Finite Group Databasebeta

← #10824916

Subgroups — #10824916

Order 768.

744 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79200C211yesmaximal, characteristic
2#79071C211yesmaximal, characteristic
3#78946C211yesmaximal, characteristic
4C2×#4789C211yesmaximal, characteristic, complemented
5C3×#2064C211yesmaximal, characteristic, complemented
6#70716C211yesmaximal, characteristic
7#70692C211yesmaximal, characteristic, complemented
8#70671C211yesmaximal, characteristic, complemented
9#70544C211yesmaximal, characteristic, complemented
10#70340C211yesmaximal, characteristic, complemented
11#70264C211yesmaximal, characteristic
12#70125C211yesmaximal, characteristic
13Dic3×#101C211yesmaximal, characteristic, complemented
14#69011C211yesmaximal, characteristic
15#68973C211yesmaximal, characteristic, complemented
16#23720—31nomaximal, Sylow
17C2×Q8×Dic3C2211yescharacteristic
18C22×#216C2211yescharacteristic
19C2×#942C2211yescharacteristic
20C2×#909C2211yescharacteristic
21#5522C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C2×C3×#108C2211yescharacteristic
24C3×#428C2211yescharacteristic
25C3×#424C2211yescharacteristic
26C3×#419C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#101C2211yescharacteristic
29C2×#836C2211yescharacteristic
30C2×#824C2211yescharacteristic
31C2×#824C2211yescharacteristic
32#5133C2211yescharacteristic
33#5129C2211yescharacteristic
34Dic3×M16C2211yescharacteristic
35Dic3×M16C2211yescharacteristic
36Dic3×M16C2211yescharacteristic
37#5110C2211yescharacteristic
38C2×#819C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#815C2211yescharacteristic
41#5004C2211yescharacteristic
42#4999C2211yescharacteristic
43#4995C2211yescharacteristic
44#4966C2211yescharacteristic
45C2×#808C2211yescharacteristic
46#4889—21no—
47#4873—21no—
48#4853—21no—
49#4852—21no—
50#4840—22no—
51#4835—22no—
52#4817—21no—
53#4790—21no—
54#4789—21no—
55#4789—22no—
56#4789—22no—
57#4788—21no—
58C3×#343C2211yescharacteristic
59#4558C2211yescharacteristic
60#4558C2211yescharacteristic
61#4554C2211yescharacteristic
62#4554C2211yescharacteristic
63#4530C2211yescharacteristic
64C4×#137—31no—
65C4×#137—31no—
66C2×#441—31no—
67#2923—31no—
68#2090—31no—
69#2079—31no—
70#2064—31no—
71#2064D311yescharacteristic, complemented
72#2052—31no—
73#2052—31no—
74#2047—31no—
75#2020—31no—
76#1778—31no—
77C4×#101—31no—
78C4×#101—31no—
79C22×C3×Q8C2311yescharacteristic
80C2×#235C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C2×C3×M16C2311yescharacteristic
83C2×C3×M16C2311yescharacteristic
84C2×C3×M16C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#117C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88Q8×Dic3—21no—
89Q8×Dic3—21no—
90Q8×Dic3—21no—
91Q8×Dic3—21no—
92C2×#216—22no—
93C2×#216—21no—
94C2×#216—22no—
95C2×#216—22no—
96#942—22no—
97#942—22no—
98C2×#215—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C2×#208C2311yescharacteristic
106C2×#208C2311yescharacteristic
107C2×#208C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#909—22no—
110#909—22no—
111C2×Dic12—21no—
112C2×#204—21no—
113#893D411yescharacteristic
114#893D411yescharacteristic
115#887—21no—
116#887—21no—
117#887—21no—
118#887—21no—
119#887D411yescharacteristic
120#887D411yescharacteristic
121#67⋊C4—21no—
122#67⋊C4—21no—
123#872—21no—
124C42×D3—21no—
125C3×#108—22no—
126C3×#107—21no—
127C3×#106—21no—
128C3×C4⋊C8—22no—
129C3×#103—22no—
130C3×#103—21no—
131C3×#103—21no—
132C3×#101—22no—
133#836—22no—
134#824—22no—
135#824—22no—
136#821—21no—
137(C2×D3)⋊C8—22no—
138(C2×D3)⋊C8—22no—
139#819—22no—
140#816—21no—
141#816—22no—
142#815—21no—
143#815—21no—
144#815—21no—
145#815D412yes—
146#815D412yes—
147C8×Dic3—21no—
148C8×Dic3—21no—
149C8×Dic3—21no—
150#810—21no—
151#810—21no—
152#810—21no—
153#809—21no—
154#808—21no—
155#808—22no—
156#807—21no—
157C4⋊(C3⋊C8)—22no—
158C2×#137—31no—
159C2×#137—31no—
160C22×Q16—31no—
161C4×#41—31no—
162C2×C4×Q8—31no—
163C2×C4×Q8—31no—
164#441—62no—
165#441—62no—
166#441—61no—
167#441—61no—
168#440—61no—
169#440—61no—
170C4×Q16—61no—
171C4×Q16—61no—
172C4×SD16—61no—
173C4×SD16—61no—
174#433—62no—
175#432—62no—
176C2×#108—31no—
177C2×#108C2×D311yescharacteristic
178#428—31no—
179#428C2×D311yescharacteristic
180C2×#106—31no—
181#424—31no—
182#424C2×D311yescharacteristic
183#419—31no—
184#419C2×D311yescharacteristic
185#417—31no—
186#417—31no—
187C2×#103—31no—
188C2×#103C2×D311yescharacteristic
189C2×#101—31no—
190C2×#101—31no—
191C2×#101—31no—
192C2×#101C2×D311yescharacteristic
193#408—31no—
194#408—31no—
195C4×M16—31no—
196C4×M16—31no—
197C4×M16—31no—
198C2×#97—31no—
199#343—31no—
200#343—31no—
201#343—31no—
202#343C2×D311yescharacteristic
203#341—31no—
204#341—31no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8#4112yes—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4C2411yescharacteristic
214C2×#67—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3#4111yescharacteristic
218C22×Dic3#4111yescharacteristic
219#235—21no—
220#235—21no—
221#235—21no—
222#235—21no—
223C2×D12C2×D411yescharacteristic
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6C2×D411yescharacteristic
233C2×Dic6#4112yes—
234C3×M16—21no—
235C3×M16—21no—
236C3×M16—21no—
237C3×M16—21no—
238C3×M16—21no—
239C3×M16#4112yes—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8C2×D412yes—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4C2×D411yescharacteristic
249C3×C4⋊C4C2×D411yescharacteristic
250C3×C22⋊C4—21no—
251C3×C42—21no—
252#216—24no—
253#216—28no—
254#215—24no—
255(C2×D3)⋊C4—21no—
256(C2×D3)⋊C4—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—41no—
260C4⋊Dic3—41no—
261Dic3⋊C4—21no—
262Dic3⋊C4—21no—
263Dic3⋊C4—41no—
264Dic3⋊C4—41no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—41no—
268C4×Dic3—41no—
269C4×Dic3—41no—
270C4×Dic3—41no—
271C4×Dic3C2×D411yescharacteristic
272C4×Dic3C2×D411yescharacteristic
273C4×Dic3C2×D411yescharacteristic
274C4×Dic3C2×D411yescharacteristic
275#208—21no—
276#208—21no—
277#208—21no—
278#208—21no—
279#208—21no—
280#208#4112yes—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8C2×D412yes—
287Dic12—24no—
288#204—24no—
289C2×#41—31no—
290C22×Q8—31no—
291C22×Q8C22×D311yescharacteristic
292#137—62no—
293#137—62no—
294#137—62no—
295#137—62no—
296C2×Q16—61no—
297C2×Q16—62no—
298C2×Q16—62no—
299C2×Q16—61no—
300C2×Q16—62no—
301C2×SD16—61no—
302C2×SD16—61no—
303C2×M16—31no—
304C2×M16—31no—
305C2×M16—31no—
306C2×M16C22×D311yescharacteristic
307C2×M16C22×D311yescharacteristic
308C2×M16C22×D311yescharacteristic
309C22×C8—31no—
310C22×C8C22×D311yescharacteristic
311C4×Q8—31no—
312C4×Q8—31no—
313C4×Q8—61no—
314C4×Q8—61no—
315C4×Q8—61no—
316C4×Q8—61no—
317C4×Q8—61no—
318C4×Q8—61no—
319C4×Q8—61no—
320C4×Q8—61no—
321C4×D4—31no—
322C4×D4—31no—
323C4×D4—61no—
324C4×D4—61no—
325#117—61no—
326#117C22×D311yescharacteristic
327C2×C4⋊C4—31no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4C22×D311yescharacteristic
331C2×C42—31no—
332C2×C42—61no—
333C2×C42—61no—
334C2×C42—61no—
335#108—22no—
336#108—62no—
337#107—21no—
338#107—61no—
339#106—21no—
340#106—62no—
341#106—61no—
342C4⋊C8—22no—
343C4⋊C8—62no—
344#103—22no—
345#103—21no—
346#103—21no—
347#103—61no—
348#103—61no—
349#103—62no—
350#103—61no—
351#103—61no—
352#102—61no—
353#102—61no—
354#101—22no—
355#101—62no—
356#101—62no—
357#101—62no—
358C22⋊C8—62no—
359C22⋊C8—62no—
360#97—32no—
361#97—32no—
362#97—61no—
363#97—61no—
364#97—61no—
365C4×C8—61no—
366C4×C8—61no—
367C4×C8—61no—
368C23×C3C2×#4111yescharacteristic
369C22×D3—21no—
370C3×Q8—22no—
371C3×Q8—22no—
372C3×Q8—41no—
373C3×Q8—41no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—81no—
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387#67—41no—
388#67—41no—
389C2×Dic3—21no—
390C2×Dic3—22no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—81no—
397C2×Dic3#12111yescharacteristic
398C2×Dic3#12111yescharacteristic
399C2×Dic3#12111yescharacteristic
400C2×Dic3#12111yescharacteristic
401D12—21no—
402D12—21no—
403C4×D3—41no—
404C4×D3—41no—
405C4×D3—81no—
406C4×D3—81no—
407Dic6—22no—
408Dic6—21no—
409Dic6—22no—
410Dic6—21no—
411Dic6—41no—
412Dic6—41no—
413C3×C8—22no—
414C3×C8—22no—
415C3×C8—41no—
416C3×C8—41no—
417C3⋊C8—22no—
418C3⋊C8—22no—
419C3⋊C8—41no—
420C3⋊C8—41no—
421#41—61no—
422#41—61no—
423#41—61no—
424#41—61no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—31no—
431C2×Q8—32no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8#23912yes—
438C2×D4—31no—
439C2×D4—61no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4C23×D311yescharacteristic
450Q16—64no—
451Q16—68no—
452Q16—64no—
453SD16—64no—
454SD16—64no—
455M16—21no—
456M16—21no—
457M16—21no—
458M16—21no—
459M16—21no—
460M16—32no—
461M16—61no—
462M16—61no—
463M16—61no—
464M16—61no—
465M16—61no—
466M16#23712yes—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—32no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8D3×D412yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—121no—
486C4⋊C4—121no—
487C4⋊C4—121no—
488C4⋊C4—121no—
489C4⋊C4D3×D411yescharacteristic
490C4⋊C4D3×D411yescharacteristic
491C22⋊C4—21no—
492C22⋊C4—61no—
493C22⋊C4—61no—
494C42—21no—
495C42—31no—
496C42—31no—
497C42—31no—
498C42—31no—
499C42—61no—
500C42—61no—
501C42—121no—
502C42—121no—
503C42—121no—
504C42—121no—
505C22×C3—21no—
506C22×C3—21no—
507C22×C3C2×#12111yescharacteristic
508C22×C3#52911yescharacteristic
509C22×C3#52911yescharacteristic
510C2×D3—41no—
511C2×D3—41no—
512C2×D3—41no—
513C3×C4—42no—
514C3×C4—41no—
515C3×C4—41no—
516C3×C4—81no—
517C3×C4#52511yescharacteristic
518C3×C4#52511yescharacteristic
519C3×C4#52511yescharacteristic
520C3×C4#53211yescharacteristic
521Dic3—21no—
522Dic3—21no—
523Dic3—21no—
524Dic3—21no—
525Dic3—42no—
526Dic3—41no—
527Dic3—81no—
528C23—61no—
529C23D3×#4111yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—41no—
533Q8—41no—
534Q8—62no—
535Q8—61no—
536Q8—62no—
537Q8—61no—
538Q8—121no—
539Q8—121no—
540D4—61no—
541D4—61no—
542D4—121no—
543D4—121no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—31no—
548C2×C4—31no—
549C2×C4—31no—
550C2×C4—31no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—62no—
555C2×C4—61no—
556C2×C4—81no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—241no—
565C2×C4—241no—
566C2×C4—241no—
567C2×C4C2×D3×D411yescharacteristic
568C2×C4C2×D3×D411yescharacteristic
569C2×C4C2×#23711yescharacteristic
570C2×C4C2×#23911yescharacteristic
571C2×C4D3×#4111yescharacteristic
572C2×C4D3×#4111yescharacteristic
573C8—22no—
574C8—22no—
575C8—41no—
576C8—41no—
577C8—62no—
578C8—62no—
579C8—121no—
580C8—121no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3#239211yescharacteristic
584C2×C3#306911yescharacteristic
585C2×C3#306911yescharacteristic
586D3—81no—
587C22—21no—
588C22—21no—
589C22—121no—
590C22—121no—
591C22—121no—
592C22D3×#12111yescharacteristic
593C22#562411yescharacteristic
594C22#570611yescharacteristic
595C4—41no—
596C4—41no—
597C4—42no—
598C4—61no—
599C4—61no—
600C4—61no—
601C4—61no—
602C4—81no—
603C4—121no—
604C4—122no—
605C4—241no—
606C4#559611yescharacteristic
607C4#560611yescharacteristic
608C4#562711yescharacteristic
609C4#562711yescharacteristic
610C3#2372011yescharacteristic, complemented, Sylow
611C2—21no—
612C2—21no—
613C2—241no—
614C2#7728111yescharacteristic
615C2#8093111yescharacteristic
616C2#8094411yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2064C2
socleC22×C3C2×#121
prime radicalC2×#235C23
hypercenter#2064D3
hypocenterC3#23720
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2064

Complements

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

NH ≅ G/Ntype
4. C2×#4789C2semidirect
5. C3×#2064C2semidirect
7. #70692C2semidirect
8. #70671C2semidirect
9. #70544C2semidirect
10. #70340C2semidirect
13. Dic3×#101C2semidirect
15. #68973C2semidirect
71. #2064D3semidirect
610. C3#23720semidirect

Maximal subgroups

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

subgroupindex
#792002
#790712
#789462
C2×#47892
C3×#20642
#707162
#706922
#706712
#705442
#703402
#702642
#701252
Dic3×#1012
#690112
#689732
#237203