Finite Group Databasebeta

← #10692734

Subgroups — #10692734

Order 768.

422 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C3×#2381C211yesmaximal, characteristic
2#83207C211yesmaximal, characteristic
3#71776C211yesmaximal, characteristic
4#69566C211yesmaximal, characteristic
5#69566C211yesmaximal, characteristic
6#67633C211yesmaximal, characteristic
7#67633C211yesmaximal, characteristic
8#9427—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#391—22no—
12C3×#387—22no—
13C3×#387—22no—
14C3×#383C2211yescharacteristic
15C3×#383C2211yescharacteristic
16#5081—22no—
17#5080C412yescomplemented
18#5078C412yescomplemented
19#4964—22no—
20#4926C2211yescharacteristic
21#4926C2211yescharacteristic
22C4×#208C2211yescharacteristic
23#4547—22no—
24#4452—22no—
25#4452—22no—
26#3165—31no—
27#2381D311yescharacteristic
28#1556—31no—
29#1515—31no—
30#1515—31no—
31#1288—31no—
32#1288—31no—
33C22×C3×Q8—21no—
34C3×C4⋊Q8—21no—
35C3×C4⋊Q8—21no—
36C3×C4⋊Q8C2×C412yes—
37C3×C4⋊Q8D412yes—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—22no—
40C2×C3×C4⋊C4D412yes—
41C2×C3×C42—22no—
42C2×C3×C42C2311yescharacteristic
43#926—21no—
44#924—21no—
45#924—21no—
46C2×#208—21no—
47#870—21no—
48#869C2×C411yescharacteristic
49#869C2×C411yescharacteristic
50C4×Dic6—21no—
51C4×Dic6—21no—
52C3×#95—24no—
53C3×#95—21no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95D412yes—
58C3×#95D412yes—
59#830—21no—
60#830—21no—
61#808—24no—
62#807—24no—
63C4⋊(C3⋊C8)—21no—
64C4⋊(C3⋊C8)—21no—
65#803—21no—
66#803—21no—
67#803—21no—
68C4×C3⋊C8C2×C412yes—
69#533—31no—
70C2×C4⋊Q8C2×D311yescharacteristic
71#501—62no—
72#500—32no—
73#498—32no—
74#432—31no—
75#432—31no—
76#428—62no—
77C4×M16—31no—
78#391—22no—
79#387—22no—
80#387—22no—
81#383C2×D311yescharacteristic
82#383C2×D311yescharacteristic
83#332—62no—
84#328—62no—
85#328—62no—
86C2×C3×Q8—22no—
87C2×C3×Q8—22no—
88C2×C3×Q8—81no—
89C2×C3×Q8—81no—
90C22×C3×C4—22no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C2×Dic6—21no—
97C2×Dic6—21no—
98C3×C4⋊C4—22no—
99C3×C4⋊C4—22no—
100C3×C4⋊C4—22no—
101C3×C4⋊C4—82no—
102C3×C4⋊C4C22⋊C414yes—
103C3×C42—82no—
104C3×C42C22×C411yescharacteristic
105C3×C42C2×D411yescharacteristic
106C3×C42#4111yescharacteristic
107C3×C42#4111yescharacteristic
108#217—41no—
109C4⋊Dic3—21no—
110C4⋊Dic3—21no—
111C4⋊Dic3—21no—
112C4⋊Dic3—21no—
113Dic3⋊C4—41no—
114Dic3⋊C4—41no—
115C4×Dic3—41no—
116#208—24no—
117C2×C3⋊C8—22no—
118C2×C3⋊C8—41no—
119C2×C3⋊C8—41no—
120C22×Q8—21no—
121C2×M16—61no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—31no—
125C4⋊Q8—31no—
126C4⋊Q8C4×D312yes—
127C4⋊Q8D1212yes—
128#125—61no—
129C22⋊Q8—61no—
130C22⋊Q8—61no—
131C4×Q8—61no—
132C4×Q8—61no—
133#117—61no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4#6712yes—
137C2×C42—22no—
138C2×C42C22×D311yescharacteristic
139#107—64no—
140#106—64no—
141C4⋊C8—61no—
142C4⋊C8—61no—
143C22⋊C8—61no—
144C22⋊C8—61no—
145#97—61no—
146#97—61no—
147#97—61no—
148C4×C8—32no—
149#95—24no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95#6712yes—
155#95#6712yes—
156C23×C3#12011yescharacteristic
157C3×Q8—82no—
158C3×Q8—82no—
159C2×C3×C4—22no—
160C2×C3×C4—24no—
161C2×C3×C4—41no—
162C2×C3×C4—41no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4C2×C22⋊C411yescharacteristic
172C2×C3×C4C4×D411yescharacteristic
173C2×C3×C4C4×D411yescharacteristic
174C2×C3×C4#12111yescharacteristic
175C2×C3×C4#12111yescharacteristic
176C2×C3×C4#12311yescharacteristic
177C2×Dic3—41no—
178C2×Dic3—41no—
179Dic6—81no—
180Dic6—81no—
181C3⋊C8—24no—
182C3⋊C8—81no—
183C3⋊C8—81no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—61no—
187C2×Q8—61no—
188C2×Q8—81no—
189C2×Q8—81no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4C2×#6711yescharacteristic
194C22×C4C2×#6711yescharacteristic
195C22×C4C2×#6711yescharacteristic
196M16—64no—
197C2×C8—62no—
198C2×C8—121no—
199C2×C8—121no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—61no—
204C4⋊C4—61no—
205C4⋊C4—61no—
206C4⋊C4—61no—
207C4⋊C4—82no—
208C4⋊C4—121no—
209C4⋊C4—121no—
210C4⋊C4(C2×D3)⋊C414yes—
211C22⋊C4—121no—
212C42—82no—
213C42—121no—
214C42C2×C4×D311yescharacteristic
215C42C2×D1211yescharacteristic
216C42#23511yescharacteristic
217C42#23511yescharacteristic
218C22×C3—21no—
219C22×C3—21no—
220C22×C3#38611yescharacteristic
221C22×C3#45612yes—
222C3×C4—21no—
223C3×C4—21no—
224C3×C4—21no—
225C3×C4—21no—
226C3×C4—21no—
227C3×C4—21no—
228C3×C4—44no—
229C3×C4—82no—
230C3×C4—82no—
231Dic3—81no—
232Dic3—81no—
233C23#95311yescharacteristic
234Q8—82no—
235Q8—82no—
236Q8—241no—
237Q8—241no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—121no—
251C2×C4—121no—
252C2×C4C2×(C2×D3)⋊C411yescharacteristic
253C2×C4C4×#6711yescharacteristic
254C2×C4C4×#6711yescharacteristic
255C2×C4#92911yescharacteristic
256C2×C4#93011yescharacteristic
257C2×C4#93011yescharacteristic
258C8—64no—
259C8—241no—
260C8—241no—
261C2×C3—22no—
262C2×C3#188611yescharacteristic
263C2×C3#190411yescharacteristic
264C2×C3#190511yescharacteristic
265C22—21no—
266C22—21no—
267C22#521311yescharacteristic
268C22#524612yes—
269C4—21no—
270C4—21no—
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—44no—
276C4—82no—
277C4—82no—
278C4—241no—
279C4—241no—
280C3#942711yescharacteristic, complemented, Sylow
281C2—22no—
282C2#7165111yescharacteristic
283C2#7166011yescharacteristic
284C2#7166211yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2381C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#2381D3
hypocenterC3#9427
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2381

Complements

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

NH ≅ G/Ntype
17. #5080C4semidirect
18. #5078C4semidirect
280. C3#9427semidirect

Maximal subgroups

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

subgroupindex
C3×#23812
#832072
#717762
#695662
#676332
#94273