Finite Group Databasebeta

← #10692666

Subgroups — #10692666

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#83348C211yesmaximal, characteristic
2#79317C211yesmaximal, characteristic
3C3×#1515C211yesmaximal, characteristic
4#71823C211yesmaximal, characteristic
5#69566C211yesmaximal, characteristic
6#67754C211yesmaximal, characteristic
7#67633C211yesmaximal, characteristic
8#9416—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#383C2211yescharacteristic
13#5093C411yescharacteristic, complemented
14#5092C411yescharacteristic, complemented
15#5080C411yescharacteristic, complemented
16#5079C411yescharacteristic, complemented
17#5052—22no—
18#5030—22no—
19#4926C2211yescharacteristic
20#4926C2211yescharacteristic
21C4×#208C2211yescharacteristic
22#4828—22no—
23#4720—22no—
24#4701—22no—
25C3×#328—22no—
26#4543—22no—
27#4452—22no—
28#3185—31no—
29#2987—31no—
30#1537—31no—
31#1515—31no—
32#1515D311yescharacteristic
33#1291—31no—
34#1288—31no—
35C3×C4⋊Q8C2×C411yescharacteristic
36C3×C4⋊Q8C2×C411yescharacteristic
37C3×#125—21no—
38C3×C22⋊Q8—21no—
39C3×C22⋊Q8—21no—
40C3×C4×Q8—21no—
41C3×C4×Q8—21no—
42C3×#117—21no—
43C2×C3×C42C2311yescharacteristic
44C2×#216—22no—
45#926—21no—
46#924—21no—
47#924D412yes—
48C2×#208—21no—
49#870—21no—
50#869C2×C411yescharacteristic
51#869C2×C411yescharacteristic
52C4×Dic6—21no—
53C4×Dic6D412yes—
54C3×C4⋊C8—21no—
55C3×C22⋊C8D412yes—
56C3×#97—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60#835—22no—
61#830—21no—
62#830—21no—
63#815—22no—
64Dic3⋊C8—24no—
65C8×Dic3—22no—
66#810—22no—
67#808—22no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70#803—21no—
71#803—21no—
72#803D412yes—
73C4×C3⋊C8C2×C411yescharacteristic
74C4×C3⋊C8C2×C411yescharacteristic
75#533—31no—
76#533C2×D311yescharacteristic
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#432—31no—
88#432C2×D311yescharacteristic
89C4×M16—31no—
90#383C2×D311yescharacteristic
91#330—62no—
92#328—22no—
93#328—62no—
94C2×C3×Q8—21no—
95C2×C3×Q8—21no—
96C22×C3×C4—41no—
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4#4111yescharacteristic
100C2×Dic6—21no—
101C2×Dic6C22⋊C412yes—
102C2×C3×C8—22no—
103C3×C4⋊C4—21no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—21no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C22⋊C4—41no—
110C3×C42—41no—
111C3×C42C22×C411yescharacteristic
112C3×C42C2×D411yescharacteristic
113C3×C42C2×D411yescharacteristic
114C3×C42#4111yescharacteristic
115#217—22no—
116#216—44no—
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3C22⋊C412yes—
121Dic3⋊C4—22no—
122Dic3⋊C4—22no—
123C4×Dic3—22no—
124#208—42no—
125C2×C3⋊C8—21no—
126C2×C3⋊C8—21no—
127C2×C3⋊C8—22no—
128C2×C3⋊C8—41no—
129C2×Q16—62no—
130C2×M16—61no—
131C4⋊Q8—31no—
132C4⋊Q8—31no—
133C4⋊Q8C4×D311yescharacteristic
134C4⋊Q8C4×D311yescharacteristic
135#125—21no—
136#125—61no—
137C22⋊Q8—21no—
138C22⋊Q8—21no—
139C22⋊Q8—32no—
140C22⋊Q8—61no—
141C4×Q8—21no—
142C4×Q8—21no—
143C4×Q8—32no—
144C4×Q8—61no—
145#117—21no—
146#117—61no—
147C2×C42C22×D311yescharacteristic
148#106—62no—
149C4⋊C8—21no—
150C4⋊C8—64no—
151C4⋊C8—61no—
152C4⋊C8—61no—
153#103—62no—
154#103—62no—
155C22⋊C8—61no—
156C22⋊C8—61no—
157C22⋊C8#6712yes—
158#97—21no—
159#97—32no—
160#97—62no—
161#97—61no—
162#97—61no—
163C4×C8—31no—
164C4×C8—31no—
165C4×C8—62no—
166#95—21no—
167#95—21no—
168#95—21no—
169C23×C3#12111yescharacteristic
170C3×Q8—42no—
171C3×Q8—81no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—81no—
178C2×C3×C4—81no—
179C2×C3×C4—81no—
180C2×C3×C4C2×C22⋊C411yescharacteristic
181C2×C3×C4C4×D411yescharacteristic
182C2×C3×C4C4×D411yescharacteristic
183C2×C3×C4#12011yescharacteristic
184C2×C3×C4#12111yescharacteristic
185C2×C3×C4#12311yescharacteristic
186C2×Dic3—22no—
187C2×Dic3—22no—
188Dic6—42no—
189Dic6—81no—
190C3×C8—42no—
191C3⋊C8—22no—
192C3⋊C8—42no—
193C3⋊C8—41no—
194C3⋊C8—81no—
195C2×Q8—21no—
196C2×Q8—21no—
197C2×Q8—32no—
198C2×Q8—61no—
199C22×C4—41no—
200C22×C4D3×D411yescharacteristic
201C22×C4#23711yescharacteristic
202C22×C4C2×#6711yescharacteristic
203Q16—124no—
204M16—122no—
205C2×C8—22no—
206C2×C8—61no—
207C2×C8—61no—
208C2×C8—62no—
209C2×C8—121no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—32no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—61no—
218C4⋊C4—62no—
219C4⋊C4—62no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C22⋊C4—41no—
223C22⋊C4—62no—
224C42—41no—
225C42—62no—
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—
240C3×C4—81no—
241C3×C4—81no—
242C3×C4—161no—
243Dic3—42no—
244Dic3—42no—
245C23#93811yescharacteristic
246Q8—42no—
247Q8—81no—
248Q8—122no—
249Q8—241no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4D3×C22⋊C411yescharacteristic
261C2×C4#67⋊C411yescharacteristic
262C2×C4#88311yescharacteristic
263C2×C4#88411yescharacteristic
264C2×C4C4×#6711yescharacteristic
265C2×C4#93711yescharacteristic
266C8—42no—
267C8—62no—
268C8—121no—
269C8—122no—
270C8—241no—
271C2×C3—41no—
272C2×C3#188411yescharacteristic
273C2×C3#190211yescharacteristic
274C2×C3#190511yescharacteristic
275C22—21no—
276C22—21no—
277C22—21no—
278C22#495611yescharacteristic
279C4—21no—
280C4—21no—
281C4—21no—
282C4—21no—
283C4—21no—
284C4—21no—
285C4—81no—
286C4—81no—
287C4—122no—
288C4—122no—
289C4—161no—
290C3#941611yescharacteristic, complemented, Sylow
291C2—41no—
292C2#7005211yescharacteristic
293C2#7034911yescharacteristic
294C2#7041711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1515C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1515D3
hypocenterC3#9416
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1515

Complements

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

NH ≅ G/Ntype
13. #5093C4semidirect
14. #5092C4semidirect
15. #5080C4semidirect
16. #5079C4semidirect
290. C3#9416semidirect

Maximal subgroups

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

subgroupindex
#833482
#793172
C3×#15152
#718232
#695662
#677542
#676332
#94163