Finite Group Databasebeta

← #10813648

Subgroups — #10813648

Order 768.

870 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82512C211yesmaximal, characteristic, complemented
2C4×#950C211yesmaximal, characteristic
3#79049C211yesmaximal, characteristic, complemented
4C2×#4824C212yesmaximal
5C3×#1806C211yesmaximal, characteristic, complemented
6#70518C211yesmaximal, characteristic, complemented
7#70376C212yesmaximal, complemented
8#69494C211yesmaximal, characteristic, complemented
9C4⋊C4×C3⋊C8C211yesmaximal, characteristic, complemented
10#68774C212yesmaximal, complemented
11#68747C212yesmaximal, complemented
12#22688—31nomaximal, Sylow
13C3×C4×#41C2211yescharacteristic
14C2×#950C2211yescharacteristic
15C2×#948C2212yes—
16C2×#907C2212yes—
17C4×#235C2211yescharacteristic
18C3×#423C2211yescharacteristic
19C2×C3×#104C2212yes—
20C3×C4×C4⋊C4C2211yescharacteristic
21#5233C2211yescharacteristic
22C2×#822C2212yes—
23C2×#822C2212yes—
24#5119C2212yes—
25C4×#216C411yescharacteristic, complemented
26C4×#216C411yescharacteristic, complemented
27C4×#215C411yescharacteristic, complemented
28C4×#215C411yescharacteristic, complemented
29C4×#214C411yescharacteristic, complemented
30C4×#214C411yescharacteristic, complemented
31C4×#213C411yescharacteristic, complemented
32C4×#213C411yescharacteristic, complemented
33#5002C2211yescharacteristic
34#4999—22no—
35#4999C2212yes—
36#4998C2211yescharacteristic
37C2×#805C2212yes—
38#4924—21no—
39#4924C2211yescharacteristic
40C2×C4⋊(C3⋊C8)—21no—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×C4×C3⋊C8C2212yes—
43C2×C4×C3⋊C8C2211yescharacteristic
44#4839—22no—
45#4837—21no—
46D3×C4⋊C8—21no—
47#4824—28no—
48C3×#339C2212yes—
49#4524C2212yes—
50#4523C2212yes—
51C2×#443—32no—
52C4×#135—31no—
53#2906—31no—
54#2906—31no—
55#1945—32no—
56#1866—32no—
57#1866—32no—
58#1848—31no—
59#1806—31no—
60#1806D311yescharacteristic, complemented
61C8×C4⋊C4—31no—
62C2×C3×#41C2311yescharacteristic
63C2×#235C2311yescharacteristic
64C2×C3×M16C2312yes—
65C3×C4×Q8C2×C411yescharacteristic
66C3×C4×Q8C2×C411yescharacteristic
67C3×C4×D4—21no—
68C3×C4×D4—21no—
69C3×C4×D4C2×C411yescharacteristic
70C3×C4×D4C2×C411yescharacteristic
71C3×#117—21no—
72C3×#117C2311yescharacteristic
73C2×C3×C4⋊C4—21no—
74C2×C3×C42—21no—
75C2×C3×C42C2312yes—
76C2×C3×C42C2311yescharacteristic
77#950D414yes—
78#950Q814yes—
79#948—24no—
80#948—42no—
81C2×#216C2×C411yescharacteristic
82C2×#215C2×C411yescharacteristic
83C2×#214C2×C411yescharacteristic
84C2×#213C2×C411yescharacteristic
85C4×#67—21no—
86C4×#67—21no—
87#926C2311yescharacteristic
88C2×#208—22no—
89C2×#208C2312yes—
90C22×C3⋊C8—22no—
91C22×C3⋊C8C2311yescharacteristic
92C22×C3⋊C8C2312yes—
93C22×C3⋊C8C2311yescharacteristic
94#907—24no—
95#907—42no—
96C2×ΓL(1,𝔽25)—22no—
97C2×D3×C8—22no—
98C4×D12C2×C411yescharacteristic
99C4×D12C2×C411yescharacteristic
100#872—21no—
101C42×D3—21no—
102C4×Dic6C2×C411yescharacteristic
103C4×Dic6C2×C411yescharacteristic
104C3×C4⋊C8—21no—
105C3×C4⋊C8—21no—
106C3×#104—24no—
107C3×#95—21no—
108#835C2×C411yescharacteristic
109#835C2×C411yescharacteristic
110#832C2×C411yescharacteristic
111#832C2×C411yescharacteristic
112#822—24no—
113#822—42no—
114Dic3⋊C8—22no—
115Dic3⋊C8—22no—
116#810C2×C411yescharacteristic
117#810C2×C411yescharacteristic
118#809C2×C411yescharacteristic
119#809C2×C411yescharacteristic
120#808C2×C411yescharacteristic
121#808C2×C411yescharacteristic
122#807C2×C411yescharacteristic
123#807C2×C411yescharacteristic
124#805—24no—
125C4⋊(C3⋊C8)—21no—
126C4⋊(C3⋊C8)—21no—
127C4⋊(C3⋊C8)—22no—
128C4⋊(C3⋊C8)—22no—
129C4⋊(C3⋊C8)D411yescharacteristic
130C4⋊(C3⋊C8)D411yescharacteristic
131C4⋊(C3⋊C8)D412yes—
132C4×C3⋊C8—24no—
133C4×C3⋊C8C2×C411yescharacteristic
134C4×C3⋊C8C2×C411yescharacteristic
135C4×C3⋊C8C2×C411yescharacteristic
136C4×C3⋊C8C2×C411yescharacteristic
137C2×#135—31no—
138C2×#131—32no—
139C2×#131—32no—
140C4×#41—31no—
141C4×#41C2×D311yescharacteristic
142#443—68no—
143C4×Q16—31no—
144C4×Q16—31no—
145C4×SD16—31no—
146C4×SD16—31no—
147C4×SD16—31no—
148C4×SD16—31no—
149C4×D8—31no—
150C4×D8—31no—
151C2×#108—32no—
152C2×#108—32no—
153#427—31no—
154#424—32no—
155#424—32no—
156#424—62no—
157#424—62no—
158#423—31no—
159#423—61no—
160#423—61no—
161#423C2×D311yescharacteristic
162C2×C4⋊C8—31no—
163C2×C4⋊C8—61no—
164C2×C4⋊C8—61no—
165C2×#104—32no—
166C2×#104C2×D312yes—
167#416—31no—
168#416—31no—
169C2×C4×C8—31no—
170C2×C4×C8—32no—
171C4×C4⋊C4C2×D311yescharacteristic
172#339—32no—
173#339C2×D312yes—
174#336—32no—
175C3×#41—21no—
176C3×#41—21no—
177C3×#41C2×D412yes—
178C3×#41C2×Q812yes—
179C2×C3×Q8C22×C411yescharacteristic
180C2×C3×D4—21no—
181C2×C3×D4C22×C411yescharacteristic
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4#4111yescharacteristic
187C22×C3×C4#4111yescharacteristic
188C22×C3×C4C2411yescharacteristic
189C2×#67—21no—
190#235—21no—
191#235—21no—
192#235C2×D412yes—
193#235C2×Q812yes—
194C2×D12C22×C411yescharacteristic
195C2×C4×D3—21no—
196C2×C4×D3—21no—
197C2×C4×D3—21no—
198C2×Dic6C22×C411yescharacteristic
199C3×M16—22no—
200C3×M16—22no—
201C2×C3×C8—22no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4C22×C411yescharacteristic
206C3×C4⋊C4C22×C411yescharacteristic
207C3×C22⋊C4—21no—
208C3×C22⋊C4—21no—
209C3×C22⋊C4—21no—
210C3×C42—21no—
211C3×C42—21no—
212C3×C42—21no—
213C3×C42—22no—
214C3×C42—22no—
215C3×C42C22×C411yescharacteristic
216C3×C42C22×C411yescharacteristic
217C3×C42C2×D411yescharacteristic
218C3×C42C2×D411yescharacteristic
219#217—21no—
220#216C4⋊C414yescomplemented
221#215C4⋊C414yescomplemented
222#214C4⋊C414yescomplemented
223#213C4⋊C414yescomplemented
224(C2×D3)⋊C4—21no—
225(C2×D3)⋊C4—21no—
226C4⋊Dic3C22×C411yescharacteristic
227C4⋊Dic3C22×C411yescharacteristic
228Dic3⋊C4—21no—
229Dic3⋊C4—21no—
230C4×Dic3—21no—
231C4×Dic3—21no—
232C4×Dic3—21no—
233#208—22no—
234#208—22no—
235#208—44no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—21no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—21no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—44no—
242C2×C3⋊C8C22×C411yescharacteristic
243C2×C3⋊C8C22×C411yescharacteristic
244C2×C3⋊C8C22×C411yescharacteristic
245C2×C3⋊C8C22×C411yescharacteristic
246C2×C3⋊C8C2×D412yes—
247C2×C3⋊C8C2×D412yes—
248C2×C3⋊C8C2×D412yes—
249C2×C3⋊C8C2×Q812yes—
250ΓL(1,𝔽25)—44no—
251D3×C8—44no—
252C2×#41—31no—
253C2×#41C22×D311yescharacteristic
254#135—34no—
255#135—34no—
256C2×Q16—31no—
257C2×SD16—31no—
258C2×SD16—31no—
259C2×D8—31no—
260#131—64no—
261#131—64no—
262#131—122no—
263#131—122no—
264C2×M16—32no—
265C2×M16—62no—
266C2×M16—62no—
267C2×M16C22×D312yes—
268C22×C8—31no—
269C22×C8—32no—
270C22×C8—31no—
271C22×C8—62no—
272C22×C8—62no—
273C4×Q8—31no—
274C4×Q8—31no—
275C4×Q8C4×D311yescharacteristic
276C4×Q8C4×D311yescharacteristic
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—31no—
280C4×D4—31no—
281C4×D4—61no—
282C4×D4—61no—
283C4×D4C4×D311yescharacteristic
284C4×D4C4×D311yescharacteristic
285#117—21no—
286#117—31no—
287#117—61no—
288#117C22×D311yescharacteristic
289C2×C4⋊C4—21no—
290C2×C42—21no—
291C2×C42—61no—
292C2×C42C22×D312yes—
293C2×C42C22×D311yescharacteristic
294#108—64no—
295#108—122no—
296#107—31no—
297#107—31no—
298#106—31no—
299#106—31no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8—31no—
303C4⋊C8—31no—
304C4⋊C8—32no—
305C4⋊C8—62no—
306C4⋊C8—62no—
307C4⋊C8—62no—
308C4⋊C8—62no—
309C4⋊C8—61no—
310C4⋊C8—61no—
311#104—24no—
312#104—64no—
313#103—31no—
314#103—31no—
315#103—31no—
316#103—31no—
317#102—31no—
318#102—31no—
319#102—31no—
320#102—31no—
321C4×C8—31no—
322C4×C8—31no—
323C4×C8—31no—
324C4×C8—31no—
325C4×C8—64no—
326#95—21no—
327C23×C3—21no—
328C23×C3C2×#4111yescharacteristic
329C22×D3—21no—
330C3×Q8—21no—
331C3×Q8C2×C4⋊C412yes—
332C3×D4—22no—
333C3×D4—22no—
334C3×D4—21no—
335C3×D4C2×C4⋊C412yes—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—24no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—41no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—81no—
348C2×C3×C4C4×D411yescharacteristic
349C2×C3×C4C4×D411yescharacteristic
350C2×C3×C4C4×D411yescharacteristic
351C2×C3×C4C4×D411yescharacteristic
352C2×C3×C4C23×C411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C22×Q811yescharacteristic
356C2×C3×C4H(5,𝔽2)11yescharacteristic
357C2×C3×C42−1+411yescharacteristic
358#67—22no—
359#67—22no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363D12—21no—
364D12C2×C4⋊C412yes—
365C4×D3—22no—
366C4×D3—22no—
367C4×D3—42no—
368C4×D3—81no—
369Dic6—21no—
370Dic6C2×C4⋊C412yes—
371C3×C8—42no—
372C3⋊C8—21no—
373C3⋊C8—21no—
374C3⋊C8—22no—
375C3⋊C8—22no—
376C3⋊C8—42no—
377C3⋊C8C2×C4⋊C412yes—
378C3⋊C8C2×C4⋊C412yes—
379#41—21no—
380#41—21no—
381#41—32no—
382#41—32no—
383#41—61no—
384#41—61no—
385#41D3×D412yes—
386#41D3×Q812yes—
387C2×Q8—31no—
388C2×Q8C2×C4×D311yescharacteristic
389C2×D4—21no—
390C2×D4—31no—
391C2×D4—61no—
392C2×D4C2×C4×D311yescharacteristic
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4#23711yescharacteristic
401C22×C4#23711yescharacteristic
402C22×C4C23×D311yescharacteristic
403Q16—34no—
404SD16—34no—
405SD16—34no—
406D8—34no—
407M16—22no—
408M16—22no—
409M16—62no—
410M16—62no—
411M16—124no—
412M16—124no—
413C2×C8—22no—
414C2×C8—31no—
415C2×C8—31no—
416C2×C8—32no—
417C2×C8—32no—
418C2×C8—31no—
419C2×C8—31no—
420C2×C8—32no—
421C2×C8—32no—
422C2×C8—62no—
423C2×C8—61no—
424C2×C8—62no—
425C2×C8—61no—
426C2×C8—62no—
427C2×C8—124no—
428C2×C8—124no—
429C4⋊C4—24no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—31no—
433C4⋊C4—31no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4C2×C4×D311yescharacteristic
437C4⋊C4C2×C4×D311yescharacteristic
438C22⋊C4—21no—
439C22⋊C4—21no—
440C22⋊C4—21no—
441C22⋊C4—61no—
442C22⋊C4—61no—
443C22⋊C4—61no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—22no—
448C42—22no—
449C42—61no—
450C42—61no—
451C42—61no—
452C42C2×C4×D311yescharacteristic
453C42C2×C4×D311yescharacteristic
454C42D3×D411yescharacteristic
455C42D3×D411yescharacteristic
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—22no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3C2×C4×D411yescharacteristic
462C22×C3#54711yescharacteristic
463C22×C3#55411yescharacteristic
464C2×D3—22no—
465C2×D3—41no—
466C2×D3—41no—
467C3×C4—22no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—44no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4#44312yescomplemented
475C3×C4C22×C4⋊C411yescharacteristic
476C3×C4#52011yescharacteristic
477C3×C4D4211yescharacteristic
478C3×C4D4×Q811yescharacteristic
479Dic3—22no—
480Dic3—41no—
481Dic3—41no—
482C23—21no—
483C23—61no—
484C23C2×#23711yescharacteristic
485Q8—21no—
486Q8—32no—
487Q8—61no—
488Q8D3×C4⋊C412yes—
489D4—22no—
490D4—22no—
491D4—21no—
492D4—32no—
493D4—62no—
494D4—62no—
495D4—61no—
496D4D3×C4⋊C412yes—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—61no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—81no—
514C2×C4—122no—
515C2×C4—241no—
516C2×C4#67⋊C411yescharacteristic
517C2×C4#67⋊C411yescharacteristic
518C2×C4#67⋊C411yescharacteristic
519C2×C4#67⋊C411yescharacteristic
520C2×C4C22×C4×D311yescharacteristic
521C2×C4C2×D3×D411yescharacteristic
522C2×C4C2×D3×D411yescharacteristic
523C2×C4C2×D3×Q811yescharacteristic
524C2×C4#100911yescharacteristic
525C2×C4#101011yescharacteristic
526C8—32no—
527C8—32no—
528C8—42no—
529C8—61no—
530C8—61no—
531C8—62no—
532C8—62no—
533C8—122no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—42no—
537C2×C3D4×C4⋊C411yescharacteristic
538C2×C3C2×#44311yescharacteristic
539C2×C3#295711yescharacteristic
540D3—42no—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—41no—
545C22—41no—
546C22—62no—
547C22—121no—
548C22—121no—
549C22C2×#67⋊C411yescharacteristic
550C22#561111yescharacteristic
551C22#563511yescharacteristic
552C4—22no—
553C4—21no—
554C4—21no—
555C4—21no—
556C4—44no—
557C4—41no—
558C4—41no—
559C4—62no—
560C4—121no—
561C4—121no—
562C4#482412yescomplemented
563C4C2×D3×C4⋊C411yescharacteristic
564C4#552311yescharacteristic
565C4#560911yescharacteristic
566C4#563211yescharacteristic
567C3#2268811yescharacteristic, complemented, Sylow
568C2—21no—
569C2—21no—
570C2—42no—
571C2—122no—
572C2#7401011yescharacteristic
573C2C2×#482411yescharacteristic
574C2#7899011yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1806C2
socleC22×C3C2×C4×D4
prime radicalC2×#950C22
cosocleC22×C3×C4C24
hypercenter#1806D3
hypocenterC3#22688
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
1. #82512C2semidirect
3. #79049C2semidirect
5. C3×#1806C2semidirect
6. #70518C2semidirect
7. #70376C2semidirect
8. #69494C2semidirect
9. C4⋊C4×C3⋊C8C2semidirect
10. #68774C2semidirect
11. #68747C2semidirect
25. C4×#216C4semidirect
26. C4×#216C4semidirect
27. C4×#215C4semidirect
28. C4×#215C4semidirect
29. C4×#214C4semidirect
30. C4×#214C4semidirect
31. C4×#213C4semidirect
32. C4×#213C4semidirect
60. #1806D3semidirect
220. #216C4⋊C4semidirect
221. #215C4⋊C4semidirect
222. #214C4⋊C4semidirect
223. #213C4⋊C4semidirect
474. C3×C4#443semidirect
562. C4#4824semidirect
567. C3#22688semidirect

Maximal subgroups

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

subgroupindex
#825122
C4×#9502
#790492
C2×#48242
C3×#18062
#705182
#703762
#694942
C4⋊C4×C3⋊C82
#687742
#687472
#226883