Finite Group Databasebeta

← #10686248

Subgroups — #10686248

Order 768.

400 conjugacy classes of subgroups.

The subgroup lattice has 340 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#70739C211yesmaximal, characteristic, complemented
5#69435C211yesmaximal, characteristic, complemented
6#67742C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#8250—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×#803C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24C3×#328C412yes—
25C3×#327C412yes—
26#4543C412yescomplemented
27#4540C412yescomplemented
28#4524—21no—
29#4524—21no—
30#4454—21no—
31C8×C3⋊C8—21no—
32#2966—31no—
33#1746—31no—
34#1524—31no—
35#1500D311yescharacteristic
36C8×M16—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—21no—
70#803—21no—
71#803—21no—
72#803C2×C412yes—
73#803C2×C412yes—
74C4×C3⋊C8—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8C2×C411yescharacteristic
77C4×C3⋊C8C2×C411yescharacteristic
78C4×C3⋊C8D411yescharacteristic
79C4×C3⋊C8D411yescharacteristic
80#529C2×D311yescharacteristic
81#446—31no—
82C8×Q8—31no—
83#436—31no—
84#435—61no—
85C8×D4—31no—
86C8×D4—61no—
87#433—61no—
88#432—31no—
89#431C2×D311yescharacteristic
90#405—61no—
91C4×M16—31no—
92C2×#97—31no—
93C2×C4×C8—31no—
94#376C2×D311yescharacteristic
95#336—61no—
96#336—61no—
97#330—32no—
98#329—32no—
99#328Dic312yes—
100#327Dic312yes—
101#322—61no—
102C82—61no—
103C2×C3×Q8C4211yescharacteristic
104C2×C3×D4—21no—
105C2×C3×D4C4211yescharacteristic
106C22×C3×C4—21no—
107C22×C3×C4—22no—
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C2×C3×C8—22no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4C4211yescharacteristic
117C3×C4⋊C4C4211yescharacteristic
118C3×C22⋊C4—21no—
119C3×C22⋊C4—21no—
120C3×C22⋊C4—21no—
121C3×C22⋊C4—21no—
122C3×C22⋊C4—21no—
123C3×C42—21no—
124C3×C42—41no—
125C3×C42—41no—
126C3×C42C22×C411yescharacteristic
127C3×C42C22×C411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130#208—24no—
131#208—42no—
132C2×C3⋊C8—22no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—22no—
136C2×C3⋊C8—41no—
137C2×C3⋊C8—41no—
138C2×C3⋊C8—42no—
139C2×C3⋊C8—41no—
140C2×C3⋊C8—41no—
141C2×C3⋊C8C22⋊C411yescharacteristic
142C2×C3⋊C8C22⋊C411yescharacteristic
143C2×C3⋊C8C22⋊C411yescharacteristic
144C2×C3⋊C8C22⋊C411yescharacteristic
145C2×M16—61no—
146C2×M16—61no—
147C22×C8—31no—
148C22×C8—31no—
149C22×C8—61no—
150C22×C8—61no—
151#126—21no—
152#125C4×D311yescharacteristic
153#124C4×D311yescharacteristic
154#123—21no—
155C22⋊Q8C2×Dic311yescharacteristic
156#121C2×Dic311yescharacteristic
157C4×Q8C4×D311yescharacteristic
158C4×D4—21no—
159C4×D4C4×D311yescharacteristic
160#117—21no—
161C2×C42—21no—
162C2×C42C22×D311yescharacteristic
163C4⋊C8—21no—
164C4⋊C8—31no—
165C4⋊C8—31no—
166C4⋊C8—61no—
167C4⋊C8—61no—
168C22⋊C8—61no—
169C22⋊C8—61no—
170C22⋊C8—61no—
171C22⋊C8C2×Dic312yes—
172#97—32no—
173#97—32no—
174#97—61no—
175#97—61no—
176#97—61no—
177C4×C8—21no—
178C4×C8—31no—
179C4×C8—31no—
180C4×C8—31no—
181C4×C8—31no—
182C4×C8—61no—
183C4×C8—61no—
184#95—21no—
185#95C2×Dic312yes—
186C23×C3—21no—
187C23×C3C4×D411yescharacteristic
188C3×Q8—41no—
189C3×D4—41no—
190C3×D4—42no—
191C2×C3×C4—22no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—42no—
203C2×C3×C4C2×C4211yescharacteristic
204C2×C3×C4C2×C22⋊C411yescharacteristic
205C2×C3×C4#11711yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4C4×D411yescharacteristic
209C3×C8—42no—
210C3⋊C8—24no—
211C3⋊C8—21no—
212C3⋊C8—21no—
213C3⋊C8—21no—
214C3⋊C8—21no—
215C3⋊C8—42no—
216C3⋊C8—41no—
217C3⋊C8—41no—
218C2×Q8C4×Dic311yescharacteristic
219C2×D4—21no—
220C2×D4C4×Dic311yescharacteristic
221C22×C4—21no—
222C22×C4—22no—
223C22×C4D3×D411yescharacteristic
224C22×C4#23711yescharacteristic
225C22×C4C22×Dic311yescharacteristic
226M16—64no—
227M16—122no—
228C2×C8—22no—
229C2×C8—31no—
230C2×C8—31no—
231C2×C8—31no—
232C2×C8—31no—
233C2×C8—62no—
234C2×C8—61no—
235C2×C8—61no—
236C2×C8—62no—
237C2×C8—121no—
238C2×C8—121no—
239C2×C8—122no—
240C2×C8—121no—
241C2×C8—121no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4C4×Dic311yescharacteristic
247C4⋊C4C4×Dic311yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—21no—
250C22⋊C4—21no—
251C22⋊C4—21no—
252C22⋊C4—21no—
253C42—21no—
254C42—41no—
255C42—41no—
256C42C2×C4×D311yescharacteristic
257C42C2×C4×D311yescharacteristic
258C42D3×D411yescharacteristic
259C42#23711yescharacteristic
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3C4×C22⋊C411yescharacteristic
266C22×C3#44312yes—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—82no—
276C3×C4#42111yescharacteristic
277C3×C4#42111yescharacteristic
278C23—21no—
279C23D4×Dic311yescharacteristic
280Q8—41no—
281D4—41no—
282D4—42no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4#87711yescharacteristic
296C2×C4D3×C22⋊C411yescharacteristic
297C2×C4#67⋊C411yescharacteristic
298C2×C4#67⋊C411yescharacteristic
299C2×C4C2×C4×Dic311yescharacteristic
300C2×C4D4×Dic311yescharacteristic
301C8—42no—
302C8—64no—
303C8—61no—
304C8—61no—
305C8—61no—
306C8—61no—
307C8—122no—
308C8—121no—
309C8—121no—
310C2×C3—22no—
311C2×C3—81no—
312C2×C3#175711yescharacteristic
313C2×C3#175911yescharacteristic
314C2×C3#176411yescharacteristic
315C22—21no—
316C22—21no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22Dic3×C22⋊C411yescharacteristic
321C22#519612yes—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—82no—
331C4#482111yescharacteristic
332C4#482111yescharacteristic
333C3#825011yescharacteristic, complemented, Sylow
334C2—22no—
335C2—81no—
336C2#7004011yescharacteristic
337C2#7028911yescharacteristic
338C2#7035411yescharacteristic
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#8250
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. #70739C2semidirect
5. #69435C2semidirect
6. #67742C2semidirect
7. #67615C2semidirect
12. #5024C4semidirect
13. Q8×C3⋊C8C4semidirect
14. #5012C4semidirect
17. D4×C3⋊C8C4semidirect
26. #4543C4semidirect
27. #4540C4semidirect
333. C3#8250semidirect

Maximal subgroups

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

subgroupindex
#831262
C3×#15002
#717462
#707392
#694352
#677422
#676152
#82503