21no—
541Q8—62no—
542Q8—62no—
543Q8—121no—
544Q8—121no—
545Q8—481no—
546Q8#89612yes—
547D4—22no—
548D4—22no—
549D4—21no—
550D4#89612yes—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—161no—
572C2×C4—161no—
573C2×C4—241no—
574C2×C4C2×#23511yescharacteristic
575C2×C4C2×#23511yescharacteristic
576C2×C4C2×D3×D411yescharacteristic
577C2×C4C2×D3×Q811yescharacteristic
578C2×C4#100711yescharacteristic
579C2×C4C22×#6711yescharacteristic
580C8—21no—
581C8—22no—
582C8—21no—
583C8—61no—
584C8—61no—
585C8—62no—
586C8—121no—
587C8—121no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—42no—
591C2×C3#244911yescharacteristic
592C2×C3#328811yescharacteristic
593C2×C3#329311yescharacteristic
594C22—21no—
595C22—21no—
596C22—22no—
597C22—41no—
598C22—41no—
599C22#551111yescharacteristic
600C22#551511yescharacteristic
601C22D4×#6711yescharacteristic
602C4—22no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—81no—
608C4—81no—
609C4—122no—
610C4—121no—
611C4—121no—
612C4—122no—
613C4—241no—
614C4#550711yescharacteristic
615C4C2×#89611yescharacteristic
616C4#579111yescharacteristic
617C4Q8×#6711yescharacteristic
618C3#2640611yescharacteristic, complemented, Sylow
619C2—21no—
620C2—21no—
621C2—42no—
622C2#8217311yescharacteristic
623C2#8400911yescharacteristic
624C2#8401811yescharacteristic
subgroupG/H
centerC22D4×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2913C2
socleC22×C3D42
prime radicalC2×C3×#41C23
hypercenter#2913D3
hypocenterC3#26406
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2913

Complements

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

NH ≅ G/Ntype
4. #79198C2semidirect
9. #70679C2semidirect
10. #70677C2semidirect
11. #70669C2semidirect
12. #70666C2semidirect
13. #70661C2semidirect
14. #70644C2semidirect
15. #70626C2semidirect
106. #951Q8semidirect
618. C3#26406semidirect

Maximal subgroups

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

subgroupindex
C3×#29132
#825712
#825592
#791982
#720192
#720172
#716662
#716642
#706792
#706772
#706692
#706662
#706612
#706442
#706262
#264063