Finite Group Databasebeta

← #10948003

Subgroups — #10948003

Order 768.

1220 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86244C211yesmaximal, characteristic
2#78689C212yesmaximal, complemented
3C3×#2126C211yesmaximal, characteristic, complemented
4#71819C211yesmaximal, characteristic, complemented
5#70061C212yesmaximal, complemented
6#69158C214yesmaximal, complemented
7#69152C214yesmaximal, complemented
8#33362—31nomaximal, Sylow
9C2×#1010C2211yescharacteristic
10C2×#1009C411yescharacteristic, complemented
11C3×#532C2211yescharacteristic
12#5830C2211yescharacteristic
13#5825C411yescharacteristic, complemented
14#5724C412yescomplemented
15#5715C2212yes—
16D3×#124C412yescomplemented
17#5668C2212yes—
18#5610—24no—
19#5552—22no—
20#5550—22no—
21#5516—22no—
22C2×#875C411yescharacteristic, complemented
23C4×#235C411yescharacteristic, complemented
24C3×#410C2212yescomplemented
25#5240C2212yescomplemented
26#5006—22no—
27#4740—24no—
28#4740C2218yescomplemented
29C3×#354C2212yescomplemented
30C3×#353C2212yescomplemented
31#4545C2212yescomplemented
32#4541C2212yescomplemented
33#4474C2218yescomplemented
34#3500—31no—
35#2884—32no—
36#2128—32no—
37#2128—34no—
38#2126—34no—
39#2126—31no—
40#2126D311yescharacteristic, complemented
41C2×C3×#41C2×C411yescharacteristic
42C2×C3×#41C2311yescharacteristic
43#1010—42no—
44#1010—42no—
45#1009—42no—
46#1009—42no—
47C2×#239C2×C412yes—
48C2×D3×Q8C2×C412yes—
49C2×#237D414yes—
50C2×#237C2312yes—
51C2×D3×D4—22no—
52C2×D3×D4C2×C412yes—
53C2×#235—21no—
54C2×#235C2×C411yescharacteristic
55C2×#235D412yes—
56C22×D12—21no—
57C22×D12C2×C411yescharacteristic
58C22×Dic6C2×C411yescharacteristic
59C22×Dic6D412yes—
60C3×C4⋊Q8C2×C411yescharacteristic
61C3×#127C2311yescharacteristic
62C3×#124C2×C411yescharacteristic
63C3×#124C2311yescharacteristic
64C3×#121—22no—
65C3×C4×D4—22no—
66C3×#117—22no—
67C2×C3×C42C2×C411yescharacteristic
68#947C2×C412yes—
69#944C2×C412yes—
70#940C2312yes—
71#939—24no—
72#937—22no—
73#937—24no—
74#936C2×C412yes—
75#935—22no—
76#930—22no—
77C4×#67—22no—
78C2×(C2×D3)⋊C4—22no—
79#926—22no—
80#926C2×C411yescharacteristic
81#924—22no—
82#896—24no—
83#894—24no—
84#887—44no—
85#885—24no—
86#884—24no—
87#67⋊C4—24no—
88D3×C22⋊C4—24no—
89#877—28no—
90#875—22no—
91#875C2×C412yes—
92#875C2×C412yes—
93C4×D12—21no—
94#872C2×C412yes—
95C42×D3C2×C412yes—
96C4×Dic6—21no—
97C4×Dic6—44no—
98C3×#99—22no—
99C3×#99C2314yes—
100#834—22no—
101#834C2314yes—
102#806—28no—
103C2×2−1+4—31no—
104C2×H(5,𝔽2)—31no—
105#548—62no—
106#546—62no—
107#546—64no—
108#536—31no—
109#536—32no—
110#535—31no—
111#535—32no—
112#532—32no—
113#532C2×D311yescharacteristic
114C2×#124—32no—
115C2×#124—31no—
116#519—62no—
117C4×#41—31no—
118#410—32no—
119#410—38no—
120#410—64no—
121#410—62no—
122#410C2×D312yescomplemented
123#354—32no—
124#354C2×D312yescomplemented
125#353—32no—
126#353—38no—
127#353C2×D312yescomplemented
128C23×D3—21no—
129C23×D3—22no—
130C3×#41—41no—
131C3×#41—41no—
132C3×#41—41no—
133C3×#41—41no—
134C2×C3×Q8C22×C411yescharacteristic
135C2×C3×Q8C22×C411yescharacteristic
136C2×C3×D4—21no—
137C2×C3×D4—22no—
138C2×C3×D4C22×C411yescharacteristic
139C2×C3×D4C2×D412yes—
140C2×C3×D4C2411yescharacteristic
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4C22×C411yescharacteristic
144C22×C3×C4C2×D412yes—
145C2×#67—24no—
146C2×#67—22no—
147C2×#67C22⋊C412yes—
148C2×#67C2×D414yes—
149C22×Dic3C22⋊C412yes—
150C22×Dic3C2×D414yes—
151#239—44no—
152D3×Q8—44no—
153#237—48no—
154#237—44no—
155D3×D4—44no—
156D3×D4—48no—
157#235—22no—
158#235—22no—
159#235—42no—
160#235—42no—
161#235—42no—
162#235—42no—
163C2×D12—22no—
164C2×D12—21no—
165C2×D12—42no—
166C2×D12—42no—
167C2×D12—41no—
168C2×D12—41no—
169C2×D12C22⋊C411yescharacteristic
170C2×D12C22×C412yes—
171C2×C4×D3—22no—
172C2×C4×D3—22no—
173C2×C4×D3C22⋊C414yes—
174C2×C4×D3C22×C412yes—
175C2×Dic6—24no—
176C2×Dic6—24no—
177C2×Dic6—41no—
178C2×Dic6—41no—
179C2×Dic6C22⋊C412yes—
180C2×Dic6C22⋊C414yes—
181C2×Dic6C22⋊C411yescharacteristic
182C2×Dic6C22×C412yes—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—42no—
185C3×C22⋊C4—22no—
186C3×C22⋊C4—22no—
187C3×C22⋊C4—24no—
188C3×C42—21no—
189C3×C42—42no—
190C3×C42C22×C411yescharacteristic
191C3×C42C22×C411yescharacteristic
192#217—21no—
193#217—24no—
194#217—24no—
195(C2×D3)⋊C4—22no—
196(C2×D3)⋊C4—24no—
197(C2×D3)⋊C4—24no—
198(C2×D3)⋊C4—24no—
199C4⋊Dic3—21no—
200C4⋊Dic3—42no—
201Dic3⋊C4—22no—
202Dic3⋊C4—24no—
203Dic3⋊C4—48no—
204C4×Dic3—22no—
205C4×Dic3—48no—
206C4×Dic3—42no—
207C4×Dic3C22×C412yes—
2082−1+4—122no—
2092−1+4—122no—
210H(5,𝔽2)—122no—
211H(5,𝔽2)—122no—
212C2×#41—34no—
213C2×#41—32no—
214C2×#41—31no—
215C2×#41—32no—
216C2×#41—32no—
217C2×#41—61no—
218C2×#41C4×D311yescharacteristic
219C2×#41C22×D311yescharacteristic
220C22×Q8—32no—
221C22×Q8—31no—
222C22×Q8—32no—
223C22×D4—32no—
224C22×D4—31no—
225C22×D4—61no—
226C22×D4—62no—
227C4⋊Q8—32no—
228C4⋊Q8C4×D311yescharacteristic
229#127—32no—
230#127C22×D311yescharacteristic
231#124—32no—
232#124—32no—
233#124—32no—
234#124—32no—
235#124—62no—
236#124—64no—
237#124C4×D311yescharacteristic
238#124C22×D311yescharacteristic
239#123—64no—
240#123—62no—
241C22⋊Q8—64no—
242C22⋊Q8—62no—
243#121—22no—
244#121—64no—
245#121—64no—
246#121—62no—
247#120—64no—
248#120—62no—
249C4×Q8—61no—
250C4×Q8—124no—
251C4×Q8—124no—
252C4×D4—22no—
253C4×D4—61no—
254C4×D4—64no—
255C4×D4—62no—
256#117—22no—
257#117—31no—
258#117—32no—
259#117—68no—
260#117—62no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—62no—
263C2×C42—32no—
264C2×C42C4×D311yescharacteristic
265#99—22no—
266#99—34no—
267#99—68no—
268#99—62no—
269#99C22×D314yes—
270C23×C3—21no—
271C23×C3C2×C22⋊C411yescharacteristic
272C23×C3C22×D412yes—
273C22×D3—22no—
274C22×D3—44no—
275C22×D3—42no—
276C22×D3—84no—
277C22×D3—82no—
278C22×D3C2×C22⋊C412yes—
279C3×Q8—41no—
280C3×Q8—41no—
281C3×D4—42no—
282C3×D4—41no—
283C3×D4—41no—
284C3×D4—42no—
285C3×D4—82no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—44no—
296C2×C3×C4—81no—
297C2×C3×C4C2×C22⋊C412yes—
298C2×C3×C4C2×C22⋊C411yescharacteristic
299C2×C3×C4C23×C411yescharacteristic
300#67—24no—
301#67—44no—
302#67—44no—
303#67—84no—
304C2×Dic3—28no—
305C2×Dic3—22no—
306C2×Dic3—42no—
307C2×Dic3—44no—
308C2×Dic3C2×C22⋊C414yes—
309C2×Dic3C2×C22⋊C412yes—
310D12—21no—
311D12—21no—
312D12—42no—
313D12—44no—
314D12—41no—
315D12—41no—
316C4×D3—24no—
317C4×D3—44no—
318C4×D3—44no—
319C4×D3—82no—
320Dic6—22no—
321Dic6—22no—
322Dic6—28no—
323Dic6—42no—
324Dic6#9912yes—
325Dic6#9912yes—
326C24—62no—
327C24—61no—
328#41—41no—
329#41—41no—
330#41—41no—
331#41—41no—
332#41—62no—
333#41—62no—
334#41—122no—
335#41—122no—
336#41—122no—
337#41—122no—
338#41—124no—
339#41—128no—
340#41—124no—
341C2×Q8—34no—
342C2×Q8—31no—
343C2×Q8—32no—
344C2×Q8—32no—
345C2×Q8—64no—
346C2×Q8—64no—
347C2×Q8—121no—
348C2×Q8—121no—
349C2×Q8—124no—
350C2×Q8C2×C4×D311yescharacteristic
351C2×Q8C2×C4×D311yescharacteristic
352C2×D4—21no—
353C2×D4—22no—
354C2×D4—32no—
355C2×D4—34no—
356C2×D4—31no—
357C2×D4—32no—
358C2×D4—62no—
359C2×D4—62no—
360C2×D4—64no—
361C2×D4—61no—
362C2×D4—121no—
363C2×D4—121no—
364C2×D4—122no—
365C2×D4—122no—
366C2×D4—124no—
367C2×D4—128no—
368C2×D4C2×C4×D311yescharacteristic
369C2×D4D3×D412yes—
370C2×D4C23×D311yescharacteristic
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—34no—
374C22×C4—32no—
375C22×C4—32no—
376C22×C4—34no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4C2×C4×D311yescharacteristic
380C22×C4D3×D412yes—
381C4⋊C4—22no—
382C4⋊C4—42no—
383C4⋊C4—62no—
384C4⋊C4—64no—
385C4⋊C4—61no—
386C4⋊C4—122no—
387C4⋊C4—128no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—24no—
391C22⋊C4—64no—
392C22⋊C4—64no—
393C22⋊C4—62no—
394C22⋊C4—64no—
395C22⋊C4—61no—
396C22⋊C4—64no—
397C22⋊C4—64no—
398C42—21no—
399C42—32no—
400C42—42no—
401C42—62no—
402C42—122no—
403C42—128no—
404C42C2×C4×D311yescharacteristic
405C42C2×C4×D311yescharacteristic
406C22×C3—22no—
407C22×C3—21no—
408C22×C3—42no—
409C22×C3—41no—
410C22×C3—42no—
411C22×C3—41no—
412C22×C3C22×C22⋊C411yescharacteristic
413C2×D3—22no—
414C2×D3—42no—
415C2×D3—44no—
416C2×D3—42no—
417C2×D3—164no—
418C2×D3—162no—
419C3×C4—22no—
420C3×C4—21no—
421C3×C4—41no—
422C3×C4—41no—
423C3×C4—41no—
424C3×C4—84no—
425C3×C4C2×#9911yescharacteristic
426C3×C4C2×#9911yescharacteristic
427Dic3—24no—
428Dic3—42no—
429Dic3—84no—
430Dic3C2×#9914yes—
431C23—21no—
432C23—32no—
433C23—62no—
434C23—122no—
435C23—124no—
436C23—242no—
437C23—244no—
438C23D3×C22⋊C411yescharacteristic
439C23C2×D3×D412yes—
440Q8—32no—
441Q8—32no—
442Q8—41no—
443Q8—41no—
444Q8—68no—
445Q8—62no—
446Q8—62no—
447Q8—122no—
448D4—42no—
449D4—41no—
450D4—41no—
451D4—42no—
452D4—64no—
453D4—61no—
454D4—61no—
455D4—82no—
456D4—122no—
457D4—124no—
458D4—124no—
459D4—124no—
460D4—121no—
461D4—121no—
462D4—244no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—32no—
471C2×C4—34no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—44no—
475C2×C4—64no—
476C2×C4—68no—
477C2×C4—62no—
478C2×C4—81no—
479C2×C4—124no—
480C2×C4—124no—
481C2×C4—122no—
482C2×C4—124no—
483C2×C4—242no—
484C2×C4D3×C22⋊C412yes—
485C2×C4D3×C22⋊C411yescharacteristic
486C2×C4C22×C4×D311yescharacteristic
487C2×C3—21no—
488C2×C3—42no—
489C2×C3—41no—
490C2×C3—81no—
491C2×C3C22×#9911yescharacteristic
492D3—42no—
493D3—82no—
494C22—22no—
495C22—21no—
496C22—41no—
497C22—42no—
498C22—42no—
499C22—41no—
500C22—62no—
501C22—122no—
502C22—124no—
503C22—122no—
504C22—482no—
505C22—484no—
506C22C2×D3×C22⋊C411yescharacteristic
507C4—21no—
508C4—22no—
509C4—34no—
510C4—41no—
511C4—41no—
512C4—41no—
513C4—64no—
514C4—84no—
515C4—122no—
516C4—244no—
517C4D3×#9911yescharacteristic
518C4D3×#9911yescharacteristic
519C3#3336211yescharacteristic, complemented, Sylow
520C2—21no—
521C2—42no—
522C2—41no—
523C2—81no—
524C2—122no—
525C2—242no—
526C2C2×D3×#9911yescharacteristic
subgroupG/H
centerC2C2×D3×#99
commutatorC2×C3×C4C23×C4
FrattiniC2×D4C23×D3
FittingC3×#2126C2
socleC2×C3C22×#99
prime radical#86244C2
cosocleC2×C3×D4C24
hypercenter#2126D3
hypocenterC3#33362
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #2126

Complements

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

NH ≅ G/Ntype
2. #78689C2semidirect
3. C3×#2126C2semidirect
4. #71819C2semidirect
5. #70061C2semidirect
6. #69158C2semidirect
7. #69152C2semidirect
10. C2×#1009C4semidirect
13. #5825C4semidirect
14. #5724C4semidirect
16. D3×#124C4semidirect
22. C2×#875C4semidirect
23. C4×#235C4semidirect
24. C3×#410C22semidirect
25. #5240C22semidirect
28. #4740C22semidirect
29. C3×#354C22semidirect
30. C3×#353C22semidirect
31. #4545C22semidirect
32. #4541C22semidirect
33. #4474C22semidirect
40. #2126D3semidirect
122. #410C2×D3semidirect
124. #354C2×D3semidirect
127. #353C2×D3semidirect
519. C3#33362semidirect

Maximal subgroups

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

subgroupindex
#862442
#786892
C3×#21262
#718192
#700612
#691582
#691522
#333623