Finite Group Databasebeta

← #10692733

Subgroups — #10692733

Order 768.

422 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 89 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#78506C211yesmaximal, characteristic
2#74265C211yesmaximal, characteristic
3#70837C211yesmaximal, characteristic
4#70822C211yesmaximal, characteristic
5#69566C211yesmaximal, characteristic
6C3×#1288C211yesmaximal, characteristic
7#67666C211yesmaximal, characteristic
8#9427—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×#383C2211yescharacteristic
14#5121—22no—
15#4937C2211yescharacteristic
16#4926C2211yescharacteristic
17#4702—22no—
18#4682C412yescomplemented
19#4680C412yescomplemented
20#4654—22no—
21#4653—22no—
22#4646—22no—
23#4486—22no—
24#4467—22no—
25#4452—22no—
26#3165—31no—
27#2381—31no—
28#1556—31no—
29#1515—31no—
30#1515—31no—
31#1288—31no—
32#1288D311yescharacteristic
33C22×Dic6—21no—
34C2×C3×M16—21no—
35C2×C3×C42C2311yescharacteristic
36#926—21no—
37C2×C4⋊Dic3—21no—
38C2×C4⋊Dic3D412yes—
39#924—21no—
40#924—21no—
41C2×Dic3⋊C4—22no—
42C2×C4×Dic3—22no—
43#870—21no—
44#869—21no—
45#869—21no—
46#869C2×C411yescharacteristic
47#869C2×C411yescharacteristic
48#869C2×C412yes—
49#869D412yes—
50C4×Dic6—21no—
51C4×Dic6—21no—
52C3×C4⋊C8—21no—
53C3×C22⋊C8—21no—
54C3×#97—21no—
55C3×#97—21no—
56C3×C4×C8C2×C412yes—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95D412yes—
60C3⋊#95—24no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95D412yes—
64#830—21no—
65#818—24no—
66#817—24no—
67C4⋊(C3⋊C8)—21no—
68#803—21no—
69#533—31no—
70C2×C4⋊Q8—31no—
71#501—62no—
72#500—32no—
73#498—32no—
74#432—31no—
75#432C2×D311yescharacteristic
76#428—62no—
77C4×M16C2×D311yescharacteristic
78#391—62no—
79#387—62no—
80#387—62no—
81#383—31no—
82#383C2×D311yescharacteristic
83#332—62no—
84#328—62no—
85#328—62no—
86C22×C3×C4—22no—
87C22×C3×C4C2×D411yescharacteristic
88C22×C3×C4C2×D411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×Dic3—22no—
91C22×Dic3—22no—
92C2×Dic6—22no—
93C2×Dic6—22no—
94C2×Dic6—21no—
95C2×Dic6—21no—
96C2×Dic6—81no—
97C2×Dic6—81no—
98C3×M16—24no—
99C2×C3×C8—22no—
100C2×C3×C8—41no—
101C3×C42C22×C411yescharacteristic
102C3×C42C2×D411yescharacteristic
103C3×C42#4111yescharacteristic
104C3×C42#4111yescharacteristic
105#217—41no—
106C4⋊Dic3—22no—
107C4⋊Dic3—22no—
108C4⋊Dic3—22no—
109C4⋊Dic3—21no—
110C4⋊Dic3—21no—
111C4⋊Dic3—21no—
112C4⋊Dic3—21no—
113C4⋊Dic3C22⋊C414yes—
114Dic3⋊C4—41no—
115Dic3⋊C4—41no—
116Dic3⋊C4—82no—
117C4×Dic3—41no—
118C4×Dic3—82no—
119C2×C3⋊C8—41no—
120C22×Q8—61no—
121C2×M16—21no—
122C4⋊Q8—32no—
123C4⋊Q8—31no—
124C4⋊Q8—31no—
125C4⋊Q8—32no—
126C4⋊Q8—61no—
127C4⋊Q8—61no—
128#125—61no—
129C22⋊Q8—61no—
130C22⋊Q8—61no—
131C4×Q8—61no—
132C4×Q8—61no—
133#117—61no—
134C2×C4⋊C4—32no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4—62no—
137C2×C42—62no—
138C2×C42C22×D311yescharacteristic
139#107—64no—
140#106—64no—
141C4⋊C8—21no—
142C4⋊C8—61no—
143C22⋊C8—21no—
144C22⋊C8—61no—
145#97—21no—
146#97—21no—
147#97—61no—
148C4×C8C4×D312yes—
149#95—21no—
150#95—21no—
151#95—32no—
152#95—64no—
153#95—61no—
154#95—61no—
155#95D1212yes—
156C23×C3#12011yescharacteristic
157C2×C3×C4—22no—
158C2×C3×C4—41no—
159C2×C3×C4—41no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4C2×C22⋊C411yescharacteristic
164C2×C3×C4C4×D411yescharacteristic
165C2×C3×C4C4×D411yescharacteristic
166C2×C3×C4#12111yescharacteristic
167C2×C3×C4#12111yescharacteristic
168C2×C3×C4#12311yescharacteristic
169C2×Dic3—24no—
170C2×Dic3—42no—
171C2×Dic3—42no—
172C2×Dic3—42no—
173C2×Dic3—42no—
174C2×Dic3—41no—
175C2×Dic3—41no—
176C2×Dic3—42no—
177Dic6—82no—
178Dic6—82no—
179Dic6—81no—
180Dic6—81no—
181C3×C8—24no—
182C3×C8—81no—
183C3⋊C8—81no—
184C2×Q8—62no—
185C2×Q8—62no—
186C2×Q8—61no—
187C2×Q8—61no—
188C2×Q8—241no—
189C2×Q8—241no—
190C22×C4—22no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4C2×D1211yescharacteristic
194C22×C4D3×D411yescharacteristic
195C22×C4D3×D411yescharacteristic
196M16—24no—
197C2×C8—22no—
198C2×C8—41no—
199C2×C8—121no—
200C4⋊C4—34no—
201C4⋊C4—62no—
202C4⋊C4—62no—
203C4⋊C4—62no—
204C4⋊C4—61no—
205C4⋊C4—61no—
206C4⋊C4—61no—
207C4⋊C4—61no—
208C4⋊C4—121no—
209C4⋊C4—121no—
210C4⋊C4—242no—
211C22⋊C4—121no—
212C42—121no—
213C42—242no—
214C42C2×C4×D311yescharacteristic
215C42#23511yescharacteristic
216C42D3×D411yescharacteristic
217C42#23911yescharacteristic
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—82no—
229Dic3—44no—
230Dic3—82no—
231Dic3—81no—
232Dic3—81no—
233C23#88211yescharacteristic
234Q8—242no—
235Q8—242no—
236Q8—241no—
237Q8—241no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—64no—
245C2×C4—122no—
246C2×C4—122no—
247C2×C4—122no—
248C2×C4—122no—
249C2×C4—121no—
250C2×C4—121no—
251C2×C4—122no—
252C2×C4C4×D1211yescharacteristic
253C2×C4D3×C22⋊C411yescharacteristic
254C2×C4#88411yescharacteristic
255C2×C4#89311yescharacteristic
256C2×C4#89411yescharacteristic
257C2×C4#89511yescharacteristic
258C8—24no—
259C8—81no—
260C8—241no—
261C2×C3—22no—
262C2×C3#188611yescharacteristic
263C2×C3#190411yescharacteristic
264C2×C3#190511yescharacteristic
265C22—21no—
266C22—21no—
267C22#466511yescharacteristic
268C22#482612yes—
269C4—21no—
270C4—21no—
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—82no—
276C4—124no—
277C4—242no—
278C4—241no—
279C4—241no—
280C3#942711yescharacteristic, complemented, Sylow
281C2—22no—
282C2#6870711yescharacteristic
283C2#6876511yescharacteristic
284C2#6902711yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1288C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1288D3
hypocenterC3#9427
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1288

Complements

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

NH ≅ G/Ntype
18. #4682C4semidirect
19. #4680C4semidirect
280. C3#9427semidirect

Maximal subgroups

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

subgroupindex
#785062
#742652
#708372
#708222
#695662
C3×#12882
#676662
#94273