Finite Group Databasebeta

← #10686241

Subgroups — #10686241

Order 768.

436 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83126C211yesmaximal, characteristic
2C3×#1500C211yesmaximal, characteristic
3#71746C211yesmaximal, characteristic
4#70741C211yesmaximal, characteristic, complemented
5#69434C211yesmaximal, characteristic, complemented
6#67742C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#8249—31nomaximal, Sylow
9C3×#529C2211yescharacteristic
10C3×#431C2211yescharacteristic
11C3×#376C2211yescharacteristic
12#5024C411yescharacteristic, complemented
13Q8×C3⋊C8C411yescharacteristic, complemented
14#5012C411yescharacteristic, complemented
15#5011—21no—
16D4×C3⋊C8—21no—
17D4×C3⋊C8C411yescharacteristic, complemented
18#5000—21no—
19#4997—21no—
20#4926C2211yescharacteristic
21C4×#208C2211yescharacteristic
22C2×#803—21no—
23C2×#803C2211yescharacteristic
24C2×C4×C3⋊C8—21no—
25C2×C4×C3⋊C8C2211yescharacteristic
26C3×#328C412yes—
27C3×#327C412yes—
28#4543C412yescomplemented
29#4540C412yescomplemented
30#4455—21no—
31#4454—21no—
32#2966—31no—
33#1726—31no—
34#1524—31no—
35#1500D311yescharacteristic
36#1451—31no—
37#1302—31no—
38#1300—31no—
39C3×#126—21no—
40C3×#125C2×C411yescharacteristic
41C3×#124C2×C411yescharacteristic
42C3×#123—21no—
43C3×C22⋊Q8C2×C411yescharacteristic
44C3×#121C2×C411yescharacteristic
45C3×C4×Q8C2×C411yescharacteristic
46C3×C4×D4—21no—
47C3×C4×D4C2×C411yescharacteristic
48C3×#117—21no—
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#208—21no—
52C2×#208—21no—
53C22×C3⋊C8—21no—
54C22×C3⋊C8—21no—
55C22×C3⋊C8D411yescharacteristic
56C22×C3⋊C8D411yescharacteristic
57C3×C4⋊C8—21no—
58C3×C22⋊C8C2×C412yes—
59C3×C4×C8—21no—
60C3×#95—21no—
61C3×#95C2×C412yes—
62#830—21no—
63#830—21no—
64#830—21no—
65C4⋊(C3⋊C8)—21no—
66C4⋊(C3⋊C8)—21no—
67C4⋊(C3⋊C8)C2×C411yescharacteristic
68C4⋊(C3⋊C8)C2×C411yescharacteristic
69#803—24no—
70#803—21no—
71#803—21no—
72#803—21no—
73#803C2×C412yes—
74#803C2×C412yes—
75C4×C3⋊C8—24no—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78C4×C3⋊C8C2×C411yescharacteristic
79C4×C3⋊C8C2×C411yescharacteristic
80C4×C3⋊C8D411yescharacteristic
81C4×C3⋊C8D411yescharacteristic
82#529C2×D311yescharacteristic
83#446—31no—
84C8×Q8—31no—
85#436—31no—
86#435—61no—
87C8×D4—31no—
88C8×D4—61no—
89#433—61no—
90#432—31no—
91#431C2×D311yescharacteristic
92#405—61no—
93C4×M16—31no—
94C2×#97—31no—
95C2×#97—61no—
96C2×C4×C8—31no—
97C2×C4×C8—61no—
98#376C2×D311yescharacteristic
99#330—32no—
100#329—32no—
101#328Dic312yes—
102#327Dic312yes—
103#322—61no—
104#322—61no—
105C2×C3×Q8C4211yescharacteristic
106C2×C3×D4—21no—
107C2×C3×D4C4211yescharacteristic
108C22×C3×C4—22no—
109C22×C3×C4—21no—
110C22×C3×C4C22×C411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C2×C3×C8—22no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4C4211yescharacteristic
119C3×C4⋊C4C4211yescharacteristic
120C3×C22⋊C4—21no—
121C3×C22⋊C4—21no—
122C3×C22⋊C4—21no—
123C3×C22⋊C4—21no—
124C3×C22⋊C4—21no—
125C3×C42—22no—
126C3×C42—22no—
127C3×C42—21no—
128C3×C42C22×C411yescharacteristic
129C3×C42C22×C411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132#208—42no—
133#208—42no—
134C2×C3⋊C8—22no—
135C2×C3⋊C8—22no—
136C2×C3⋊C8—22no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—21no—
140C2×C3⋊C8—21no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—42no—
143C2×C3⋊C8C22⋊C411yescharacteristic
144C2×C3⋊C8C22⋊C411yescharacteristic
145C2×C3⋊C8C22⋊C411yescharacteristic
146C2×C3⋊C8C22⋊C411yescharacteristic
147C2×M16—61no—
148C2×M16—61no—
149C22×C8—31no—
150C22×C8—31no—
151C22×C8—61no—
152C22×C8—61no—
153#126—21no—
154#125C4×D311yescharacteristic
155#124C4×D311yescharacteristic
156#123—21no—
157C22⋊Q8C2×Dic311yescharacteristic
158#121C2×Dic311yescharacteristic
159C4×Q8C4×D311yescharacteristic
160C4×D4—21no—
161C4×D4C4×D311yescharacteristic
162#117—21no—
163C2×C42—21no—
164C2×C42C22×D311yescharacteristic
165C4⋊C8—21no—
166C4⋊C8—31no—
167C4⋊C8—31no—
168C4⋊C8—61no—
169C4⋊C8—61no—
170C22⋊C8—61no—
171C22⋊C8—61no—
172C22⋊C8—61no—
173C22⋊C8C2×Dic312yes—
174#97—32no—
175#97—32no—
176#97—61no—
177#97—64no—
178#97—61no—
179#97—61no—
180C4×C8—21no—
181C4×C8—31no—
182C4×C8—31no—
183C4×C8—31no—
184C4×C8—31no—
185C4×C8—61no—
186C4×C8—64no—
187C4×C8—61no—
188#95—21no—
189#95C2×Dic312yes—
190C23×C3—21no—
191C23×C3C4×D411yescharacteristic
192C3×Q8—41no—
193C3×D4—41no—
194C3×D4—42no—
195C2×C3×C4—22no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—24no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4C2×C4211yescharacteristic
208C2×C3×C4C2×C22⋊C411yescharacteristic
209C2×C3×C4#11711yescharacteristic
210C2×C3×C4C4×D411yescharacteristic
211C2×C3×C4C4×D411yescharacteristic
212C2×C3×C4C4×D411yescharacteristic
213C3×C8—42no—
214C3⋊C8—24no—
215C3⋊C8—21no—
216C3⋊C8—21no—
217C3⋊C8—42no—
218C3⋊C8—41no—
219C3⋊C8—41no—
220C3⋊C8#10412yes—
221C3⋊C8#10412yes—
222C2×Q8C4×Dic311yescharacteristic
223C2×D4—21no—
224C2×D4C4×Dic311yescharacteristic
225C22×C4—22no—
226C22×C4—21no—
227C22×C4D3×D411yescharacteristic
228C22×C4#23711yescharacteristic
229C22×C4C22×Dic311yescharacteristic
230M16—122no—
231M16—122no—
232C2×C8—22no—
233C2×C8—31no—
234C2×C8—31no—
235C2×C8—31no—
236C2×C8—31no—
237C2×C8—62no—
238C2×C8—62no—
239C2×C8—62no—
240C2×C8—62no—
241C2×C8—62no—
242C2×C8—61no—
243C2×C8—61no—
244C2×C8—62no—
245C2×C8—122no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4C4×Dic311yescharacteristic
251C4⋊C4C4×Dic311yescharacteristic
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—21no—
257C42—22no—
258C42—22no—
259C42—21no—
260C42C2×C4×D311yescharacteristic
261C42C2×C4×D311yescharacteristic
262C42D3×D411yescharacteristic
263C42#23711yescharacteristic
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3C4×C22⋊C411yescharacteristic
270C22×C3#44412yes—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—44no—
277C3×C4—41no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4C2×#10411yescharacteristic
281C3×C4C2×#10411yescharacteristic
282C23—21no—
283C23D4×Dic311yescharacteristic
284Q8—41no—
285D4—41no—
286D4—42no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4#87711yescharacteristic
300C2×C4D3×C22⋊C411yescharacteristic
301C2×C4#67⋊C411yescharacteristic
302C2×C4#67⋊C411yescharacteristic
303C2×C4C2×C4×Dic311yescharacteristic
304C2×C4D4×Dic311yescharacteristic
305C8—32no—
306C8—32no—
307C8—42no—
308C8—64no—
309C8—61no—
310C8—61no—
311C8—122no—
312C8—121no—
313C8—121no—
314C2×C3—22no—
315C2×C3—81no—
316C2×C3#175711yescharacteristic
317C2×C3C4×#10411yescharacteristic
318C2×C3#176511yescharacteristic
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22Dic3×C22⋊C411yescharacteristic
325C22#519712yes—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—44no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4D3×#10411yescharacteristic
336C4D3×#10411yescharacteristic
337C3#824911yescharacteristic, complemented, Sylow
338C2—22no—
339C2—81no—
340C2#7004011yescharacteristic
341C2#7029011yescharacteristic
342C2Dic3×#10411yescharacteristic
subgroupG/H
centerC2×C4D3×C22⋊C4
commutatorC2×C3×C4C2×C42
FrattiniC2×C42C22×D3
FittingC3×#1500C2
socleC22×C3C4×C22⋊C4
prime radicalC3×#121C2×C4
cosocleC2×C3×C42C23
hypercenter#1500D3
hypocenterC3#8249
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1500

Complements

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

NH ≅ G/Ntype
4. #70741C2semidirect
5. #69434C2semidirect
6. #67742C2semidirect
7. #67615C2semidirect
12. #5024C4semidirect
13. Q8×C3⋊C8C4semidirect
14. #5012C4semidirect
17. D4×C3⋊C8C4semidirect
28. #4543C4semidirect
29. #4540C4semidirect
337. C3#8249semidirect

Maximal subgroups

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

subgroupindex
#831262
C3×#15002
#717462
#707412
#694342
#677422
#676152
#82493