Finite Group Databasebeta

← #10692670

Subgroups — #10692670

Order 768.

380 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79317C211yesmaximal, characteristic
2#78540C211yesmaximal, characteristic
3#70837C211yesmaximal, characteristic
4#70837C211yesmaximal, characteristic
5#70815C211yesmaximal, characteristic
6C3×#1291C211yesmaximal, characteristic
7#67666C211yesmaximal, characteristic
8#9416—31nomaximal, Sylow
9#5470C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5137—22no—
15#4937C2211yescharacteristic
16#4926C2211yescharacteristic
17#4828—22no—
18#4720—22no—
19#4716—22no—
20#4701—22no—
21#4699C411yescharacteristic, complemented
22#4681C411yescharacteristic, complemented
23#4681C411yescharacteristic, complemented
24#4680C411yescharacteristic, complemented
25#4467—22no—
26#4467—22no—
27#4465—22no—
28#3185—31no—
29#2987—31no—
30#1537—31no—
31#1515—31no—
32#1515—31no—
33#1291D311yescharacteristic
34#1288—31no—
35C2×C3×M16—21no—
36C2×C3×C42C2311yescharacteristic
37#926—21no—
38#926—21no—
39#924—21no—
40#924—21no—
41#924—21no—
42#924D412yes—
43C2×Dic12—22no—
44#870—21no—
45#870—21no—
46#869C2×C411yescharacteristic
47#869C2×C411yescharacteristic
48#869C2×C411yescharacteristic
49#869C2×C411yescharacteristic
50C4×Dic6—21no—
51C4×Dic6—21no—
52C4×Dic6—21no—
53C4×Dic6D412yes—
54C3×C4⋊C8—21no—
55C3×C4⋊C8—21no—
56C3×C22⋊C8—21no—
57C3×C22⋊C8D412yes—
58C3×#97—21no—
59C3×#97—21no—
60C3×#97—21no—
61C3×C4×C8C2×C411yescharacteristic
62C3×C4×C8C2×C411yescharacteristic
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66#830—21no—
67#817—22no—
68#816—22no—
69#816—22no—
70#815—22no—
71Dic3⋊C8—24no—
72C8×Dic3—22no—
73C4⋊(C3⋊C8)—21no—
74#803D412yes—
75#533—31no—
76#533—31no—
77#498—31no—
78#494—31no—
79#487—31no—
80#487—31no—
81#470—62no—
82#446—62no—
83#446—62no—
84#441—62no—
85#433—62no—
86#432—31no—
87#432C2×D311yescharacteristic
88#432C2×D311yescharacteristic
89C4×M16C2×D311yescharacteristic
90#383—31no—
91#330—62no—
92#328—62no—
93#328—62no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4#4111yescharacteristic
97C22×Dic3—41no—
98C2×Dic6—21no—
99C2×Dic6—21no—
100C2×Dic6—21no—
101C2×Dic6C22⋊C412yes—
102C3×M16—42no—
103C2×C3×C8—21no—
104C2×C3×C8—21no—
105C2×C3×C8—22no—
106C2×C3×C8—41no—
107C3×C42C22×C411yescharacteristic
108C3×C42C2×D411yescharacteristic
109C3×C42C2×D411yescharacteristic
110C3×C42#4111yescharacteristic
111#217—22no—
112#217—41no—
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3—21no—
116C4⋊Dic3—21no—
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3C22⋊C412yes—
121Dic3⋊C4—22no—
122Dic3⋊C4—22no—
123Dic3⋊C4—41no—
124Dic3⋊C4—41no—
125C4×Dic3—22no—
126C4×Dic3—41no—
127C2×C3⋊C8—22no—
128Dic12—44no—
129C2×Q16—62no—
130C2×M16—21no—
131C4⋊Q8—31no—
132C4⋊Q8—31no—
133C4⋊Q8—31no—
134C4⋊Q8—31no—
135#125—61no—
136#125—61no—
137C22⋊Q8—32no—
138C22⋊Q8—61no—
139C22⋊Q8—61no—
140C22⋊Q8—61no—
141C4×Q8—32no—
142C4×Q8—61no—
143C4×Q8—61no—
144C4×Q8—61no—
145#117—61no—
146#117—61no—
147C2×C42C22×D311yescharacteristic
148#106—62no—
149C4⋊C8—21no—
150C4⋊C8—21no—
151C4⋊C8—64no—
152C4⋊C8—61no—
153#103—62no—
154#103—62no—
155C22⋊C8—21no—
156C22⋊C8—61no—
157C22⋊C8D1212yes—
158#97—21no—
159#97—21no—
160#97—21no—
161#97—32no—
162#97—62no—
163C4×C8—62no—
164C4×C8C4×D311yescharacteristic
165C4×C8C4×D311yescharacteristic
166#95—61no—
167#95—61no—
168#95—61no—
169C23×C3#12111yescharacteristic
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4C2×C22⋊C411yescharacteristic
174C2×C3×C4C4×D411yescharacteristic
175C2×C3×C4C4×D411yescharacteristic
176C2×C3×C4#12011yescharacteristic
177C2×C3×C4#12111yescharacteristic
178C2×C3×C4#12311yescharacteristic
179C2×Dic3—22no—
180C2×Dic3—22no—
181C2×Dic3—41no—
182C2×Dic3—41no—
183C2×Dic3—81no—
184C2×Dic3—81no—
185C2×Dic3—81no—
186Dic6—42no—
187Dic6—42no—
188Dic6—81no—
189Dic6—81no—
190C3×C8—22no—
191C3×C8—42no—
192C3×C8—41no—
193C3×C8—81no—
194C3⋊C8—42no—
195C2×Q8—32no—
196C2×Q8—61no—
197C2×Q8—61no—
198C2×Q8—61no—
199C22×C4—121no—
200C22×C4C2×D1211yescharacteristic
201C22×C4D3×D411yescharacteristic
202C22×C4#23911yescharacteristic
203Q16—124no—
204M16—42no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—22no—
208C2×C8—41no—
209C2×C8—62no—
210C4⋊C4—32no—
211C4⋊C4—61no—
212C4⋊C4—61no—
213C4⋊C4—62no—
214C4⋊C4—61no—
215C4⋊C4—62no—
216C4⋊C4—61no—
217C4⋊C4—61no—
218C4⋊C4—61no—
219C4⋊C4—61no—
220C4⋊C4—121no—
221C4⋊C4—121no—
222C22⋊C4—62no—
223C22⋊C4—121no—
224C42—62no—
225C42—121no—
226C42C2×C4×D311yescharacteristic
227C42#23511yescharacteristic
228C42D3×D411yescharacteristic
229C42D3×D411yescharacteristic
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3#38611yescharacteristic
234C3×C4—21no—
235C3×C4—21no—
236C3×C4—21no—
237C3×C4—21no—
238C3×C4—21no—
239C3×C4—21no—
240Dic3—42no—
241Dic3—42no—
242Dic3—81no—
243Dic3—81no—
244Dic3—161no—
245C23#89511yescharacteristic
246Q8—122no—
247Q8—122no—
248Q8—241no—
249Q8—241no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—62no—
254C2×C4—62no—
255C2×C4—121no—
256C2×C4—121no—
257C2×C4—241no—
258C2×C4—241no—
259C2×C4—241no—
260C2×C4C4×D1211yescharacteristic
261C2×C4D3×C22⋊C411yescharacteristic
262C2×C4#88211yescharacteristic
263C2×C4#88411yescharacteristic
264C2×C4#89311yescharacteristic
265C2×C4#89411yescharacteristic
266C8—22no—
267C8—41no—
268C8—42no—
269C8—81no—
270C8—122no—
271C2×C3—41no—
272C2×C3#188411yescharacteristic
273C2×C3#190211yescharacteristic
274C2×C3#190511yescharacteristic
275C22—21no—
276C22—21no—
277C22—21no—
278C22#466511yescharacteristic
279C4—21no—
280C4—21no—
281C4—21no—
282C4—21no—
283C4—21no—
284C4—21no—
285C4—122no—
286C4—122no—
287C4—241no—
288C4—241no—
289C4—481no—
290C3#941611yescharacteristic, complemented, Sylow
291C2—41no—
292C2#6872511yescharacteristic
293C2#6877711yescharacteristic
294C2#6903011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1291C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1291D3
hypocenterC3#9416
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1291

Complements

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

NH ≅ G/Ntype
21. #4699C4semidirect
22. #4681C4semidirect
23. #4681C4semidirect
24. #4680C4semidirect
290. C3#9416semidirect

Maximal subgroups

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

subgroupindex
#793172
#785402
#708372
#708152
C3×#12912
#676662
#94163