Finite Group Databasebeta

← #10695799

Subgroups — #10695799

Order 768.

716 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 83 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#84084C212yesmaximal
2#78617C211yesmaximal, characteristic, complemented
3C3×#1780C211yesmaximal, characteristic
4#69923C212yesmaximal, complemented
5#67679C211yesmaximal, characteristic, complemented
6#9976—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C2×#951—22no—
9C2×#924C2212yes—
10C2×C3×C22⋊C8—22no—
11C2×C3×C22⋊C8C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13#5239—24no—
14#5232—22no—
15#5228—22no—
16#5218—22no—
17#5207C2212yes—
18#5116—22no—
19#4739—22no—
20#4722—22no—
21#4469—24no—
22#3189—31no—
23#3007—32no—
24#1780D311yescharacteristic
25#1472—32no—
26#1293—31no—
27C2×C3×#41—22no—
28C2×C3×#41—22no—
29C2×C3×#41—44no—
30C22×C3×Q8D411yescharacteristic
31C22×C3×D4—21no—
32C22×C3×D4D411yescharacteristic
33C23×C3×C4—21no—
34C23×C3×C4C2311yescharacteristic
35C22×Dic6D412yes—
36C22×C3×C8—21no—
37C22×C3×C8D412yes—
38C2×#217—22no—
39#951—48no—
40C2×#216—24no—
41C2×#214—24no—
42C2×C4⋊Dic3D412yes—
43#924—22no—
44#924—22no—
45#924—22no—
46#924—22no—
47C2×Dic3⋊C4—22no—
48C2×#208—22no—
49C2×Dic12—22no—
50C3×C22⋊C8—22no—
51C3×C22⋊C8—22no—
52C3×C22⋊C8—44no—
53#835—24no—
54#832—24no—
55#830—22no—
56#830D414yes—
57#818—22no—
58#816—24no—
59C22×#41C2×D311yescharacteristic
60C2×#137—62no—
61C2×C22⋊Q8—32no—
62#484—62no—
63#467—62no—
64#452—64no—
65#451—62no—
66#451—62no—
67#450—62no—
68#417—62no—
69#407—32no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8C2×D311yescharacteristic
72C2×C22⋊C8C2×D311yescharacteristic
73#324—64no—
74C24×C3—21no—
75C24×C3#4111yescharacteristic
76C3×#41—28no—
77C3×#41—44no—
78C3×#41—44no—
79C3×#41—44no—
80C2×C3×Q8—22no—
81C2×C3×Q8—42no—
82C2×C3×D4—22no—
83C2×C3×D4—42no—
84C2×C3×D4—42no—
85C2×C3×D4—42no—
86C2×C3×D4—42no—
87C22×C3×C4—24no—
88C22×C3×C4—21no—
89C22×C3×C4—42no—
90C22×C3×C4—42no—
91C22×C3×C4—41no—
92C22×C3×C4—41no—
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D412yes—
95C22×C3×C4C2×D412yes—
96C22×C3×C4C2×D411yescharacteristic
97C22×Dic3—22no—
98C22×Dic3—22no—
99C2×Dic6—22no—
100C2×Dic6—22no—
101C2×Dic6—44no—
102C2×C3×C8—22no—
103C2×C3×C8—42no—
104C2×C3×C8—42no—
105C2×C3×C8—41no—
106#217—42no—
107#217—42no—
108#216—48no—
109#214—48no—
110C4⋊Dic3—22no—
111C4⋊Dic3—22no—
112Dic3⋊C4—42no—
113Dic3⋊C4—42no—
114#208—44no—
115C2×C3⋊C8—24no—
116Dic12—82no—
117C2×#41—22no—
118C2×#41—22no—
119C2×#41—44no—
120C22×Q8—32no—
121C22×Q8D1211yescharacteristic
122C22×D4—21no—
123C22×D4D1211yescharacteristic
124C23×C4—21no—
125C23×C4C22×D311yescharacteristic
126#137—128no—
127C2×Q16—62no—
128C2×Q16—64no—
129C2×SD16—64no—
130C2×M16—62no—
131C22×C8—21no—
132C22×C8#6712yes—
133C22⋊Q8—62no—
134C22⋊Q8—62no—
135C22⋊Q8—62no—
136C22⋊Q8—62no—
137C2×C4⋊C4—32no—
138C2×C4⋊C4—62no—
139C2×C22⋊C4—62no—
140#107—62no—
141#103—64no—
142#103—64no—
143#102—64no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—34no—
147C22⋊C8—44no—
148C22⋊C8—62no—
149C23×C3—22no—
150C23×C3—22no—
151C23×C3—41no—
152C23×C3—41no—
153C23×C3—41no—
154C23×C3—42no—
155C23×C3—41no—
156C23×C3—42no—
157C23×C3#12112yes—
158C23×C3#12111yescharacteristic
159C3×Q8—24no—
160C3×Q8—42no—
161C3×D4—24no—
162C3×D4—42no—
163C3×D4—48no—
164C2×C3×C4—24no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—41no—
168C2×C3×C4—42no—
169C2×C3×C4—44no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—44no—
173C2×C3×C4#12012yes—
174C2×C3×C4#12011yescharacteristic
175C2×C3×C4#12711yescharacteristic
176C2×Dic3—44no—
177C2×Dic3—42no—
178C2×Dic3—42no—
179C2×Dic3—82no—
180Dic6—44no—
181Dic6—44no—
182C3×C8—42no—
183C3×C8—81no—
184C3⋊C8—44no—
185C24—21no—
186C24#23511yescharacteristic
187#41—28no—
188#41—44no—
189#41—44no—
190#41—44no—
191C2×Q8—22no—
192C2×Q8—42no—
193C2×Q8—62no—
194C2×Q8—62no—
195C2×Q8—124no—
196C2×D4—22no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C22×C4—24no—
202C22×C4—21no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—62no—
208C22×C4—62no—
209C22×C4C2×D1211yescharacteristic
210C22×C4D3×D412yes—
211C22×C4D3×D412yes—
212C22×C4C2×#6711yescharacteristic
213Q16—128no—
214Q16—242no—
215SD16—128no—
216M16—124no—
217C2×C8—22no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—41no—
221C2×C8—64no—
222C4⋊C4—62no—
223C4⋊C4—62no—
224C4⋊C4—122no—
225C4⋊C4—122no—
226C22⋊C4—122no—
227C22⋊C4—122no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—22no—
231C22×C3—22no—
232C22×C3—22no—
233C22×C3—44no—
234C22×C3—42no—
235C22×C3—82no—
236C22×C3—82no—
237C22×C3—81no—
238C22×C3—81no—
239C22×C3#39211yescharacteristic
240C22×C3#47312yes—
241C3×C4—22no—
242C3×C4—21no—
243C3×C4—42no—
244C3×C4B(4,𝔽2)12yes—
245Dic3—82no—
246Dic3—82no—
247C23—22no—
248C23—22no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—42no—
253C23—42no—
254C23—41no—
255C23#88412yes—
256C23#93011yescharacteristic
257Q8—24no—
258Q8—42no—
259Q8—124no—
260Q8—124no—
261D4—24no—
262D4—42no—
263D4—48no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—44no—
273C2×C4—124no—
274C2×C4—122no—
275C2×C4—122no—
276C2×C4—242no—
277C2×C4#88212yes—
278C2×C4#93711yescharacteristic
279C2×C4#94011yescharacteristic
280C8—42no—
281C8—81no—
282C8—124no—
283C2×C3—22no—
284C2×C3—42no—
285C2×C3—82no—
286C2×C3#201711yescharacteristic
287C2×C3#202011yescharacteristic
288C2×C3#202111yescharacteristic
289C22—21no—
290C22—21no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—44no—
295C22—42no—
296C22—82no—
297C22—82no—
298C22—81no—
299C22—81no—
300C22#496011yescharacteristic
301C22#514312yes—
302C4—21no—
303C4—22no—
304C4—42no—
305C4—242no—
306C4—242no—
307C4#503212yes—
308C3#997611yescharacteristic, complemented, Sylow
309C2—22no—
310C2—42no—
311C2—82no—
312C2#6950011yescharacteristic
313C2#7000811yescharacteristic
314C2#7001311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1780C2
socleC22×C3#392
prime radicalC22×C3×#41C22
hypercenter#1780D3
hypocenterC3#9976
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1780

Complements

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

NH ≅ G/Ntype
2. #78617C2semidirect
4. #69923C2semidirect
5. #67679C2semidirect
308. C3#9976semidirect

Maximal subgroups

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

subgroupindex
#840842
#786172
C3×#17802
#699232
#676792
#99763