Finite Group Databasebeta

← #10849401

Subgroups — #10849401

Order 768.

1092 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85850C211yesmaximal, characteristic
2C3×#3079C211yesmaximal, characteristic, complemented
3#83253C211yesmaximal, characteristic, complemented
4#83199C211yesmaximal, characteristic, complemented
5C4×#837C211yesmaximal, characteristic, complemented
6#70680C214yesmaximal, complemented
7#70524C211yesmaximal, characteristic, complemented
8#70512C211yesmaximal, characteristic, complemented
9#70311C214yesmaximal, complemented
10#25666—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C3×C4×#41C2211yescharacteristic
13#5833C2212yes—
14C2×#944—22no—
15#5584—21no—
16#5579—21no—
17#5565—22no—
18#5563—21no—
19#5552—22no—
20#5551—21no—
21#5534—22no—
22#5523C2212yes—
23#5477—21no—
24#5477C2211yescharacteristic
25#5470—21no—
26C2×#869C2211yescharacteristic
27C3×#477—21no—
28C3×#476—21no—
29C3×#475—21no—
30C3×#474—21no—
31C3×#428C2212yes—
32C3×#423C2211yescharacteristic
33C3×#417C2212yes—
34C2×#837C2212yes—
35C2×#825C2214yes—
36#5093—21no—
37#5089—22no—
38#5088—21no—
39#4975—22no—
40#4975C2214yes—
41#4966C2214yes—
42C42×Dic3C2211yescharacteristic
43C4×#208C2211yescharacteristic
44#4554C2214yes—
45#4532C2212yes—
46#3511—31no—
47#3141—31no—
48#3079D311yescharacteristic, complemented
49#2868—31no—
50#2039—34no—
51#1997—31no—
52#1962—31no—
53#1859—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#944—48no—
79#939—24no—
80#938—22no—
81#930—22no—
82#930—21no—
83#930—21no—
84#930—22no—
85#926—22no—
86C2×C4⋊Dic3D412yes—
87C2×C4⋊Dic3C2312yes—
88#924—22no—
89#924—22no—
90C2×Dic3⋊C4—28no—
91C2×C4×Dic3—21no—
92C2×C4×Dic3—21no—
93C2×C4×Dic3C2312yes—
94C2×#208C2312yes—
95#898—24no—
96#896—24no—
97#892—22no—
98D3×C4⋊C4—22no—
99#885—24no—
100#883—24no—
101#67⋊C4—24no—
102#875—22no—
103#875D412yes—
104#874D411yescharacteristic
105C4×D12—22no—
106C4×D12—22no—
107C4×D12—21no—
108C4×D12—21no—
109#870—22no—
110#869—21no—
111#869—21no—
112#869—21no—
113#869—21no—
114#869D411yescharacteristic
115C4×Dic6—22no—
116C4×Dic6—22no—
117C3×#107—22no—
118C3×#106—22no—
119C3×C4⋊C8—21no—
120C3×C4⋊C8—21no—
121C3×#103—22no—
122C3×#102—22no—
123#837—24no—
124#825—28no—
125#810—24no—
126#809—24no—
127#803—21no—
128C4×C3⋊C8—21no—
129#556—61no—
130#555—61no—
131#554—61no—
132#553—62no—
133#548—62no—
134#547—61no—
135#542—62no—
136#537—32no—
137#533—61no—
138#532—31no—
139#532—61no—
140C2×C4⋊Q8—31no—
141C2×C4⋊Q8—62no—
142C2×C4⋊Q8C2×D311yescharacteristic
143#520—32no—
144C4×#41C2×D311yescharacteristic
145#489—62no—
146#487—61no—
147#486—61no—
148#477—21no—
149#476—21no—
150#475—21no—
151#474—21no—
152#428C2×D312yes—
153#423C2×D311yescharacteristic
154C2×#104—32no—
155C2×#104—34no—
156#417—34no—
157#417C2×D312yes—
158C4×M16—31no—
159#384—34no—
160#384—62no—
161C43—31no—
162#343—34no—
163#339—32no—
164C3×#41—22no—
165C3×#41—22no—
166C2×C3×Q8—44no—
167C2×C3×Q8—41no—
168C2×C3×Q8—41no—
169C2×C3×Q8C2×D411yescharacteristic
170C2×C3×D4—21no—
171C2×C3×D4C2×D411yescharacteristic
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2411yescharacteristic
179C2×#67—22no—
180C2×#67—24no—
181C22×Dic3—22no—
182C22×Dic3C2×D414yes—
183#235—24no—
184#235—24no—
185#235—84no—
186C2×D12—21no—
187C2×D12—21no—
188C2×D12C2×D412yes—
189C2×C4×D3—22no—
190C2×C4×D3—24no—
191C2×Dic6—22no—
192C2×Dic6—22no—
193C2×Dic6—44no—
194C2×Dic6C2×D412yes—
195C3×M16—24no—
196C2×C3×C8—22no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—41no—
202C3×C4⋊C4—41no—
203C3×C4⋊C4C2×D411yescharacteristic
204C3×C4⋊C4C2×D411yescharacteristic
205C3×C22⋊C4—21no—
206C3×C22⋊C4—21no—
207C3×C22⋊C4—21no—
208C3×C42—21no—
209C3×C42—21no—
210C3×C42—21no—
211C3×C42C2×D411yescharacteristic
212C3×C42C2×D411yescharacteristic
213C3×C42C2×D411yescharacteristic
214C3×C42C2×D411yescharacteristic
215#217—24no—
216(C2×D3)⋊C4—24no—
217(C2×D3)⋊C4—24no—
218(C2×D3)⋊C4—22no—
219(C2×D3)⋊C4—22no—
220C4⋊Dic3—21no—
221C4⋊Dic3—22no—
222C4⋊Dic3—22no—
223C4⋊Dic3—21no—
224C4⋊Dic3—22no—
225C4⋊Dic3—22no—
226Dic3⋊C4—24no—
227Dic3⋊C4—24no—
228Dic3⋊C4—416no—
229C4×Dic3—24no—
230C4×Dic3—24no—
231C4×Dic3—24no—
232C4×Dic3—22no—
233C4×Dic3#4114yescomplemented
234#208—22no—
235#208—22no—
236C2×C3⋊C8—22no—
237C2×#41—32no—
238C2×#41—62no—
239C2×#41C22×D311yescharacteristic
240C22×Q8—21no—
241C22×Q8—32no—
242C2×M16—32no—
243C2×M16C22×D312yes—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—31no—
249C4⋊Q8—61no—
250C4⋊Q8—61no—
251C4⋊Q8—61no—
252C4⋊Q8—61no—
253C4⋊Q8—128no—
254#127—31no—
255#126—64no—
256#125—62no—
257#124—32no—
258#124—62no—
259#124—64no—
260#123—64no—
261C22⋊Q8—64no—
262C22⋊Q8—62no—
263C22⋊Q8—62no—
264C22⋊Q8—62no—
265#121—62no—
266#121—62no—
267#121—64no—
268#121—62no—
269#121—61no—
270#121—61no—
271C4×Q8—62no—
272C4×Q8—62no—
273C4×Q8#6711yescharacteristic
274C4×Q8#6711yescharacteristic
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—62no—
278C4×D4—62no—
279C4×D4—64no—
280C4×D4—61no—
281C4×D4—61no—
282C4×D4#6711yescharacteristic
283C4×D4#6711yescharacteristic
284#117—21no—
285#117—62no—
286#117—62no—
287#117C22×D311yescharacteristic
288C2×C4⋊C4—21no—
289C2×C4⋊C4—32no—
290C2×C4⋊C4—32no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—68no—
293C2×C4⋊C4C22×D312yes—
294C2×C42—21no—
295C2×C42—32no—
296C2×C42—61no—
297C2×C42—61no—
298C2×C42C22×D311yescharacteristic
299#107—22no—
300#106—22no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303#104—68no—
304#104—64no—
305#103—22no—
306#103—64no—
307#102—22no—
308#102—64no—
309#97—61no—
310C4×C8—61no—
311C23×C3—21no—
312C23×C3C22×D411yescharacteristic
313C22×D3—22no—
314C3×Q8—21no—
315C3×Q8—21no—
316C3×Q8—82no—
317C3×Q8—82no—
318C3×D4—21no—
319C3×D4—21no—
320C3×D4—42no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—44no—
330C2×C3×C4—42no—
331C2×C3×C4#12011yescharacteristic
332C2×C3×C4#12011yescharacteristic
333C2×C3×C4#12011yescharacteristic
334C2×C3×C4#12011yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C2×#4112yes—
338C2×C3×C42−1+412yes—
339#67—44no—
340#67—84no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—22no—
345C2×Dic3—24no—
346C2×Dic3—44no—
347D12—24no—
348D12—42no—
349D12—42no—
350C4×D3—42no—
351C4×D3—42no—
352C4×D3—84no—
353Dic6—24no—
354Dic6—44no—
355Dic6—82no—
356C3×C8—42no—
357C3⋊C8—42no—
358#41—22no—
359#41—22no—
360#41—64no—
361#41—64no—
362#41—244no—
363C2×Q8—32no—
364C2×Q8—44no—
365C2×Q8—41no—
366C2×Q8—41no—
367C2×Q8—62no—
368C2×Q8—62no—
369C2×Q8—124no—
370C2×Q8C2×#6711yescharacteristic
371C2×D4—21no—
372C2×D4—32no—
373C2×D4—62no—
374C2×D4—64no—
375C2×D4—61no—
376C2×D4—61no—
377C2×D4C2×#6711yescharacteristic
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—34no—
383C22×C4—64no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4D3×D411yescharacteristic
387C22×C4D3×D411yescharacteristic
388C22×C4C23×D311yescharacteristic
389M16—24no—
390M16—62no—
391M16—62no—
392C2×C8—22no—
393C2×C8—62no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—64no—
401C4⋊C4—64no—
402C4⋊C4—62no—
403C4⋊C4—62no—
404C4⋊C4—61no—
405C4⋊C4—61no—
406C4⋊C4—62no—
407C4⋊C4—62no—
408C4⋊C4—1216no—
409C4⋊C4C2×#6711yescharacteristic
410C4⋊C4C2×#6711yescharacteristic
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—64no—
415C22⋊C4—64no—
416C22⋊C4—64no—
417C22⋊C4—62no—
418C22⋊C4—62no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—34no—
423C42—64no—
424C42—64no—
425C42—64no—
426C42—62no—
427C42D3×D411yescharacteristic
428C42D3×D411yescharacteristic
429C42C2×#6711yescharacteristic
430C42C2×#6711yescharacteristic
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3C2×#12011yescharacteristic
437C22×C3#54812yes—
438C2×D3—42no—
439C2×D3—42no—
440C2×D3—42no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—41no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—82no—
447C3×C4#45312yes—
448C3×C4#45612yes—
449C3×C4#53211yescharacteristic
450C3×C4#53611yescharacteristic
451C3×C4#54712yes—
452Dic3—28no—
453Dic3—42no—
454Dic3—44no—
455C23—21no—
456C23—62no—
457C23C2×D3×D411yescharacteristic
458Q8—21no—
459Q8—21no—
460Q8—64no—
461Q8—82no—
462Q8—82no—
463Q8—124no—
464Q8—242no—
465D4—21no—
466D4—21no—
467D4—42no—
468D4—64no—
469D4—124no—
470D4—122no—
471D4—122no—
472D4—244no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—64no—
484C2×C4—64no—
485C2×C4—64no—
486C2×C4—62no—
487C2×C4—64no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—124no—
491C2×C4—244no—
492C2×C4#93711yescharacteristic
493C2×C4#93711yescharacteristic
494C2×C4#93711yescharacteristic
495C2×C4#93711yescharacteristic
496C2×C4C2×D3×D411yescharacteristic
497C2×C4#100712yes—
498C2×C4D3×#4112yes—
499C2×C4C22×#6711yescharacteristic
500C8—42no—
501C8—122no—
502C2×C3—22no—
503C2×C3—81no—
504C2×C3#243211yescharacteristic
505C2×C3C2×#45311yescharacteristic
506C2×C3C2×#45611yescharacteristic
507D3—82no—
508C22—21no—
509C22—21no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—122no—
514C22—122no—
515C22—122no—
516C22#563312yes—
517C22C2×#93711yescharacteristic
518C4—21no—
519C4—21no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—68no—
524C4—82no—
525C4—122no—
526C4—124no—
527C4#519812yes—
528C4#520512yes—
529C4#563112yes—
530C4#581611yescharacteristic
531C4#583211yescharacteristic
532C3#2566611yescharacteristic, complemented, Sylow
533C2—22no—
534C2—81no—
535C2—242no—
536C2#8275911yescharacteristic
537C2C2×#519811yescharacteristic
538C2C2×#520511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3079C2
socleC22×C3C2×#120
prime radical#85850C2
hypercenter#3079D3
hypocenterC3#25666
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. #83253C2semidirect
4. #83199C2semidirect
5. C4×#837C2semidirect
6. #70680C2semidirect
7. #70524C2semidirect
8. #70512C2semidirect
9. #70311C2semidirect
48. #3079D3semidirect
233. C4×Dic3#41semidirect
532. C3#25666semidirect

Maximal subgroups

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

subgroupindex
#858502
C3×#30792
#832532
#831992
C4×#8372
#706802
#705242
#705122
#703112
#256663