Finite Group Databasebeta

← #11660210

Subgroups — #11660210

Order 768.

1702 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#235C211yesmaximal, characteristic
2C3×#2984C211yesmaximal, characteristic, complemented
3#82435C211yesmaximal, characteristic, complemented
4#79967C213yesmaximal, complemented
5#79965C213yesmaximal, complemented
6#79960C213yesmaximal, complemented
7#79958C213yesmaximal, complemented
8#79921C211yesmaximal, characteristic, complemented
9#79920C211yesmaximal, characteristic, complemented
10Q8×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
11D3×C8×Q8C211yesmaximal, characteristic, complemented
12#79334C213yesmaximal
13#79316C213yesmaximal, complemented
14#79283C213yesmaximal
15#79267C213yesmaximal, complemented
16#60253—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18Q8×#67C411yescharacteristic
19Q8×#67C411yescharacteristic
20#5579C413yes—
21#5578C2213yes—
22#5578C2213yes—
23Q8×D12C411yescharacteristic
24C4×D3×Q8C2211yescharacteristic
25C4×D3×Q8C2211yescharacteristic
26#5567C413yes—
27Q8×Dic6C411yescharacteristic
28#5511C413yes—
29#5511C413yes—
30C4×#235C2213yes—
31#5470C2213yes—
32C3×#446C2213yes—
33C3×#446C2213yes—
34C3×C8×Q8C2211yescharacteristic
35C3×C8×Q8C2211yescharacteristic
36C3×#432C2213yes—
37C3×#431C2213yes—
38Q8×C3⋊C8C2211yescharacteristic
39Q8×C3⋊C8C2211yescharacteristic
40Q8×C3⋊C8C2213yes—
41Q8×C3⋊C8C2213yes—
42#4925C2213yes—
43#4925C2213yes—
44#4840C2213yes—
45#4840C2213yes—
46#4839C2213yes—
47#4839C2213yes—
48#4837C2213yes—
49#4835C2213yes—
50#4833C2213yes—
51D3×C4⋊C8C2213yes—
52#4831C2213yes—
53#4831C2213yes—
54#4831C2216yes—
55#4830C2213yes—
56#4830C2213yes—
57#4830C2216yes—
58#4727C2211yescharacteristic
59#4727C2213yes—
60#4720C2213yes—
61#4720C2216yes—
62#4720C2213yes—
63#4709C2216yes—
64#4708C2216yes—
65#4701C2216yes—
66#4701C2216yes—
67C4×ΓL(1,𝔽25)C2213yes—
68#4685C2213yes—
69#4685C2213yes—
70C4×D3×C8C2213yes—
71#4679C2213yes—
72#4679C2213yes—
73C8×Dic6C2213yes—
74C8×Dic6C2213yes—
75Q8×#41—31no—
76#2986—33no—
77#2986—33no—
78#2985—33no—
79#2985—33no—
80#2984—31no—
81#2984D311yescharacteristic, complemented
82#2982—31no—
83#2982—31no—
84Q8×M16—31no—
85#2962—33no—
86#2960—33no—
87C2×C8×Q8—31no—
88#2925—33no—
89#2922—33no—
90C22×C3×Q8C2×C411yescharacteristic
91C2×D3×Q8C2×C411yescharacteristic
92C2×D3×Q8C2×C411yescharacteristic
93C2×#235C2311yescharacteristic
94C3×C4×Q8C2×C411yescharacteristic
95C3×C4×Q8C2×C413yes—
96C3×C4×Q8C2311yescharacteristic
97C3×C4×Q8C2313yes—
98C2×C3×C4⋊C4C2×C413yes—
99C2×C3×C42C2313yes—
100#946C2×C413yes—
101#946C2×C413yes—
102Q8×Dic3C2×C411yescharacteristic
103Q8×Dic3C2×C411yescharacteristic
104#944C2×C413yes—
105#944C2×C413yes—
106C4×#67C2×C413yes—
107C4×#67C2×C413yes—
108C4×#67C2×C413yes—
109C4×#67C2×C413yes—
110#926C2313yes—
111#924C2×C416yes—
112#924C2×C413yes—
113#924C2×C413yes—
114C2×ΓL(1,𝔽25)C2311yescharacteristic
115C2×ΓL(1,𝔽25)C2313yes—
116C2×D3×C8C2311yescharacteristic
117C2×D3×C8C2313yes—
118#897C2×C413yes—
119#897C2×C413yes—
120#896C2×C416yes—
121#896C2×C416yes—
122D3×C4⋊C4C2×C413yes—
123D3×C4⋊C4C2×C413yes—
124#889C2×C416yes—
125#889C2×C416yes—
126#888C2×C413yes—
127#888C2×C413yes—
128#887C2×C413yes—
129#887C2×C413yes—
130C4×D12C2×C413yes—
131C4×D12C2×C413yes—
132#872C2313yes—
133#872C2313yes—
134C42×D3C2313yes—
135C42×D3C2313yes—
136#870C2×C416yes—
137#869C2×C413yes—
138#869C2×C413yes—
139C4×Dic6C2×C413yes—
140C4×Dic6C2×C413yes—
141C4×Dic6C2313yes—
142C4×Dic6C2313yes—
143C4×Dic6C2316yes—
144C3×C4⋊C8C2313yes—
145C3×C4⋊C8C2313yes—
146C3×C4⋊C8C2313yes—
147C3×C4⋊C8C2313yes—
148C3×C22⋊C8C2311yescharacteristic
149C3×C22⋊C8C2313yes—
150C3×#97C2316yes—
151C3×C4×C8C2313yes—
152C3×C4×C8C2313yes—
153#830C2313yes—
154#830C2311yescharacteristic
155(C2×D3)⋊C8C2311yescharacteristic
156(C2×D3)⋊C8C2311yescharacteristic
157(C2×D3)⋊C8C2313yes—
158(C2×D3)⋊C8C2313yes—
159#815C2313yes—
160#815C2316yes—
161#815C2313yes—
162Dic3⋊C8C2313yes—
163Dic3⋊C8C2316yes—
164Dic3⋊C8C2313yes—
165Dic3⋊C8C2316yes—
166Dic3⋊C8C2313yes—
167Dic3⋊C8C2313yes—
168C8×Dic3C2313yes—
169C8×Dic3C2316yes—
170C8×Dic3C2313yes—
171C4⋊(C3⋊C8)C2313yes—
172C4⋊(C3⋊C8)C2313yes—
173C4⋊(C3⋊C8)C2313yes—
174C4⋊(C3⋊C8)C2313yes—
175C4×C3⋊C8C2313yes—
176C4×C3⋊C8C2313yes—
177C4×C3⋊C8C2316yes—
178Q82—31no—
179#557—33no—
180#554—33no—
181#554—33no—
182#554—33no—
183D4×Q8—31no—
184D4×Q8—31no—
185D4×Q8—31no—
186#533—33no—
187#533—33no—
188#533—33no—
189C4×#41—33no—
190C2×C4×Q8—31no—
191C2×C4×Q8—31no—
192C2×C4×Q8C2×D311yescharacteristic
193#446—33no—
194#446—36no—
195#446—33no—
196#446—36no—
197#446—36no—
198#446—33no—
199#446—33no—
200#446C2×D313yes—
201#446C2×D313yes—
202C8×Q8—31no—
203C8×Q8—33no—
204C8×Q8—33no—
205C8×Q8—31no—
206C8×Q8—33no—
207C8×Q8—33no—
208C8×Q8—36no—
209C8×Q8—33no—
210C8×Q8—33no—
211C8×Q8C2×D311yescharacteristic
212C8×Q8C2×D311yescharacteristic
213#433—33no—
214#433—33no—
215#433—33no—
216#433—33no—
217#433—36no—
218#433—36no—
219#432—33no—
220#432C2×D313yes—
221#431—33no—
222#431—33no—
223#431—33no—
224#431—33no—
225#431—33no—
226#431C2×D313yes—
227#424—33no—
228#424—33no—
229#423—33no—
230C2×C4⋊C8—33no—
231#408—31no—
232#408—33no—
233#405—36no—
234C4×M16—33no—
235C2×C4×C8—33no—
236C2×C3×Q8—24no—
237C2×C3×Q8C22×C413yes—
238C2×C3×Q8C22×C411yescharacteristic
239C22×C3×C4C22×C413yes—
240C22×C3×C4C22×C413yes—
241C22×C3×C4C2411yescharacteristic
242C2×#67C22×C411yescharacteristic
243C2×#67C22×C411yescharacteristic
244D3×Q8—24no—
245D3×Q8—24no—
246#235—22no—
247#235—22no—
248C2×D12C22×C411yescharacteristic
249C2×C4×D3C22×C413yes—
250C2×C4×D3C22×C413yes—
251C2×C4×D3C22×C413yes—
252C2×C4×D3C22×C413yes—
253C2×C4×D3C2411yescharacteristic
254C2×C4×D3C2411yescharacteristic
255C2×Dic6C22×C411yescharacteristic
256C2×Dic6C22×C413yes—
257C2×Dic6C22×C413yes—
258C2×Dic6C22×C416yes—
259C2×C3×C8C2411yescharacteristic
260C2×C3×C8C2413yes—
261C2×C3×C8C2413yes—
262C2×C3×C8C2411yescharacteristic
263C3×C4⋊C4C22×C413yes—
264C3×C4⋊C4C22×C413yes—
265C3×C4⋊C4C22×C416yes—
266C3×C42C22×C413yes—
267C3×C42C22×C413yes—
268C3×C42C2413yes—
269C3×C42C2413yes—
270#217C22×C413yes—
271#217C22×C413yes—
272(C2×D3)⋊C4C22×C413yes—
273(C2×D3)⋊C4C22×C413yes—
274(C2×D3)⋊C4C22×C413yes—
275(C2×D3)⋊C4C22×C413yes—
276C4⋊Dic3C22×C413yes—
277C4⋊Dic3C22×C413yes—
278C4⋊Dic3C22×C413yes—
279C4⋊Dic3C22×C413yes—
280C4⋊Dic3C22×C416yes—
281Dic3⋊C4C22×C413yes—
282Dic3⋊C4C22×C413yes—
283Dic3⋊C4C22×C413yes—
284Dic3⋊C4C22×C413yes—
285Dic3⋊C4C22×C416yes—
286Dic3⋊C4C22×C416yes—
287Dic3⋊C4C22×C416yes—
288Dic3⋊C4C22×C416yes—
289C4×Dic3C22×C413yes—
290C4×Dic3C22×C413yes—
291C4×Dic3C22×C413yes—
292C4×Dic3C22×C413yes—
293C4×Dic3C2413yes—
294C4×Dic3C2413yes—
295C2×C3⋊C8C2411yescharacteristic
296C2×C3⋊C8C2413yes—
297C2×C3⋊C8C2411yescharacteristic
298C2×C3⋊C8C2413yes—
299ΓL(1,𝔽25)—22no—
300ΓL(1,𝔽25)—26no—
301D3×C8—26no—
302D3×C8—22no—
303C2×#41—31no—
304C22×Q8—31no—
305C22×Q8—31no—
306C22×Q8C4×D311yescharacteristic
307C2×M16—31no—
308C2×M16—33no—
309C22×C8—31no—
310C22×C8—33no—
311C4⋊Q8—33no—
312C4⋊Q8—33no—
313C4⋊Q8—33no—
314C4⋊Q8—33no—
315C4⋊Q8—33no—
316C4⋊Q8—33no—
317#125—36no—
318#125—36no—
319#125—36no—
320C22⋊Q8—33no—
321C22⋊Q8—33no—
322C22⋊Q8—33no—
323C22⋊Q8—33no—
324C22⋊Q8—36no—
325C22⋊Q8—33no—
326C22⋊Q8—33no—
327C22⋊Q8—36no—
328C22⋊Q8—36no—
329C4×Q8—31no—
330C4×Q8—31no—
331C4×Q8—33no—
332C4×Q8—33no—
333C4×Q8—33no—
334C4×Q8—33no—
335C4×Q8—33no—
336C4×Q8—33no—
337C4×Q8—36no—
338C4×Q8C4×D311yescharacteristic
339C4×Q8C4×D313yes—
340C4×Q8C22×D311yescharacteristic
341C4×Q8C22×D313yes—
342C4×D4—33no—
343C4×D4—33no—
344C4×D4—33no—
345C4×D4—33no—
346C4×D4—33no—
347C4×D4—33no—
348#117—33no—
349#117—33no—
350#117—33no—
351C2×C4⋊C4—33no—
352C2×C4⋊C4—33no—
353C2×C4⋊C4C4×D313yes—
354C2×C42—33no—
355C2×C42—33no—
356C2×C42C22×D313yes—
357C4⋊C8—36no—
358C4⋊C8—36no—
359C4⋊C8—33no—
360C4⋊C8—33no—
361C4⋊C8—33no—
362C4⋊C8—33no—
363C4⋊C8—33no—
364C4⋊C8—33no—
365C4⋊C8—33no—
366C4⋊C8—33no—
367C4⋊C8C22×D313yes—
368C4⋊C8C22×D313yes—
369C4⋊C8C22×D313yes—
370C4⋊C8C22×D313yes—
371C22⋊C8—31no—
372C22⋊C8—33no—
373C22⋊C8—33no—
374C22⋊C8—31no—
375C22⋊C8—31no—
376C22⋊C8—33no—
377C22⋊C8C22×D311yescharacteristic
378C22⋊C8C22×D313yes—
379#97—33no—
380#97—33no—
381#97—36no—
382#97C22×D316yes—
383C4×C8—33no—
384C4×C8—33no—
385C4×C8—36no—
386C4×C8—33no—
387C4×C8—33no—
388C4×C8—36no—
389C4×C8C22×D313yes—
390C4×C8C22×D313yes—
391C23×C3C23×C411yescharacteristic
392C22×D3C23×C411yescharacteristic
393C22×D3C23×C411yescharacteristic
394C3×Q8—26no—
395C3×Q8#13114yes—
396C2×C3×C4—26no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—43no—
400C2×C3×C4C23×C413yes—
401C2×C3×C4C23×C413yes—
402C2×C3×C4C23×C413yes—
403C2×C3×C4C23×C411yescharacteristic
404C2×C3×C4C23×C413yes—
405C2×C3×C4C2511yescharacteristic
406#67—22no—
407#67—22no—
408C2×Dic3C23×C411yescharacteristic
409C2×Dic3C23×C413yes—
410C2×Dic3C23×C413yes—
411C2×Dic3C23×C413yes—
412C2×Dic3C23×C413yes—
413C2×Dic3C23×C411yescharacteristic
414D12—22no—
415C4×D3—26no—
416C4×D3—26no—
417C4×D3—21no—
418C4×D3—21no—
419C4×D3—43no—
420C4×D3—43no—
421C4×D32−1+412yes—
422C4×D32−1+412yes—
423Dic6—26no—
424Dic6—26no—
425Dic6—22no—
426Dic6—46no—
427C3×C8—21no—
428C3×C8—23no—
429C3×C8—23no—
430C3×C8—21no—
431C3⋊C8—23no—
432C3⋊C8—21no—
433C3⋊C8—21no—
434C3⋊C8—23no—
435#41—62no—
436#41—62no—
437C2×Q8—24no—
438C2×Q8—31no—
439C2×Q8—33no—
440C2×Q8—33no—
441C2×Q8—36no—
442C2×Q8—64no—
443C2×Q8—64no—
444C2×Q8C2×C4×D313yes—
445C2×Q8C2×C4×D311yescharacteristic
446C2×D4—31no—
447C2×D4—31no—
448C2×D4—31no—
449C22×C4—31no—
450C22×C4—33no—
451C22×C4—33no—
452C22×C4—31no—
453C22×C4—33no—
454C22×C4—33no—
455C22×C4C2×C4×D313yes—
456C22×C4C2×C4×D313yes—
457C22×C4C23×D311yescharacteristic
458M16—62no—
459M16—66no—
460C2×C8—33no—
461C2×C8—31no—
462C2×C8—31no—
463C2×C8—33no—
464C2×C8—66no—
465C2×C8—62no—
466C2×C8C23×D311yescharacteristic
467C2×C8C23×D313yes—
468C2×C8C23×D313yes—
469C2×C8C23×D311yescharacteristic
470C4⋊C4—36no—
471C4⋊C4—36no—
472C4⋊C4—33no—
473C4⋊C4—33no—
474C4⋊C4—33no—
475C4⋊C4—33no—
476C4⋊C4—33no—
477C4⋊C4—33no—
478C4⋊C4—33no—
479C4⋊C4—33no—
480C4⋊C4—36no—
481C4⋊C4—36no—
482C4⋊C4—36no—
483C4⋊C4C2×C4×D313yes—
484C4⋊C4C2×C4×D316yes—
485C4⋊C4C2×C4×D313yes—
486C22⋊C4—33no—
487C22⋊C4—33no—
488C22⋊C4—33no—
489C22⋊C4—33no—
490C22⋊C4—33no—
491C22⋊C4—33no—
492C42—33no—
493C42—33no—
494C42—33no—
495C42—33no—
496C42—33no—
497C42—33no—
498C42C2×C4×D313yes—
499C42C2×C4×D313yes—
500C42C23×D313yes—
501C42C23×D313yes—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3C24×C411yescharacteristic
506C2×D3—21no—
507C2×D3—21no—
508C2×D3—21no—
509C2×D3—21no—
510C2×D3—21no—
511C2×D3#51912yes—
512C3×C4—21no—
513C3×C4—23no—
514C3×C4—23no—
515C3×C4—23no—
516C3×C4C2×#13116yes—
517C3×C4C2×2−1+411yescharacteristic
518C3×C4C2×2−1+411yescharacteristic
519Dic3—21no—
520Dic3—23no—
521Dic3—23no—
522Dic3—23no—
523Dic3—23no—
524Dic3#51912yes—
525C23—31no—
526C23—31no—
527C23C22×C4×D311yescharacteristic
528Q8—26no—
529Q8—62no—
530Q8—66no—
531Q8—66no—
532Q8—126no—
533Q8#90714yes—
534D4—62no—
535D4—62no—
536D4—62no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—26no—
540C2×C4—31no—
541C2×C4—32no—
542C2×C4—32no—
543C2×C4—31no—
544C2×C4—33no—
545C2×C4—33no—
546C2×C4—33no—
547C2×C4—33no—
548C2×C4—43no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—66no—
552C2×C4—66no—
553C2×C4—123no—
554C2×C4—123no—
555C2×C4C22×C4×D313yes—
556C2×C4C22×C4×D311yescharacteristic
557C2×C4C22×C4×D313yes—
558C2×C4C22×C4×D313yes—
559C2×C4C22×C4×D313yes—
560C2×C4C24×D311yescharacteristic
561C8—21no—
562C8—23no—
563C8—23no—
564C8—21no—
565C8—63no—
566C8—61no—
567C8—61no—
568C8—63no—
569C2×C3—41no—
570C2×C3C2×#51911yescharacteristic
571C2×C3C22×#13111yescharacteristic
572C2×C3#357311yescharacteristic
573D3—22no—
574D3—41no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—32no—
579C22—61no—
580C22—61no—
581C22—61no—
582C22—61no—
583C22—61no—
584C22C23×C4×D311yescharacteristic
585C4—23no—
586C4—21no—
587C4—23no—
588C4—23no—
589C4—32no—
590C4—63no—
591C4—63no—
592C4—61no—
593C4—63no—
594C4—63no—
595C4C2×#90716yes—
596C4D3×2−1+411yescharacteristic
597C4D3×2−1+411yescharacteristic
598C3#6025311yescharacteristic, complemented, Sylow
599C2—41no—
600C2—62no—
601C2—121no—
602C2D3×#51911yescharacteristic
603C2C22×#90711yescharacteristic
604C2#8716011yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D3
commutator, socleC22×C3C24×C4
FittingC3×#2984C2
prime radicalC2×#235C23
cosocleC2×C3×C4C25
hypercenter#2984D3
hypocenterC3#60253
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #2984

Complements

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

NH ≅ G/Ntype
2. C3×#2984C2semidirect
3. #82435C2semidirect
4. #79967C2semidirect
5. #79965C2semidirect
6. #79960C2semidirect
7. #79958C2semidirect
8. #79921C2semidirect
9. #79920C2semidirect
10. Q8×ΓL(1,𝔽25)C2semidirect
11. D3×C8×Q8C2semidirect
13. #79316C2semidirect
15. #79267C2semidirect
81. #2984D3semidirect
598. C3#60253semidirect

Maximal subgroups

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

subgroupindex
Q8×#2352
C3×#29842
#824352
#799672
#799652
#799602
#799582
#799212
#799202
Q8×ΓL(1,𝔽25)2
D3×C8×Q82
#793342
#793162
#792832
#792672
#602533