Finite Group Databasebeta

← #10849410

Subgroups — #10849410

Order 768.

1020 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85850C211yesmaximal, characteristic
2C3×#3079C211yesmaximal, characteristic, complemented
3#83252C211yesmaximal, characteristic, complemented
4#83199C211yesmaximal, characteristic, complemented
5C4×#837C211yesmaximal, characteristic, complemented
6#70681C214yesmaximal, complemented
7#70524C211yesmaximal, characteristic, complemented
8#70512C211yesmaximal, characteristic, complemented
9#70312C214yesmaximal, complemented
10#25667—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C3×C4×#41C2211yescharacteristic
13#5833C2212yes—
14#5584—21no—
15#5579—21no—
16#5565—22no—
17#5563—21no—
18#5552—22no—
19#5551—21no—
20#5534—22no—
21#5523C2212yes—
22#5477—21no—
23#5477C2211yescharacteristic
24#5470—21no—
25C2×#869C2211yescharacteristic
26C3×#477—21no—
27C3×#476—21no—
28C3×#475—21no—
29C3×#474—21no—
30C3×#428C2212yes—
31C3×#423C2211yescharacteristic
32C3×#417C2212yes—
33C2×#837C2212yes—
34#5229—22no—
35C2×#825C2214yes—
36#5093—21no—
37#5089—22no—
38#5088—21no—
39#4977C2214yes—
40#4973—22no—
41#4966C2214yes—
42C42×Dic3C2211yescharacteristic
43C4×#208C2211yescharacteristic
44#4554C2214yes—
45#4532C2212yes—
46#3511—31no—
47#3141—31no—
48#3079D311yescharacteristic, complemented
49#2862—31no—
50#2040—34no—
51#1997—31no—
52#1962—31no—
53#1860—34no—
54C4×#104—31no—
55C2×C3×#41C2311yescharacteristic
56C22×C3×Q8—21no—
57C2×#235—22no—
58C2×#235C2312yes—
59C22×Dic6D412yes—
60C2×C3×M16C2312yes—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8—21no—
65C3×C4×Q8D411yescharacteristic
66C3×C4×Q8D411yescharacteristic
67C3×C4×D4—21no—
68C3×C4×D4—21no—
69C3×C4×D4D411yescharacteristic
70C3×C4×D4D411yescharacteristic
71C3×#117—21no—
72C3×#117C2311yescharacteristic
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C42—21no—
76C2×C3×C42C2311yescharacteristic
77#946—22no—
78#939—24no—
79#938—22no—
80#930—22no—
81#930—21no—
82#930—21no—
83#930—22no—
84#926—22no—
85C2×C4⋊Dic3D412yes—
86C2×C4⋊Dic3C2312yes—
87#924—22no—
88#924—22no—
89C2×C4×Dic3—21no—
90C2×C4×Dic3—21no—
91C2×C4×Dic3C2312yes—
92C2×#208C2312yes—
93#898—24no—
94#896—24no—
95#892—22no—
96D3×C4⋊C4—22no—
97#885—24no—
98#883—24no—
99#67⋊C4—24no—
100#875—22no—
101#875D412yes—
102#874D411yescharacteristic
103C4×D12—22no—
104C4×D12—22no—
105C4×D12—21no—
106C4×D12—21no—
107#870—22no—
108#869—21no—
109#869—21no—
110#869—21no—
111#869—21no—
112#869D411yescharacteristic
113C4×Dic6—22no—
114C4×Dic6—22no—
115C3×#107—22no—
116C3×#106—22no—
117C3×C4⋊C8—21no—
118C3×C4⋊C8—21no—
119C3×#103—22no—
120C3×#102—22no—
121#837—24no—
122C3⋊#95—28no—
123#825—28no—
124#810—24no—
125#809—24no—
126#803—21no—
127C4×C3⋊C8—21no—
128#556—61no—
129#555—61no—
130#554—61no—
131#553—62no—
132#548—62no—
133#547—61no—
134#542—62no—
135#537—32no—
136#533—61no—
137#532—31no—
138#532—61no—
139C2×C4⋊Q8—31no—
140C2×C4⋊Q8C2×D311yescharacteristic
141#520—32no—
142C4×#41C2×D311yescharacteristic
143#489—62no—
144#487—61no—
145#486—61no—
146#477—21no—
147#476—21no—
148#475—21no—
149#474—21no—
150#428C2×D312yes—
151#423C2×D311yescharacteristic
152C2×#104—32no—
153C2×#104—34no—
154#417—34no—
155#417C2×D312yes—
156C4×M16—31no—
157#391—62no—
158#391—62no—
159#382—34no—
160C43—31no—
161#343—34no—
162#339—32no—
163C3×#41—22no—
164C3×#41—22no—
165C2×C3×Q8—44no—
166C2×C3×Q8—41no—
167C2×C3×Q8—41no—
168C2×C3×Q8C2×D411yescharacteristic
169C2×C3×D4—21no—
170C2×C3×D4C2×D411yescharacteristic
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2411yescharacteristic
178C2×#67—22no—
179C2×#67—24no—
180C22×Dic3—22no—
181C22×Dic3C2×D414yes—
182#235—24no—
183#235—24no—
184#235—84no—
185C2×D12—21no—
186C2×D12—21no—
187C2×D12C2×D412yes—
188C2×C4×D3—22no—
189C2×C4×D3—24no—
190C2×Dic6—22no—
191C2×Dic6—22no—
192C2×Dic6—44no—
193C2×Dic6C2×D412yes—
194C3×M16—24no—
195C2×C3×C8—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—41no—
201C3×C4⋊C4—41no—
202C3×C4⋊C4C2×D411yescharacteristic
203C3×C4⋊C4C2×D411yescharacteristic
204C3×C22⋊C4—21no—
205C3×C22⋊C4—21no—
206C3×C22⋊C4—21no—
207C3×C42—21no—
208C3×C42—21no—
209C3×C42—21no—
210C3×C42C2×D411yescharacteristic
211C3×C42C2×D411yescharacteristic
212C3×C42C2×D411yescharacteristic
213C3×C42C2×D411yescharacteristic
214#217—24no—
215(C2×D3)⋊C4—24no—
216(C2×D3)⋊C4—24no—
217(C2×D3)⋊C4—22no—
218(C2×D3)⋊C4—22no—
219C4⋊Dic3—21no—
220C4⋊Dic3—22no—
221C4⋊Dic3—22no—
222C4⋊Dic3—21no—
223C4⋊Dic3—22no—
224C4⋊Dic3—22no—
225Dic3⋊C4—24no—
226Dic3⋊C4—24no—
227C4×Dic3—24no—
228C4×Dic3—22no—
229C4×Dic3—42no—
230C4×Dic3—42no—
231C4×Dic3#4114yescomplemented
232#208—22no—
233#208—22no—
234C2×C3⋊C8—22no—
235C2×#41—32no—
236C2×#41—62no—
237C2×#41C22×D311yescharacteristic
238C22×Q8—21no—
239C22×Q8—32no—
240C2×M16—32no—
241C2×M16C22×D312yes—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—31no—
247C4⋊Q8—61no—
248C4⋊Q8—61no—
249C4⋊Q8—61no—
250C4⋊Q8—61no—
251#127—31no—
252#126—64no—
253#125—62no—
254#124—32no—
255#124—62no—
256#124—64no—
257#123—64no—
258C22⋊Q8—64no—
259C22⋊Q8—62no—
260C22⋊Q8—62no—
261C22⋊Q8—62no—
262#121—62no—
263#121—62no—
264#121—64no—
265#121—62no—
266#121—61no—
267#121—61no—
268C4×Q8—62no—
269C4×Q8—62no—
270C4×Q8#6711yescharacteristic
271C4×Q8#6711yescharacteristic
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—62no—
275C4×D4—62no—
276C4×D4—64no—
277C4×D4—61no—
278C4×D4—61no—
279C4×D4#6711yescharacteristic
280C4×D4#6711yescharacteristic
281#117—21no—
282#117—62no—
283#117—62no—
284#117C22×D311yescharacteristic
285C2×C4⋊C4—21no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4—32no—
288C2×C4⋊C4—62no—
289C2×C4⋊C4C22×D312yes—
290C2×C42—21no—
291C2×C42—32no—
292C2×C42—61no—
293C2×C42—61no—
294C2×C42C22×D311yescharacteristic
295#107—22no—
296#106—22no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299#104—68no—
300#104—64no—
301#103—22no—
302#103—64no—
303#102—22no—
304#102—64no—
305#97—61no—
306C4×C8—61no—
307#95—68no—
308C23×C3—21no—
309C23×C3C22×D411yescharacteristic
310C22×D3—22no—
311C3×Q8—21no—
312C3×Q8—21no—
313C3×Q8—82no—
314C3×Q8—82no—
315C3×D4—21no—
316C3×D4—21no—
317C3×D4—42no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—44no—
327C2×C3×C4—42no—
328C2×C3×C4#12011yescharacteristic
329C2×C3×C4#12011yescharacteristic
330C2×C3×C4#12011yescharacteristic
331C2×C3×C4#12011yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C2×#4112yes—
335C2×C3×C42−1+412yes—
336#67—44no—
337#67—84no—
338C2×Dic3—22no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—44no—
344D12—24no—
345D12—42no—
346D12—42no—
347C4×D3—42no—
348C4×D3—42no—
349C4×D3—84no—
350Dic6—24no—
351Dic6—44no—
352Dic6—82no—
353C3×C8—42no—
354C3⋊C8—42no—
355#41—22no—
356#41—22no—
357#41—64no—
358#41—64no—
359#41—244no—
360C2×Q8—32no—
361C2×Q8—44no—
362C2×Q8—41no—
363C2×Q8—41no—
364C2×Q8—62no—
365C2×Q8—62no—
366C2×Q8—124no—
367C2×Q8C2×#6711yescharacteristic
368C2×D4—21no—
369C2×D4—32no—
370C2×D4—62no—
371C2×D4—64no—
372C2×D4—61no—
373C2×D4—61no—
374C2×D4C2×#6711yescharacteristic
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—34no—
380C22×C4—64no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4D3×D411yescharacteristic
384C22×C4D3×D411yescharacteristic
385C22×C4C23×D311yescharacteristic
386M16—24no—
387M16—62no—
388M16—62no—
389C2×C8—22no—
390C2×C8—62no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—64no—
398C4⋊C4—64no—
399C4⋊C4—62no—
400C4⋊C4—62no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4C2×#6711yescharacteristic
406C4⋊C4C2×#6711yescharacteristic
407C22⋊C4—21no—
408C22⋊C4—21no—
409C22⋊C4—21no—
410C22⋊C4—64no—
411C22⋊C4—64no—
412C22⋊C4—64no—
413C22⋊C4—62no—
414C22⋊C4—62no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—34no—
419C42—64no—
420C42—62no—
421C42—122no—
422C42—122no—
423C42D3×D411yescharacteristic
424C42D3×D411yescharacteristic
425C42C2×#6711yescharacteristic
426C42C2×#6711yescharacteristic
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—41no—
430C22×C3—41no—
431C22×C3—41no—
432C22×C3C2×#12011yescharacteristic
433C22×C3#54812yes—
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—42no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—41no—
440C3×C4—41no—
441C3×C4—41no—
442C3×C4—82no—
443C3×C4#45412yes—
444C3×C4#45512yes—
445C3×C4#53211yescharacteristic
446C3×C4#53611yescharacteristic
447C3×C4#54712yes—
448Dic3—44no—
449Dic3—42no—
450Dic3—44no—
451C23—21no—
452C23—62no—
453C23C2×D3×D411yescharacteristic
454Q8—21no—
455Q8—21no—
456Q8—64no—
457Q8—82no—
458Q8—82no—
459Q8—124no—
460Q8—242no—
461D4—21no—
462D4—21no—
463D4—42no—
464D4—64no—
465D4—124no—
466D4—122no—
467D4—122no—
468D4—244no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—64no—
480C2×C4—62no—
481C2×C4—64no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—122no—
485C2×C4—122no—
486C2×C4—124no—
487C2×C4—244no—
488C2×C4#93711yescharacteristic
489C2×C4#93711yescharacteristic
490C2×C4#93711yescharacteristic
491C2×C4#93711yescharacteristic
492C2×C4C2×D3×D411yescharacteristic
493C2×C4#100712yes—
494C2×C4D3×#4112yes—
495C2×C4C22×#6711yescharacteristic
496C8—42no—
497C8—122no—
498C2×C3—22no—
499C2×C3—81no—
500C2×C3#243211yescharacteristic
501C2×C3C2×#45511yescharacteristic
502C2×C3C2×#45411yescharacteristic
503D3—82no—
504C22—21no—
505C22—21no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—122no—
510C22—122no—
511C22—122no—
512C22#563312yes—
513C22C2×#93711yescharacteristic
514C4—21no—
515C4—21no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—82no—
520C4—124no—
521C4—124no—
522C4—122no—
523C4#520112yes—
524C4#520212yes—
525C4#563112yes—
526C4#581611yescharacteristic
527C4#583211yescharacteristic
528C3#2566711yescharacteristic, complemented, Sylow
529C2—22no—
530C2—81no—
531C2—242no—
532C2#8275911yescharacteristic
533C2C2×#520111yescharacteristic
534C2C2×#520211yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3079C2
socleC22×C3C2×#120
prime radical#85850C2
hypercenter#3079D3
hypocenterC3#25667
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3079

Complements

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

NH ≅ G/Ntype
2. C3×#3079C2semidirect
3. #83252C2semidirect
4. #83199C2semidirect
5. C4×#837C2semidirect
6. #70681C2semidirect
7. #70524C2semidirect
8. #70512C2semidirect
9. #70312C2semidirect
48. #3079D3semidirect
231. C4×Dic3#41semidirect
528. C3#25667semidirect

Maximal subgroups

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

subgroupindex
#858502
C3×#30792
#832522
#831992
C4×#8372
#706812
#705242
#705122
#703122
#256673