Finite Group Databasebeta

← #10690462

Subgroups — #10690462

Order 768.

822 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 147 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#83173C211yesmaximal, characteristic, complemented
2C2×#5073C211yesmaximal, characteristic
3C2×C3×#331C211yesmaximal, characteristic, complemented
4#70457C212yesmaximal, complemented
5#67704C212yesmaximal, complemented
6#9061—31nomaximal, Sylow
7C2×C3×#127C2211yescharacteristic
8C2×D4×Dic3—22no—
9C22×#213—22no—
10C2×#874C2211yescharacteristic
11C2×C3×C4⋊C8C2212yes—
12#5221—22no—
13#5220—22no—
14C2×#821—22no—
15C2×#821—22no—
16#5073—22no—
17#5073C414yescomplemented
18C4×C4⋊Dic3C2212yes—
19C2×C4×C3⋊C8C2211yescharacteristic
20C3×#331—22no—
21C2×#493—31no—
22#2366—31no—
23#1982—32no—
24C2×#331D311yescharacteristic, complemented
25#1279—32no—
26C22×C3×D4—22no—
27C22×C3×D4D412yes—
28C23×Dic3—22no—
29C22×D12—21no—
30C22×D12—22no—
31C22×C3×C8—22no—
32C3×#127—22no—
33C3×#127C2×C412yes—
34C3×#127D412yes—
35C2×C3×C42C2311yescharacteristic
36C2×#217—24no—
37C2×#217—22no—
38D4×Dic3—44no—
39D4×Dic3—44no—
40C2×#213—28no—
41C2×#213—44no—
42C2×#213—44no—
43C2×C4⋊Dic3D412yes—
44C2×C4⋊Dic3D412yes—
45C2×C4×Dic3—22no—
46C22×C3⋊C8—21no—
47#874—21no—
48#874—22no—
49#874C2×C412yescomplemented
50C3×C4⋊C8—22no—
51C3×C4⋊C8—22no—
52C3⋊#95—22no—
53#821—44no—
54#821—44no—
55C4×C3⋊C8—21no—
56C4×C3⋊C8C2×C412yescomplemented
57C22×D8—62no—
58C2×#127—31no—
59C2×#127C2×D311yescharacteristic
60C2×C4×D4—62no—
61#493—34no—
62#493—62no—
63C2×C4⋊C8C2×D312yes—
64C2×#102—62no—
65C2×#102—62no—
66C2×C4×C8—31no—
67#390—62no—
68#386—62no—
69C4×C4⋊C4—32no—
70#331—22no—
71C24×C3—22no—
72C23×D3—41no—
73C2×C3×D4—24no—
74C2×C3×D4—24no—
75C2×C3×D4—22no—
76C2×C3×D4—42no—
77C2×C3×D4—42no—
78C2×C3×D4—42no—
79C2×C3×D4—42no—
80C2×C3×D4—42no—
81C2×C3×D4—42no—
82C2×C3×D4C22⋊C414yes—
83C22×C3×C4C2×D411yescharacteristic
84C22×C3×C4C2×D412yes—
85C22×Dic3—22no—
86C22×Dic3—22no—
87C22×Dic3—48no—
88C22×Dic3—82no—
89C2×D12—22no—
90C2×D12—24no—
91C2×D12—42no—
92C2×D12—42no—
93C2×D12—42no—
94C2×D12—42no—
95C2×D12—42no—
96C2×D12—42no—
97C2×D12—42no—
98C2×D12—41no—
99C2×C3×C8—42no—
100C2×C3×C8—42no—
101C2×C3×C8—42no—
102C3×C42C22×C411yescharacteristic
103C3×C42C2×D411yescharacteristic
104C3×C42#4112yes—
105#217—48no—
106#217—82no—
107#213—416no—
108C4⋊Dic3—22no—
109C4⋊Dic3—22no—
110C4⋊Dic3—22no—
111C4⋊Dic3D814yescomplemented
112C4×Dic3—42no—
113C4×Dic3—42no—
114C2×C3⋊C8—22no—
115C2×C3⋊C8—22no—
116C2×C3⋊C8—22no—
117C22×D4—22no—
118C22×D4—61no—
119C22×D4—62no—
120C22×D4D1212yes—
121C23×C4—62no—
122C2×D8—68no—
123C2×D8—124no—
124C2×D8—124no—
125C22×C8—22no—
126C22×C8—61no—
127#127—22no—
128#127—32no—
129#127—62no—
130#127—61no—
131#127C4×D312yes—
132#127#6712yes—
133C4×D4—124no—
134C4×D4—124no—
135C2×C4⋊C4—32no—
136C2×C4⋊C4—32no—
137C2×C22⋊C4—64no—
138C2×C22⋊C4—62no—
139C2×C42—62no—
140C2×C42C22×D311yescharacteristic
141C4⋊C8—22no—
142C4⋊C8—22no—
143#102—124no—
144#102—124no—
145C4×C8—32no—
146C4×C8—61no—
147#95—62no—
148C23×C3—24no—
149C23×C3—44no—
150C23×C3—44no—
151C23×C3—42no—
152C23×C3—42no—
153C23×C3#12011yescharacteristic
154C22×D3—42no—
155C22×D3—82no—
156C22×D3—82no—
157C22×D3—81no—
158C22×D3—81no—
159C3×D4—44no—
160C3×D4—44no—
161C3×D4—44no—
162C3×D4—44no—
163C3×D4—44no—
164C3×D4—82no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4C2×C22⋊C411yescharacteristic
168C2×C3×C4C4×D412yes—
169C2×C3×C4#12112yes—
170C2×C3×C4#12311yescharacteristic
171C2×C3×C4C2×D812yes—
172C2×C3×C4#13512yes—
173C2×C3×C4AGL(1,ℤ/8)12yes—
174C2×C3×C4AGL(1,ℤ/8)12yes—
175C2×Dic3—24no—
176C2×Dic3—42no—
177C2×Dic3—42no—
178C2×Dic3—42no—
179C2×Dic3—42no—
180C2×Dic3—42no—
181C2×Dic3—88no—
182D12—44no—
183D12—44no—
184D12—44no—
185D12—82no—
186D12—82no—
187D12—82no—
188C3×C8—82no—
189C3⋊C8—24no—
190C24—22no—
191C24—121no—
192C2×D4—24no—
193C2×D4—22no—
194C2×D4—24no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—64no—
202C2×D4—62no—
203C2×D4—122no—
204C2×D4—122no—
205C2×D4—121no—
206C2×D4—122no—
207C2×D4—122no—
208C2×D4—122no—
209C2×D4—122no—
210C2×D4—122no—
211C2×D4(C2×D3)⋊C414yes—
212C22×C4—62no—
213C22×C4—62no—
214C22×C4—128no—
215C22×C4—242no—
216C22×C4C2×D1211yescharacteristic
217C22×C4D3×D412yes—
218D8—1216no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—62no—
223C2×C8—62no—
224C2×C8—62no—
225C4⋊C4—34no—
226C4⋊C4—62no—
227C4⋊C4—62no—
228C4⋊C4—62no—
229C22⋊C4—128no—
230C22⋊C4—242no—
231C42—122no—
232C42—122no—
233C42C2×C4×D311yescharacteristic
234C42#23712yes—
235C42C2×#6711yescharacteristic
236C22×C3—48no—
237C22×C3—44no—
238C22×C3—82no—
239C22×C3—82no—
240C22×C3—84no—
241C22×C3#38611yescharacteristic
242C22×C3#44712yes—
243C22×C3#44912yes—
244C22×C3#45311yescharacteristic
245C22×C3#45311yescharacteristic
246C2×D3—84no—
247C2×D3—82no—
248C2×D3—161no—
249C2×D3—161no—
250C2×D3—162no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4C4×D812yes—
254C3×C4#44212yes—
255C3×C4#45912yes—
256C3×C4#46312yes—
257Dic3—44no—
258Dic3—82no—
259C23—24no—
260C23—42no—
261C23—42no—
262C23—44no—
263C23—44no—
264C23—122no—
265C23—241no—
266C23—241no—
267C23—242no—
268C23—242no—
269C23#88211yescharacteristic
270D4—44no—
271D4—44no—
272D4—44no—
273D4—44no—
274D4—44no—
275D4—82no—
276D4—124no—
277D4—124no—
278D4—124no—
279D4—242no—
280D4—242no—
281D4—242no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—64no—
285C2×C4—122no—
286C2×C4—122no—
287C2×C4—122no—
288C2×C4—122no—
289C2×C4—122no—
290C2×C4—248no—
291C2×C4#67⋊C412yes—
292C2×C4#88611yescharacteristic
293C2×C4D3×D812yes—
294C2×C4#91112yes—
295C2×C4#91412yes—
296C2×C4#91612yes—
297C2×C4C2×(C2×D3)⋊C411yescharacteristic
298C2×C4#93912yes—
299C8—64no—
300C8—82no—
301C2×C3—84no—
302C2×C3#161911yescharacteristic
303C2×C3#162011yescharacteristic
304C2×C3#162112yes—
305C2×C3#187912yes—
306C2×C3#188711yescharacteristic
307D3—162no—
308C22—48no—
309C22—44no—
310C22—82no—
311C22—82no—
312C22—84no—
313C22—244no—
314C22—242no—
315C22—481no—
316C22—481no—
317C22—482no—
318C22#474711yescharacteristic
319C22#474711yescharacteristic
320C22#477312yes—
321C22#478112yes—
322C22#495811yescharacteristic
323C4—21no—
324C4—21no—
325C4—124no—
326C4—242no—
327C4#475612yes—
328C4#478512yes—
329C4#515012yes—
330C4#516512yes—
331C3#906111yescharacteristic, complemented, Sylow
332C2—84no—
333C2—482no—
334C2#6839011yescharacteristic
335C2#6840211yescharacteristic
336C2#6840312yes—
337C2#7007011yescharacteristic
338C2#7017212yes—
subgroupG/H
centerC23#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#331C2
socleC23×C3#120
prime radicalC2×#5073C2
cosocleC2×C3×C42C23
hypercenterC2×#331D3
hypocenterC3#9061
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#331

Complements

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

NH ≅ G/Ntype
1. #83173C2semidirect
3. C2×C3×#331C2semidirect
4. #70457C2semidirect
5. #67704C2semidirect
17. #5073C4semidirect
24. C2×#331D3semidirect
49. #874C2×C4semidirect
56. C4×C3⋊C8C2×C4semidirect
111. C4⋊Dic3D8semidirect
331. C3#9061semidirect

Maximal subgroups

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

subgroupindex
#831732
C2×#50732
C2×C3×#3312
#704572
#677042
#90613