Finite Group Databasebeta

← #10947995

Subgroups — #10947995

Order 768.

1188 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86340C211yesmaximal, characteristic
2#78689C212yesmaximal, complemented
3C3×#2129C211yesmaximal, characteristic, complemented
4#71871C211yesmaximal, characteristic, complemented
5#70060C212yesmaximal, complemented
6#69159C214yesmaximal, complemented
7#69158C214yesmaximal, complemented
8#33360—31nomaximal, Sylow
9C2×#1010C2211yescharacteristic
10C2×#1007C411yescharacteristic, complemented
11C3×#536C2211yescharacteristic
12#5832C2211yescharacteristic
13C2×#944C411yescharacteristic, complemented
14#5724C412yescomplemented
15D3×C4⋊Q8C412yescomplemented
16#5675C2212yes—
17#5668C2212yes—
18#5655—24no—
19#5632—22no—
20#5631—22no—
21#5526C411yescharacteristic, complemented
22#5522C411yescharacteristic, complemented
23#5516—22no—
24C3×#410C2212yes—
25#5240C2212yes—
26#5006—22no—
27#4740—24no—
28#4740C2218yes—
29C3×#354C2214yes—
30#4541C2214yes—
31#4474C2214yes—
32#4473C2214yes—
33#3499—31no—
34#2884—32no—
35#2129—34no—
36#2129—31no—
37#2129D311yescharacteristic, complemented
38#2126—32no—
39#2126—34no—
40C2×C3×#41C2311yescharacteristic
41C22×C3×Q8C2×C411yescharacteristic
42#1010—44no—
43#1007—44no—
44C2×#239—21no—
45C2×#239C2×C412yes—
46C2×D3×Q8—21no—
47C2×D3×Q8C2×C412yes—
48C2×D3×Q8C2×C412yes—
49C2×#237D414yes—
50C2×#237C2312yes—
51C2×#235—22no—
52C2×#235C2×C411yescharacteristic
53C2×#235C2×C411yescharacteristic
54C2×#235D412yes—
55C22×Dic6C2×C411yescharacteristic
56C22×Dic6D412yes—
57C3×C4⋊Q8C2×C412yes—
58C3×#124C2312yes—
59C3×#123—22no—
60C3×C22⋊Q8—22no—
61C3×#117—22no—
62C3×#117C2×C411yescharacteristic
63#947C2×C411yescharacteristic
64#944—22no—
65#944C2×C412yes—
66#944C2×C412yes—
67#944C2×C411yescharacteristic
68#940C2311yescharacteristic
69#939—22no—
70#936C2311yescharacteristic
71C4×#67—22no—
72#926—22no—
73C2×Dic3⋊C4—22no—
74C2×C4×Dic3C2×C411yescharacteristic
75#897—22no—
76#896—24no—
77#894—24no—
78#893—21no—
79D3×C4⋊C4—24no—
80#888—24no—
81#887—21no—
82#887—44no—
83#886—22no—
84#884—24no—
85#883—24no—
86#67⋊C4—22no—
87#67⋊C4—24no—
88#878—24no—
89#877—28no—
90#875C2×C412yes—
91#872C2×C412yes—
92C42×D3C2×C412yes—
93#869C2×C412yes—
94C4×Dic6—44no—
95C3×#99—22no—
96C3×#99C2314yes—
97#834—22no—
98#834C2314yes—
99#806—28no—
100C2×2−1+4—31no—
101C2×2−1+4—31no—
102D4×Q8—62no—
103#547—62no—
104#547—64no—
105#536—32no—
106#536—32no—
107#536—31no—
108#536C2×D311yescharacteristic
109#532—31no—
110#532—32no—
111C2×C4⋊Q8—31no—
112C2×C4⋊Q8—32no—
113#519—62no—
114C4×#41—31no—
115#410—32no—
116#410—38no—
117#410—64no—
118#410—62no—
119#410C2×D312yes—
120#354—34no—
121#354—34no—
122#354C2×D314yes—
123#353—34no—
124C3×#41—42no—
125C2×C3×Q8—21no—
126C2×C3×Q8—42no—
127C2×C3×Q8C22×C412yes—
128C2×C3×Q8C22×C411yescharacteristic
129C2×C3×D4C2×D412yes—
130C2×C3×D4C2411yescharacteristic
131C22×C3×C4—21no—
132C22×C3×C4C22×C411yescharacteristic
133C22×C3×C4C2×D412yes—
134C2×#67—21no—
135C2×#67—22no—
136C2×#67C22⋊C412yes—
137C2×#67C2×D414yes—
138C22×Dic3—21no—
139C22×Dic3C22⋊C412yes—
140C22×Dic3C2×D414yes—
141#239—44no—
142#239—44no—
143D3×Q8—44no—
144D3×Q8—44no—
145D3×Q8—44no—
146#237—48no—
147#237—44no—
148#235—24no—
149#235—42no—
150#235—42no—
151#235—48no—
152#235—42no—
153C2×D12—22no—
154C2×D12C22⋊C411yescharacteristic
155C2×D12C22×C411yescharacteristic
156C2×C4×D3—22no—
157C2×C4×D3—22no—
158C2×C4×D3—21no—
159C2×C4×D3—22no—
160C2×C4×D3C22⋊C414yes—
161C2×C4×D3C22×C412yes—
162C2×Dic6—24no—
163C2×Dic6—24no—
164C2×Dic6—22no—
165C2×Dic6—42no—
166C2×Dic6C22⋊C414yes—
167C2×Dic6C22⋊C412yes—
168C2×Dic6C22⋊C411yescharacteristic
169C2×Dic6C22×C411yescharacteristic
170C2×Dic6C22×C412yes—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—42no—
174C3×C22⋊C4—21no—
175C3×C22⋊C4—24no—
176C3×C22⋊C4—24no—
177C3×C42—42no—
178C3×C42C22×C412yes—
179#217—22no—
180#217—24no—
181(C2×D3)⋊C4—22no—
182(C2×D3)⋊C4—22no—
183(C2×D3)⋊C4—24no—
184C4⋊Dic3—24no—
185C4⋊Dic3—42no—
186Dic3⋊C4—22no—
187Dic3⋊C4—22no—
188Dic3⋊C4—24no—
189Dic3⋊C4—24no—
190Dic3⋊C4—48no—
191C4×Dic3—21no—
192C4×Dic3—22no—
193C4×Dic3—48no—
194C4×Dic3—42no—
195C4×Dic3C22×C411yescharacteristic
196C4×Dic3C22×C411yescharacteristic
1972−1+4—124no—
1982−1+4—124no—
199C2×#41—34no—
200C2×#41—32no—
201C2×#41—31no—
202C2×#41—32no—
203C2×#41—32no—
204C2×#41—31no—
205C2×#41—62no—
206C2×#41—61no—
207C2×#41C22×D311yescharacteristic
208C22×Q8—32no—
209C22×Q8—31no—
210C22×Q8—32no—
211C22×Q8—32no—
212C22×Q8—61no—
213C22×Q8C4×D311yescharacteristic
214C4⋊Q8—32no—
215C4⋊Q8—31no—
216C4⋊Q8—32no—
217C4⋊Q8—32no—
218C4⋊Q8—64no—
219C4⋊Q8—62no—
220C4⋊Q8C4×D312yes—
221#127—31no—
222#124—31no—
223#124—31no—
224#124—32no—
225#124C22×D312yes—
226#123—22no—
227#123—62no—
228#123—64no—
229#123—64no—
230C22⋊Q8—22no—
231C22⋊Q8—64no—
232C22⋊Q8—64no—
233C22⋊Q8—62no—
234#121—64no—
235#121—62no—
236C4×Q8—61no—
237C4×Q8—124no—
238C4×Q8—124no—
239C4×D4—64no—
240C4×D4—62no—
241C4×D4—61no—
242C4×D4—62no—
243#117—22no—
244#117—32no—
245#117—68no—
246#117—62no—
247#117C4×D311yescharacteristic
248C2×C4⋊C4—64no—
249C2×C4⋊C4—62no—
250C2×C42—31no—
251C2×C42—32no—
252#99—22no—
253#99—34no—
254#99—68no—
255#99—62no—
256#99C22×D314yes—
257C23×C3C2×C22⋊C411yescharacteristic
258C23×C3C22×D412yes—
259C22×D3—21no—
260C22×D3C2×C22⋊C412yes—
261C3×Q8—41no—
262C3×Q8—42no—
263C3×Q8—42no—
264C3×D4—42no—
265C3×D4—41no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—42no—
272C2×C3×C4—44no—
273C2×C3×C4C2×C22⋊C412yes—
274C2×C3×C4C2×C22⋊C411yescharacteristic
275C2×C3×C4C23×C411yescharacteristic
276#67—24no—
277#67—44no—
278#67—44no—
279#67—42no—
280C2×Dic3—21no—
281C2×Dic3—28no—
282C2×Dic3—21no—
283C2×Dic3—21no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—44no—
287C2×Dic3C2×C22⋊C414yes—
288C2×Dic3C2×C22⋊C412yes—
289D12—22no—
290D12—41no—
291D12—44no—
292C4×D3—24no—
293C4×D3—44no—
294C4×D3—44no—
295C4×D3—44no—
296C4×D3—42no—
297C4×D3—44no—
298Dic6—22no—
299Dic6—22no—
300Dic6—28no—
301Dic6—42no—
302Dic6—44no—
303Dic6—41no—
304Dic6#9914yes—
305#41—42no—
306#41—64no—
307#41—122no—
308#41—122no—
309#41—122no—
310#41—128no—
311#41—124no—
312#41—128no—
313#41—124no—
314#41—124no—
315C2×Q8—21no—
316C2×Q8—34no—
317C2×Q8—31no—
318C2×Q8—32no—
319C2×Q8—32no—
320C2×Q8—31no—
321C2×Q8—42no—
322C2×Q8—62no—
323C2×Q8—64no—
324C2×Q8—64no—
325C2×Q8—122no—
326C2×Q8—124no—
327C2×Q8—124no—
328C2×Q8—124no—
329C2×Q8C2×C4×D312yes—
330C2×Q8C2×C4×D311yescharacteristic
331C2×D4—32no—
332C2×D4—34no—
333C2×D4—31no—
334C2×D4—31no—
335C2×D4—62no—
336C2×D4—61no—
337C2×D4—62no—
338C2×D4D3×D412yes—
339C2×D4C23×D311yescharacteristic
340C22×C4—21no—
341C22×C4—34no—
342C22×C4—32no—
343C22×C4—32no—
344C22×C4—34no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4C2×C4×D311yescharacteristic
351C22×C4D3×D412yes—
352C4⋊C4—24no—
353C4⋊C4—21no—
354C4⋊C4—42no—
355C4⋊C4—64no—
356C4⋊C4—64no—
357C4⋊C4—64no—
358C4⋊C4—62no—
359C4⋊C4—62no—
360C4⋊C4—122no—
361C4⋊C4—128no—
362C22⋊C4—21no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—64no—
366C22⋊C4—62no—
367C22⋊C4—62no—
368C22⋊C4—62no—
369C22⋊C4—64no—
370C42—31no—
371C42—31no—
372C42—42no—
373C42—62no—
374C42—61no—
375C42—122no—
376C42—128no—
377C42C2×C4×D312yes—
378C22×C3—22no—
379C22×C3—21no—
380C22×C3—41no—
381C22×C3—42no—
382C22×C3C22×C22⋊C411yescharacteristic
383C2×D3—22no—
384C2×D3—42no—
385C2×D3—42no—
386C2×D3—41no—
387C3×C4—22no—
388C3×C4—21no—
389C3×C4—42no—
390C3×C4—42no—
391C3×C4—84no—
392C3×C4C2×#9912yes—
393Dic3—22no—
394Dic3—24no—
395Dic3—41no—
396Dic3—41no—
397Dic3—84no—
398Dic3C2×#9914yes—
399C23—32no—
400C23—61no—
401C23D3×C22⋊C411yescharacteristic
402C23C2×D3×D412yes—
403Q8—34no—
404Q8—41no—
405Q8—42no—
406Q8—42no—
407Q8—68no—
408Q8—62no—
409Q8—62no—
410Q8—122no—
411Q8—121no—
412Q8—124no—
413D4—42no—
414D4—41no—
415D4—64no—
416D4—62no—
417D4—121no—
418D4—124no—
419D4—124no—
420D4—124no—
421D4—122no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—32no—
428C2×C4—34no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—64no—
432C2×C4—68no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—61no—
436C2×C4—124no—
437C2×C4—124no—
438C2×C4—124no—
439C2×C4—124no—
440C2×C4—122no—
441C2×C4—122no—
442C2×C4—122no—
443C2×C4—124no—
444C2×C4D3×C22⋊C412yes—
445C2×C4D3×C22⋊C411yescharacteristic
446C2×C4C22×C4×D311yescharacteristic
447C2×C3—21no—
448C2×C3—42no—
449C2×C3—41no—
450C2×C3C22×#9911yescharacteristic
451D3—42no—
452D3—81no—
453C22—22no—
454C22—21no—
455C22—41no—
456C22—42no—
457C22—62no—
458C22—122no—
459C22—122no—
460C22—121no—
461C22C2×D3×C22⋊C411yescharacteristic
462C4—21no—
463C4—22no—
464C4—34no—
465C4—42no—
466C4—42no—
467C4—62no—
468C4—64no—
469C4—84no—
470C4—121no—
471C4—121no—
472C4—244no—
473C4D3×#9912yes—
474C3#3336011yescharacteristic, complemented, Sylow
475C2—21no—
476C2—42no—
477C2—41no—
478C2—122no—
479C2—241no—
480C2C2×D3×#9911yescharacteristic
subgroupG/H
centerC2C2×D3×#99
commutatorC2×C3×C4C23×C4
FrattiniC2×D4C23×D3
FittingC3×#2129C2
socleC2×C3C22×#99
prime radical#86340C2
cosocleC2×C3×D4C24
hypercenter#2129D3
hypocenterC3#33360
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #2129

Complements

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

NH ≅ G/Ntype
2. #78689C2semidirect
3. C3×#2129C2semidirect
4. #71871C2semidirect
5. #70060C2semidirect
6. #69159C2semidirect
7. #69158C2semidirect
10. C2×#1007C4semidirect
13. C2×#944C4semidirect
14. #5724C4semidirect
15. D3×C4⋊Q8C4semidirect
21. #5526C4semidirect
22. #5522C4semidirect
37. #2129D3semidirect
474. C3#33360semidirect

Maximal subgroups

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

subgroupindex
#863402
#786892
C3×#21292
#718712
#700602
#691592
#691582
#333603