Finite Group Databasebeta

← #10858706

Subgroups — #10858706

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
1#83785C211yesmaximal, characteristic
2#79101C211yesmaximal, characteristic, complemented
3#78823C211yesmaximal, characteristic
4#78821C211yesmaximal, characteristic
5C3×#1866C211yesmaximal, characteristic
6#72019C211yesmaximal, characteristic
7#70717C211yesmaximal, characteristic, complemented
8#70701C211yesmaximal, characteristic, complemented
9#70532C211yesmaximal, characteristic, complemented
10#70365C211yesmaximal, characteristic
11#70363C211yesmaximal, characteristic
12#68764C211yesmaximal, characteristic, complemented
13#68735C211yesmaximal, characteristic, complemented
14#68691C211yesmaximal, characteristic, complemented
15#68645C211yesmaximal, characteristic, complemented
16#26406—31nomaximal, Sylow
17C2×C3×#131C2211yescharacteristic
18#5827C2211yescharacteristic
19#5827C2211yescharacteristic
20C2×#951C2211yescharacteristic
21#5519C2211yescharacteristic
22#5516C2211yescharacteristic
23C2×C3×#108C2211yescharacteristic
24C3×#424C2211yescharacteristic
25C2×C3×#104C2211yescharacteristic
26C3×#416C2211yescharacteristic
27#5232C2211yescharacteristic
28C2×#824C2211yescharacteristic
29#5121—21no—
30#5121—21no—
31#5121C2211yescharacteristic
32#5121C2211yescharacteristic
33#5121C2211yescharacteristic
34#5120C2211yescharacteristic
35#5106—21no—
36#5106C2211yescharacteristic
37C2×#818—21no—
38C2×#818C2211yescharacteristic
39C2×#818C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×#816C2211yescharacteristic
42#5102C2211yescharacteristic
43#5101C2211yescharacteristic
44#4995C2211yescharacteristic
45#4995C2211yescharacteristic
46#4935C2211yescharacteristic
47#4929C2211yescharacteristic
48#4851—21no—
49#4850—21no—
50#4829—21no—
51#4829—21no—
52#4796—21no—
53#4793—21no—
54#4790—21no—
55#4789—21no—
56#4766—21no—
57#4759—21no—
58#4758—21no—
59#4757—21no—
60C3×#339C2211yescharacteristic
61C3×#336C2211yescharacteristic
62#4554C2211yescharacteristic
63#4554C2211yescharacteristic
64#4554C2211yescharacteristic
65#4552C2211yescharacteristic
66#4523C2211yescharacteristic
67#4523C2211yescharacteristic
68#3083—31no—
69#3082—31no—
70#2947—31no—
71#2913—31no—
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#1866D311yescharacteristic
81#1859—31no—
82#1857—31no—
83C2×C3×#41C2311yescharacteristic
84C22×Dic6C2311yescharacteristic
85C3×#131—22no—
86C3×#131—22no—
87C2×C3×M16—21no—
88C2×C3×M16C2311yescharacteristic
89C2×C3×M16C2311yescharacteristic
90C22×C3×C8—21no—
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C3×#117C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95#951—22no—
96#951Q814yescomplemented
97Q8×Dic3—21no—
98Q8×Dic3—21no—
99C2×#216—21no—
100D4×Dic3—21no—
101D4×Dic3—21no—
102D4×Dic3—41no—
103D4×Dic3—41no—
104C2×#214—21no—
105#926—21no—
106#926—21no—
107#926C2311yescharacteristic
108#926C2311yescharacteristic
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3—41no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×#208C2311yescharacteristic
116C2×#208C2311yescharacteristic
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×#108—22no—
128C3×C4⋊C8—21no—
129C3×C4⋊C8—21no—
130C3×#104—22no—
131C3×#103D411yescharacteristic
132C3×#103D411yescharacteristic
133C3×#102D411yescharacteristic
134C3×#102D411yescharacteristic
135#835—21no—
136#832—21no—
137C3⋊#95—41no—
138#824—22no—
139#818—21no—
140#818—21no—
141#818—21no—
142#818—21no—
143#818—21no—
144#818—21no—
145#818—21no—
146#818D411yescharacteristic
147#818D411yescharacteristic
148#817—21no—
149#817—21no—
150#817—21no—
151#817D411yescharacteristic
152#817D411yescharacteristic
153#816—21no—
154#816—21no—
155#816—21no—
156#816—21no—
157#816—21no—
158#816—21no—
159#815—21no—
160Dic3⋊C8—21no—
161Dic3⋊C8—21no—
162C8×Dic3—21no—
163#808—21no—
164#808—21no—
165#807—21no—
166#807—21no—
167C2×#137—31no—
168C2×#131C2×D311yescharacteristic
169#544—31no—
170#520—31no—
171#520—31no—
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×#108C2×D311yescharacteristic
186#428—31no—
187#428—31no—
188#428—31no—
189#428—31no—
190#428—31no—
191#428—61no—
192#428—61no—
193#427—31no—
194#427—61no—
195C2×#107—31no—
196C2×#107—31no—
197C2×#107—61no—
198#424—31no—
199#424C2×D311yescharacteristic
200C2×#104C2×D311yescharacteristic
201#419—31no—
202#417—31no—
203#416C2×D311yescharacteristic
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#339C2×D311yescharacteristic
217#336C2×D311yescharacteristic
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×C4C2411yescharacteristic
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3—21no—
232C22×Dic3—41no—
233C22×Dic3—41no—
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—21no—
244C3×M16—21no—
245C3×M16#4112yes—
246C2×C3×C8—24no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8C2×D411yescharacteristic
254C2×C3×C8C2×D411yescharacteristic
255C2×C3×C8C2×D411yescharacteristic
256C2×C3×C8C2×D411yescharacteristic
257C3×C4⋊C4C2×D411yescharacteristic
258C3×C4⋊C4C2×D411yescharacteristic
259C3×C22⋊C4—21no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42—21no—
263#217—21no—
264#217—21no—
265#217—41no—
266#217—41no—
267#216—24no—
268#214—24no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—41no—
276C4⋊Dic3—41no—
277C4⋊Dic3—41no—
278C4⋊Dic3—82no—
279C4⋊Dic3C2×D411yescharacteristic
280C4⋊Dic3C2×D411yescharacteristic
281C4⋊Dic3C2×D411yescharacteristic
282C4⋊Dic3C2×D411yescharacteristic
283Dic3⋊C4—41no—
284Dic3⋊C4—41no—
285Dic3⋊C4—41no—
286Dic3⋊C4—41no—
287C4×Dic3—21no—
288C4×Dic3—21no—
289C4×Dic3—41no—
290C4×Dic3—41no—
291C4×Dic3—41no—
292C4×Dic3—41no—
293#208—21no—
294#208—21no—
295#208—21no—
296#208C2×Q812yes—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
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×M16C22×D311yescharacteristic
311C2×M16C22×D311yescharacteristic
312C22×C8—21no—
313C22×C8C22×D311yescharacteristic
314C22×C8C22×D311yescharacteristic
315C4⋊Q8—61no—
316C4⋊Q8—61no—
317#125—61no—
318#125—61no—
319C22⋊Q8—121no—
320C4×Q8—61no—
321C4×Q8—61no—
322C4×Q8—61no—
323C4×Q8—61no—
324C4×Q8—61no—
325C4×Q8—61no—
326C4×D4—61no—
327C4×D4—61no—
328C4×D4—121no—
329C4×D4—121no—
330#117—31no—
331#117—31no—
332#117—61no—
333#117—61no—
334#117—121no—
335#117C22×D311yescharacteristic
336C2×C4⋊C4—31no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—121no—
342C2×C42C22×D311yescharacteristic
343#108—22no—
344#107—31no—
345#107—31no—
346#107—61no—
347#107—61no—
348#107—61no—
349#107—61no—
350#107—61no—
351#107—61no—
352#107—61no—
353#107—61no—
354#107—61no—
355#106—31no—
356#106—31no—
357#106—61no—
358#106—61no—
359#106—61no—
360#106—61no—
361#106—61no—
362C4⋊C8—21no—
363C4⋊C8—21no—
364C4⋊C8—61no—
365C4⋊C8—61no—
366#104—22no—
367#103—61no—
368#103—61no—
369#103—61no—
370#103—61no—
371#103—61no—
372#103—61no—
373#103—61no—
374#103D1211yescharacteristic
375#103D1211yescharacteristic
376#102—61no—
377#102D1211yescharacteristic
378#102D1211yescharacteristic
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—22no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C22×Q811yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C42−1+411yescharacteristic
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—22no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—81no—
416C2×Dic3—81no—
417C2×Dic3—81no—
418C2×Dic3—161no—
419C2×Dic3—161no—
420Dic6—22no—
421Dic6—22no—
422Dic6—41no—
423Dic6—41no—
424Dic6—161no—
425C3×C8—21no—
426C3×C8—22no—
427C3×C8—21no—
428C3×C8—21no—
429C3×C8—21no—
430C3×C8—41no—
431C3⋊C8—22no—
432C3⋊C8—41no—
433#41—21no—
434#41—21no—
435#41#23712yes—
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×Q8C2×D1211yescharacteristic
445C2×D4—21no—
446C2×D4C2×D1211yescharacteristic
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—121no—
454C22×C4—121no—
455C22×C4C23×D311yescharacteristic
456Q16—64no—
457SD16—64no—
458M16—24no—
459M16—21no—
460M16—21no—
461M16—21no—
462M16—32no—
463M16—61no—
464M16—61no—
465M16—61no—
466M16#23712yes—
467C2×C8—24no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8C2×D1211yescharacteristic
477C2×C8C2×D1211yescharacteristic
478C2×C8D3×D411yescharacteristic
479C2×C8D3×D411yescharacteristic
480C4⋊C4—31no—
481C4⋊C4—31no—
482C4⋊C4—31no—
483C4⋊C4—31no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—121no—
491C4⋊C4—121no—
492C4⋊C4—121no—
493C4⋊C4—121no—
494C4⋊C4—121no—
495C4⋊C4—121no—
496C4⋊C4—121no—
497C4⋊C4—242no—
498C4⋊C4C2×D1211yescharacteristic
499C4⋊C4C2×D1211yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—61no—
502C22⋊C4—61no—
503C22⋊C4—121no—
504C22⋊C4—121no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—61no—
509C42—61no—
510C42—121no—
511C42—121no—
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—42no—
524C3×C4—41no—
525C3×C4—41no—
526C3×C4C2×C22⋊Q811yescharacteristic
527C3×C4#52511yescharacteristic
528C3×C4#54711yescharacteristic
529C3×C4D4×Q811yescharacteristic
530Dic3—42no—
531Dic3—41no—
532Dic3—41no—
533Dic3—41no—
534Dic3—41no—
535Dic3—81no—
536Dic3—81no—
537Dic3—81no—
538C23—21no—
539C23#100911yescharacteristic
540Q8—21no—
541Q8—62no—
542Q8—62no—
543Q8—121no—
544Q8—121no—
545Q8—481no—
546Q8#89712yes—
547D4—22no—
548D4—22no—
549D4—21no—
550D4#89712yes—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—62no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—241no—
570C2×C4—241no—
571C2×C4—241no—
572C2×C4—481no—
573C2×C4—481no—
574C2×C4C22×D1211yescharacteristic
575C2×C4C2×D3×D411yescharacteristic
576C2×C4C2×#23711yescharacteristic
577C2×C4C2×#23711yescharacteristic
578C2×C4C2×D3×Q811yescharacteristic
579C2×C4#101011yescharacteristic
580C8—21no—
581C8—21no—
582C8—21no—
583C8—22no—
584C8—21no—
585C8—41no—
586C8—62no—
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—
599C22D4×D1211yescharacteristic
600C22#563511yescharacteristic
601C22#566511yescharacteristic
602C4—22no—
603C4—42no—
604C4—41no—
605C4—41no—
606C4—121no—
607C4—121no—
608C4—121no—
609C4—121no—
610C4—122no—
611C4—241no—
612C4—241no—
613C4—241no—
614C4C2×#89711yescharacteristic
615C4#555511yescharacteristic
616C4#558611yescharacteristic
617C4#563211yescharacteristic
618C3#2640611yescharacteristic, complemented, Sylow
619C2—21no—
620C2—21no—
621C2—42no—
622C2#7442211yescharacteristic
623C2#7989511yescharacteristic
624C2#7990011yescharacteristic
subgroupG/H
centerC22D4×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1866C2
socleC22×C3D42
prime radicalC2×C3×#41C23
hypercenter#1866D3
hypocenterC3#26406
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1866

Complements

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

NH ≅ G/Ntype
2. #79101C2semidirect
7. #70717C2semidirect
8. #70701C2semidirect
9. #70532C2semidirect
12. #68764C2semidirect
13. #68735C2semidirect
14. #68691C2semidirect
15. #68645C2semidirect
96. #951Q8semidirect
618. C3#26406semidirect

Maximal subgroups

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

subgroupindex
#837852
#791012
#788232
#788212
C3×#18662
#720192
#707172
#707012
#705322
#703652
#703632
#687642
#687352
#686912
#686452
#264063