Finite Group Databasebeta

← #10824915

Subgroups — #10824915

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
1C4×#942C211yesmaximal, characteristic
2C2×#5030C211yesmaximal, characteristic, complemented
3C4×#909C211yesmaximal, characteristic
4#78520C211yesmaximal, characteristic
5C3×C4×#101C211yesmaximal, characteristic, complemented
6#70126C211yesmaximal, characteristic
7#70125C211yesmaximal, characteristic
8#70120C211yesmaximal, characteristic, complemented
9#70109C211yesmaximal, characteristic, complemented
10#69873C211yesmaximal, characteristic, complemented
11C4×#824C211yesmaximal, characteristic, complemented
12#69011C211yesmaximal, characteristic
13#68999C211yesmaximal, characteristic, complemented
14#68987C211yesmaximal, characteristic
15#68981C211yesmaximal, characteristic, complemented
16#23720—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C22×#216C2211yescharacteristic
19C2×#942C2211yescharacteristic
20C2×#909C2211yescharacteristic
21C4×#235C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#101C2211yescharacteristic
24C2×C3×#101C2211yescharacteristic
25C3×C4×M16C2211yescharacteristic
26C3×C4×M16C2211yescharacteristic
27#5225C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#824C2211yescharacteristic
30C2×#824C2211yescharacteristic
31#5133C2211yescharacteristic
32#5129C2211yescharacteristic
33#5129C2211yescharacteristic
34C2×#822C2211yescharacteristic
35C2×#822C2211yescharacteristic
36#5121C2211yescharacteristic
37#5120C2211yescharacteristic
38#5119C2211yescharacteristic
39#5119C2211yescharacteristic
40#5030—21no—
41#5030—22no—
42#5030—22no—
43C4×#216—21no—
44#5026—21no—
45C4×#215—21no—
46C2×#810C2211yescharacteristic
47#4966C2211yescharacteristic
48#4964C2211yescharacteristic
49C2×#808C2211yescharacteristic
50C4×#208C2211yescharacteristic
51C2×#803C2211yescharacteristic
52#4716—21no—
53#4710—21no—
54#4705—22no—
55C4×Dic12—21no—
56C4×#204—21no—
57#4690—22no—
58C3×#343C2211yescharacteristic
59C3×#343C2211yescharacteristic
60#4558C2211yescharacteristic
61#4558C2211yescharacteristic
62#4554C2211yescharacteristic
63#4554C2211yescharacteristic
64C4×#137—31no—
65C4×#137—31no—
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×#101D311yescharacteristic, complemented
79C22×C3×Q8C2311yescharacteristic
80C2×#235C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C2×C3×M16C2311yescharacteristic
83C2×C3×M16C2311yescharacteristic
84C2×C3×M16C2311yescharacteristic
85C2×C3×M16C2311yescharacteristic
86C3×C4×Q8—21no—
87C3×C4×Q8—21no—
88C3×C4×Q8—21no—
89C3×C4×Q8—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#216—22no—
95C2×#216—21no—
96C2×#216—22no—
97C2×#216—22no—
98#942—22no—
99#942—22no—
100C2×#215—21no—
101C4×#67—21no—
102C4×#67—21no—
103#926C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C2×#208C2311yescharacteristic
108C2×#208C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111#909—22no—
112#909—22no—
113C2×Dic12—21no—
114C2×#204—21no—
115C4×D12D411yescharacteristic
116C4×D12D411yescharacteristic
117#872—21no—
118C42×D3—21no—
119C4×Dic6—21no—
120C4×Dic6—21no—
121C4×Dic6—21no—
122C4×Dic6—21no—
123C4×Dic6D411yescharacteristic
124C4×Dic6D411yescharacteristic
125C3×#101—22no—
126C3×#101—22no—
127C3×#97—21no—
128C3×#97—21no—
129C3×C4×C8—21no—
130C3×C4×C8—21no—
131#835—21no—
132#835—21no—
133#835—22no—
134#824—22no—
135#824—22no—
136#822—22no—
137#822—22no—
138#821—21no—
139(C2×D3)⋊C8—22no—
140(C2×D3)⋊C8—22no—
141#818—21no—
142#817—21no—
143#816—21no—
144#816—21no—
145#816—21no—
146Dic3⋊C8—22no—
147Dic3⋊C8—22no—
148#810—21no—
149#810—22no—
150#809—21no—
151#808—21no—
152#808—22no—
153#807—21no—
154#803—21no—
155#803D412yes—
156#803D412yes—
157C4×C3⋊C8—21no—
158C2×#137—31no—
159C2×#137—31no—
160C22×Q16—31no—
161C4×#41—31no—
162C2×C4×Q8—31no—
163C2×C4×Q8C2×D311yescharacteristic
164#441—61no—
165#441—62no—
166#441—62no—
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×#108—31no—
178#428—31no—
179#428—31no—
180C2×#106—31no—
181#424—31no—
182#424—31no—
183#419—31no—
184#419—31no—
185#417—31no—
186#417—31no—
187C2×#103—31no—
188C2×#103—31no—
189C2×#101—31no—
190C2×#101—31no—
191C2×#101C2×D311yescharacteristic
192C2×#101C2×D311yescharacteristic
193#408—31no—
194#408—31no—
195C4×M16—31no—
196C4×M16C2×D311yescharacteristic
197C4×M16C2×D311yescharacteristic
198C2×#97—31no—
199#343—31no—
200#343—31no—
201#343C2×D311yescharacteristic
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×C4#4111yescharacteristic
214C22×C3×C4#4111yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C2×#67—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
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—21no—
240C3×M16—21no—
241C3×M16#4112yes—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—41no—
249C3×C4⋊C4—41no—
250C3×C42—41no—
251C3×C42—41no—
252C3×C42C2×D411yescharacteristic
253C3×C42C2×D411yescharacteristic
254C3×C42C2×D411yescharacteristic
255C3×C42C2×D411yescharacteristic
256#217—21no—
257#216—24no—
258#216—28no—
259#215—24no—
260(C2×D3)⋊C4—21no—
261(C2×D3)⋊C4—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3C2×D411yescharacteristic
265C4⋊Dic3C2×D411yescharacteristic
266Dic3⋊C4—21no—
267Dic3⋊C4—21no—
268Dic3⋊C4—41no—
269Dic3⋊C4—41no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272C4×Dic3—21no—
273C4×Dic3—41no—
274C4×Dic3—41no—
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⋊C8C2×D412yes—
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×M16C22×D311yescharacteristic
306C2×M16C22×D311yescharacteristic
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—31no—
316C4×Q8—31no—
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—31no—
326#117—61no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4C22×D311yescharacteristic
331C2×C42—21no—
332C2×C42—61no—
333C2×C42—61no—
334C2×C42C22×D311yescharacteristic
335#108—62no—
336#108—62no—
337#107—61no—
338#107—61no—
339#106—61no—
340#106—62no—
341#106—61no—
342C4⋊C8—62no—
343C4⋊C8—62no—
344#103—62no—
345#103—61no—
346#103—61no—
347#103—61no—
348#103—61no—
349#103—61no—
350#103—61no—
351#103—62no—
352#102—61no—
353#102—61no—
354#101—22no—
355#101—22no—
356#101—62no—
357#101—62no—
358C22⋊C8—62no—
359C22⋊C8—62no—
360#97—21no—
361#97—21no—
362#97—32no—
363#97—32no—
364#97—61no—
365C4×C8—21no—
366C4×C8—21no—
367C4×C8—61no—
368C23×C3C2×#4111yescharacteristic
369C22×D3—21no—
370C3×Q8—22no—
371C3×Q8—22no—
372C3×Q8—41no—
373C3×Q8—41no—
374C2×C3×C4—22no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—81no—
381C2×C3×C4#12111yescharacteristic
382C2×C3×C4#12111yescharacteristic
383C2×C3×C4#12111yescharacteristic
384C2×C3×C4#12111yescharacteristic
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391#67—41no—
392#67—41no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—22no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—81no—
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—41no—
415C3×C8—41no—
416C3×C8—41no—
417C3⋊C8—22no—
418C3⋊C8—22no—
419C3⋊C8—22no—
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#23512yes—
438C2×D4—31no—
439C2×D4—61no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4#23511yescharacteristic
448C22×C4#23511yescharacteristic
449C22×C4C23×D311yescharacteristic
450Q16—64no—
451Q16—68no—
452Q16—64no—
453SD16—64no—
454SD16—64no—
455M16—21no—
456M16—21no—
457M16—21no—
458M16—21no—
459M16—21no—
460M16—21no—
461M16—21no—
462M16—32no—
463M16—61no—
464M16—61no—
465M16—61no—
466M16#23512yes—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—32no—
472C2×C8—32no—
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—31no—
482C4⋊C4—31no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—121no—
490C4⋊C4—121no—
491C22⋊C4—61no—
492C22⋊C4—61no—
493C22⋊C4—61no—
494C42—41no—
495C42—41no—
496C42—61no—
497C42—61no—
498C42—61no—
499C42—121no—
500C42—121no—
501C42D3×D411yescharacteristic
502C42D3×D411yescharacteristic
503C42D3×D411yescharacteristic
504C42D3×D411yescharacteristic
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—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—42no—
518C3×C4—81no—
519C3×C4#52511yescharacteristic
520C3×C4#52511yescharacteristic
521C3×C4#52511yescharacteristic
522C3×C4#53211yescharacteristic
523Dic3—42no—
524Dic3—41no—
525Dic3—41no—
526Dic3—41no—
527Dic3—81no—
528C23—61no—
529C23C2×#23511yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—41no—
533Q8—41no—
534Q8—62no—
535Q8—62no—
536Q8—61no—
537Q8—61no—
538Q8—121no—
539Q8—121no—
540D4—61no—
541D4—61no—
542D4—121no—
543D4—121no—
544C2×C4—22no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—62no—
554C2×C4—81no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—241no—
561C2×C4—241no—
562C2×C4—241no—
563C2×C4#88411yescharacteristic
564C2×C4#88411yescharacteristic
565C2×C4#88411yescharacteristic
566C2×C4#88411yescharacteristic
567C2×C4C2×#23511yescharacteristic
568C2×C4C2×#23511yescharacteristic
569C2×C4C2×D3×D411yescharacteristic
570C2×C4C2×D3×D411yescharacteristic
571C2×C4D3×#4111yescharacteristic
572C2×C4D3×#4111yescharacteristic
573C8—22no—
574C8—41no—
575C8—41no—
576C8—41no—
577C8—62no—
578C8—62no—
579C8—62no—
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—
592C22#547911yescharacteristic
593C22C2×#88411yescharacteristic
594C22#552811yescharacteristic
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—42no—
600C4—81no—
601C4—121no—
602C4—122no—
603C4—121no—
604C4—121no—
605C4—241no—
606C4#554711yescharacteristic
607C4#554711yescharacteristic
608C4#554711yescharacteristic
609C4#554811yescharacteristic
610C3#2372011yescharacteristic, complemented, Sylow
611C2—21no—
612C2—21no—
613C2—241no—
614C2#7358111yescharacteristic
615C2#7871211yescharacteristic
616C2#7872111yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#101C2
socleC22×C3C2×#121
prime radicalC2×#235C23
hypercenterC4×#101D3
hypocenterC3#23720
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
2. C2×#5030C2semidirect
5. C3×C4×#101C2semidirect
8. #70120C2semidirect
9. #70109C2semidirect
10. #69873C2semidirect
11. C4×#824C2semidirect
13. #68999C2semidirect
15. #68981C2semidirect
78. C4×#101D3semidirect
610. C3#23720semidirect

Maximal subgroups

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

subgroupindex
C4×#9422
C2×#50302
C4×#9092
#785202
C3×C4×#1012
#701262
#701252
#701202
#701092
#698732
C4×#8242
#690112
#689992
#689872
#689812
#237203