Finite Group Databasebeta

← #10824913

Subgroups — #10824913

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
1C3×C4×#137C211yesmaximal, characteristic
2C4×#951C211yesmaximal, characteristic
3#82514C211yesmaximal, characteristic
4C2×#5030C211yesmaximal, characteristic, complemented
5#72109C211yesmaximal, characteristic
6#72105C211yesmaximal, characteristic
7C4×#836C211yesmaximal, characteristic, complemented
8#70679C211yesmaximal, characteristic, complemented
9#70671C211yesmaximal, characteristic, complemented
10#70340C211yesmaximal, characteristic, complemented
11#70337C211yesmaximal, characteristic, complemented
12#70310C211yesmaximal, characteristic
13#70302C211yesmaximal, characteristic
14#69873C211yesmaximal, characteristic, complemented
15C4×#824C211yesmaximal, characteristic, complemented
16#23720—31nomaximal, Sylow
17C2×C3×#137C2211yescharacteristic
18C3×C4×#41C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×#951C2211yescharacteristic
21C22×#216C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#441—21no—
24C3×#440—21no—
25C3×C4×Q16—21no—
26C3×C4×SD16—21no—
27C3×#428C2211yescharacteristic
28C3×#419C2211yescharacteristic
29C3×#417C2211yescharacteristic
30C3×C4×M16C2211yescharacteristic
31#5234C2211yescharacteristic
32#5234C2211yescharacteristic
33#5232C2211yescharacteristic
34C2×#836C2211yescharacteristic
35C2×#836C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#824C2211yescharacteristic
38C2×#824C2211yescharacteristic
39C2×#822C2211yescharacteristic
40C2×#822C2211yescharacteristic
41#5030—21no—
42#5030—22no—
43#5030—22no—
44C4×#216—21no—
45#5018—21no—
46C4×#214—21no—
47#5004C2211yescharacteristic
48#5000—22no—
49#4999C2211yescharacteristic
50#4999C2211yescharacteristic
51#4995C2211yescharacteristic
52C2×#810C2211yescharacteristic
53C2×#808C2211yescharacteristic
54#4926—22no—
55C4×#208C2211yescharacteristic
56C4×#208C2211yescharacteristic
57C2×#803C2211yescharacteristic
58#4558C2211yescharacteristic
59#4558C2211yescharacteristic
60#4554C2211yescharacteristic
61#4554C2211yescharacteristic
62#4530C2211yescharacteristic
63#4530C2211yescharacteristic
64C4×#137—31no—
65C4×#137D311yescharacteristic
66C2×#441—31no—
67#2923—31no—
68#2090—31no—
69#2079—31no—
70#2064—31no—
71#2064—31no—
72#2052—31no—
73#2052—31no—
74#2047—31no—
75#2020—31no—
76#1778—31no—
77C4×#101—31no—
78C4×#101—31no—
79C2×C3×#41C2311yescharacteristic
80C22×C3×Q8C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C3×#137—22no—
83C3×#137—22no—
84C2×C3×Q16—21no—
85C2×C3×SD16—21no—
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C3×C4×Q8—21no—
89C3×C4×Q8—21no—
90C3×C4×Q8—21no—
91C3×C4×Q8—21no—
92C3×C4×Q8D411yescharacteristic
93C3×C4×Q8D411yescharacteristic
94C3×C4×D4—21no—
95C3×C4×D4—21no—
96C3×C4×D4D411yescharacteristic
97C3×C4×D4D411yescharacteristic
98C3×#117—21no—
99C3×#117C2311yescharacteristic
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C42—21no—
103C2×C3×C42—21no—
104C2×C3×C42C2311yescharacteristic
105#951—22no—
106#951—22no—
107C2×#216—21no—
108C2×#216—22no—
109C2×#216—22no—
110C2×#216—22no—
111C2×#214—21no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C2×#208C2311yescharacteristic
116C2×#208C2311yescharacteristic
117C2×#208C2311yescharacteristic
118C2×#208C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C4×Dic6—21no—
122C4×Dic6—21no—
123C4×Dic6—21no—
124C4×Dic6—21no—
125C3×#107—21no—
126C3×#106—21no—
127C3×#103—21no—
128C3×#103—21no—
129C3×#103—21no—
130C3×#102—21no—
131C3×#97—21no—
132C3×C4×C8—21no—
133#836—22no—
134#836—22no—
135#835—21no—
136#835—22no—
137#832—21no—
138#830—22no—
139#830—22no—
140#824—22no—
141#824—22no—
142#822—22no—
143#822—22no—
144#810—21no—
145#810—21no—
146#810—22no—
147#808—21no—
148#808—22no—
149#807—21no—
150C4⋊(C3⋊C8)—22no—
151C4⋊(C3⋊C8)—22no—
152#803—21no—
153#803—21no—
154#803D412yes—
155#803D412yes—
156C4×C3⋊C8—21no—
157C4×C3⋊C8—21no—
158C2×#137—31no—
159C2×#137C2×D311yescharacteristic
160C22×Q16—31no—
161C4×#41C2×D311yescharacteristic
162C2×C4×Q8—31no—
163C2×C4×Q8C2×D311yescharacteristic
164#441—21no—
165#441—62no—
166#441—62no—
167#441—61no—
168#440—21no—
169#440—61no—
170C4×Q16—21no—
171C4×Q16—61no—
172C4×SD16—21no—
173C4×SD16—61no—
174#433—62no—
175#432—62no—
176C2×#108—31no—
177C2×#108—31no—
178#428—31no—
179#428C2×D311yescharacteristic
180C2×#106—31no—
181#424—31no—
182#424—31no—
183#419—31no—
184#419C2×D311yescharacteristic
185#417—31no—
186#417C2×D311yescharacteristic
187C2×#103—31no—
188C2×#103—31no—
189C2×#101—31no—
190C2×#101—31no—
191C2×#101—31no—
192C2×#101—31no—
193#408—31no—
194#408—31no—
195C4×M16—31no—
196C4×M16—31no—
197C4×M16C2×D311yescharacteristic
198C2×#97—31no—
199#343—31no—
200#343—31no—
201#343—31no—
202#343—31no—
203#341—31no—
204#341—31no—
205C3×#41—21no—
206C3×#41—21no—
207C3×#41—21no—
208C3×#41—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8C2×D411yescharacteristic
215C2×C3×Q8#4112yes—
216C2×C3×D4—21no—
217C2×C3×D4C2×D411yescharacteristic
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4#4111yescharacteristic
224C22×C3×C4#4111yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C22×Dic3—21no—
227C22×Dic3—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6#4112yes—
234C3×Q16—24no—
235C3×SD16—24no—
236C3×M16—21no—
237C3×M16—21no—
238C3×M16—21no—
239C3×M16#4112yes—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—41no—
247C3×C4⋊C4—41no—
248C3×C4⋊C4C2×D411yescharacteristic
249C3×C4⋊C4C2×D411yescharacteristic
250C3×C22⋊C4—21no—
251C3×C22⋊C4—21no—
252C3×C22⋊C4—21no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42—41no—
257C3×C42—41no—
258C3×C42C2×D411yescharacteristic
259C3×C42C2×D411yescharacteristic
260C3×C42C2×D411yescharacteristic
261C3×C42C2×D411yescharacteristic
262#216—28no—
263#216—24no—
264#214—24no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267Dic3⋊C4—41no—
268Dic3⋊C4—41no—
269C4×Dic3—41no—
270C4×Dic3—41no—
271#208—21no—
272#208—21no—
273#208—21no—
274#208—21no—
275#208—21no—
276#208—21no—
277#208—21no—
278#208#4112yes—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8C2×D412yes—
288C2×C3⋊C8C2×D412yes—
289C2×#41C22×D311yescharacteristic
290C22×Q8—31no—
291C22×Q8C22×D311yescharacteristic
292#137—22no—
293#137—22no—
294#137—62no—
295#137—62no—
296C2×Q16—21no—
297C2×Q16—62no—
298C2×Q16—62no—
299C2×Q16—62no—
300C2×Q16—61no—
301C2×SD16—21no—
302C2×SD16—61no—
303C2×M16—31no—
304C2×M16—31no—
305C2×M16—31no—
306C2×M16—31no—
307C2×M16C22×D311yescharacteristic
308C2×M16C22×D311yescharacteristic
309C22×C8—31no—
310C22×C8—31no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×Q8—21no—
315C4×Q8—61no—
316C4×Q8—61no—
317C4×Q8—61no—
318C4×Q8—61no—
319C4×Q8#6711yescharacteristic
320C4×Q8#6711yescharacteristic
321C4×D4—21no—
322C4×D4—21no—
323C4×D4#6711yescharacteristic
324C4×D4#6711yescharacteristic
325#117—21no—
326#117C22×D311yescharacteristic
327C2×C4⋊C4—21no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4C22×D311yescharacteristic
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—61no—
334C2×C42C22×D311yescharacteristic
335#108—62no—
336#108—62no—
337#107—21no—
338#107—61no—
339#106—21no—
340#106—62no—
341#106—61no—
342C4⋊C8—62no—
343C4⋊C8—62no—
344#103—21no—
345#103—21no—
346#103—21no—
347#103—62no—
348#103—62no—
349#103—61no—
350#103—61no—
351#103—61no—
352#102—21no—
353#102—61no—
354#101—62no—
355#101—62no—
356#101—62no—
357#101—62no—
358C22⋊C8—62no—
359C22⋊C8—62no—
360#97—21no—
361#97—32no—
362#97—32no—
363#97—61no—
364#97—61no—
365C4×C8—21no—
366C4×C8—61no—
367C4×C8—61no—
368C23×C3—21no—
369C23×C3C2×#4111yescharacteristic
370C3×Q8—22no—
371C3×Q8—21no—
372C3×Q8—22no—
373C3×Q8—21no—
374C3×Q8—41no—
375C3×Q8—41no—
376C3×D4—21no—
377C3×D4—21no—
378C3×D4—41no—
379C3×D4—41no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—81no—
392C2×C3×C4—81no—
393C2×C3×C4—81no—
394C2×C3×C4#12111yescharacteristic
395C2×C3×C4#12111yescharacteristic
396C2×C3×C4#12111yescharacteristic
397C2×C3×C4#12111yescharacteristic
398C2×C3×C4C22×D411yescharacteristic
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×Dic3—22no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—81no—
409Dic6—22no—
410Dic6—22no—
411Dic6—41no—
412Dic6—41no—
413C3×C8—22no—
414C3×C8—41no—
415C3⋊C8—22no—
416C3⋊C8—22no—
417C3⋊C8—22no—
418C3⋊C8—41no—
419C3⋊C8—41no—
420C3⋊C8—41no—
421#41—21no—
422#41—21no—
423#41—21no—
424#41—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—32no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8#23512yes—
437C2×Q8C2×#6711yescharacteristic
438C2×D4—21no—
439C2×D4C2×#6711yescharacteristic
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4#23711yescharacteristic
448C22×C4#23711yescharacteristic
449C22×C4C23×D311yescharacteristic
450Q16—24no—
451Q16—68no—
452Q16—64no—
453SD16—24no—
454SD16—64no—
455M16—21no—
456M16—21no—
457M16—21no—
458M16—32no—
459M16—61no—
460M16—61no—
461M16—61no—
462M16—61no—
463M16—61no—
464M16—61no—
465M16—61no—
466M16#23512yes—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—32no—
470C2×C8—32no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—121no—
488C4⋊C4—121no—
489C4⋊C4C2×#6711yescharacteristic
490C4⋊C4C2×#6711yescharacteristic
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—41no—
498C42—41no—
499C42—121no—
500C42—121no—
501C42D3×D411yescharacteristic
502C42D3×D411yescharacteristic
503C42C2×#6711yescharacteristic
504C42C2×#6711yescharacteristic
505C22×C3—21no—
506C22×C3—21no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3C2×#12111yescharacteristic
511C22×C3#52911yescharacteristic
512C22×C3#52911yescharacteristic
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—41no—
518C3×C4—42no—
519C3×C4—41no—
520C3×C4—41no—
521C3×C4—81no—
522C3×C4#52511yescharacteristic
523C3×C4#52511yescharacteristic
524C3×C4#52511yescharacteristic
525C3×C4#53211yescharacteristic
526Dic3—42no—
527Dic3—81no—
528C23—21no—
529C23C2×#23711yescharacteristic
530Q8—22no—
531Q8—21no—
532Q8—22no—
533Q8—21no—
534Q8—41no—
535Q8—41no—
536Q8—62no—
537Q8—62no—
538Q8—121no—
539Q8—121no—
540D4—21no—
541D4—21no—
542D4—41no—
543D4—41no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—62no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—241no—
563C2×C4#93911yescharacteristic
564C2×C4#93911yescharacteristic
565C2×C4#93911yescharacteristic
566C2×C4#93911yescharacteristic
567C2×C4C2×#23511yescharacteristic
568C2×C4C2×#23511yescharacteristic
569C2×C4C2×D3×D411yescharacteristic
570C2×C4D3×#4111yescharacteristic
571C2×C4D3×#4111yescharacteristic
572C2×C4C22×#6711yescharacteristic
573C8—22no—
574C8—41no—
575C8—62no—
576C8—62no—
577C8—62no—
578C8—121no—
579C8—121no—
580C8—121no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3—81no—
584C2×C3#239211yescharacteristic
585C2×C3#306911yescharacteristic
586C2×C3#306911yescharacteristic
587C22—21no—
588C22—21no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22#553811yescharacteristic
593C22#555711yescharacteristic
594C22C2×#93911yescharacteristic
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—81no—
604C4—122no—
605C4—241no—
606C4#554711yescharacteristic
607C4#554811yescharacteristic
608C4#579111yescharacteristic
609C4#582811yescharacteristic
610C3#2372011yescharacteristic, complemented, Sylow
611C2—21no—
612C2—21no—
613C2—81no—
614C2#8234511yescharacteristic
615C2#8397811yescharacteristic
616C2#8400311yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#137C2
socleC22×C3C2×#121
prime radicalC2×C3×#41C23
hypercenterC4×#137D3
hypocenterC3#23720
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#137

Complements

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

NH ≅ G/Ntype
4. C2×#5030C2semidirect
7. C4×#836C2semidirect
8. #70679C2semidirect
9. #70671C2semidirect
10. #70340C2semidirect
11. #70337C2semidirect
14. #69873C2semidirect
15. C4×#824C2semidirect
610. C3#23720semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#1372
C4×#9512
#825142
C2×#50302
#721092
#721052
C4×#8362
#706792
#706712
#703402
#703372
#703102
#703022
#698732
C4×#8242
#237203