Finite Group Databasebeta

← #10714870

Subgroups — #10714870

Order 768.

484 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 81 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#76421C211yesmaximal, characteristic
2#70021C212yesmaximal
3#69642C211yesmaximal, characteristic
4C3×#1393C211yesmaximal, characteristic
5#67694C212yesmaximal
6#13463—31nomaximal, Sylow
7#5798C2211yescharacteristic
8C2×C3×#99C2212yes—
9C3×#390C2211yescharacteristic
10#5220C2212yes—
11C2×#833C2211yescharacteristic
12#4986—21no—
13#4964—21no—
14#4950—22no—
15#4950—22no—
16#4947—22no—
17#4929—21no—
18#4527—22no—
19#4523—21no—
20#2804—31no—
21#1867—32no—
22#1856—31no—
23#1393D311yescharacteristic
24#1390—32no—
25C22×C3×D4C2311yescharacteristic
26C23×Dic3D412yes—
27C22×Dic6—21no—
28C2×C3×C4⋊C4D411yescharacteristic
29C2×C3×C22⋊C4—22no—
30C2×C3×C22⋊C4—22no—
31C2×C3×C42D411yescharacteristic
32C2×#217—22no—
33C2×#217—22no—
34C2×#217D412yes—
35D4×Dic3—41no—
36D4×Dic3—41no—
37#926—21no—
38C2×C4⋊Dic3—21no—
39C2×C4⋊Dic3—21no—
40C2×C4⋊Dic3—42no—
41C2×#208—21no—
42C3×#99—22no—
43C3×#99Q814yescomplemented
44#833—21no—
45#833Q812yescomplemented
46C3⋊#95—22no—
47C3⋊#95—21no—
48C3⋊#95—22no—
49C3⋊#95—22no—
50C3⋊#95—42no—
51#808—41no—
52#807—41no—
53#518—31no—
54#428—61no—
55C2×#100—31no—
56C2×#99C2×D312yes—
57#400—61no—
58#393—62no—
59#391—61no—
60#390C2×D311yescharacteristic
61#386—32no—
62#385—62no—
63#385—62no—
64#342—62no—
65#339—61no—
66C24×C3C2×D412yes—
67C2×C3×D4—41no—
68C2×C3×D4—41no—
69C2×C3×D4C2×Q812yes—
70C2×C3×D4C2×Q811yescharacteristic
71C2×C3×D4#4111yescharacteristic
72C22×C3×C4—21no—
73C22×C3×C4—21no—
74C22×C3×C4—42no—
75C22×C3×C4C2×D411yescharacteristic
76C22×Dic3—22no—
77C22×Dic3—22no—
78C22×Dic3—22no—
79C22×Dic3—22no—
80C22×Dic3—42no—
81C22×Dic3—82no—
82C2×Dic6—42no—
83C2×Dic6—81no—
84C2×Dic6—81no—
85C3×C4⋊C4—41no—
86C3×C22⋊C4—24no—
87C3×C22⋊C4—44no—
88C3×C22⋊C4—42no—
89C3×C42—41no—
90#217—42no—
91#217—41no—
92#217—42no—
93#217—82no—
94C4⋊Dic3—21no—
95C4⋊Dic3—21no—
96C4⋊Dic3—81no—
97C4⋊Dic3—81no—
98C4⋊Dic3—84no—
99C4×Dic3—41no—
100#208—22no—
101#208—41no—
102C2×C3⋊C8—41no—
103C22×Q8—61no—
104C22×D4C22×D311yescharacteristic
105C23×C4—32no—
106C2×M16—61no—
107C4×D4—121no—
108C4×D4—121no—
109#117—61no—
110C2×C4⋊C4—61no—
111C2×C4⋊C4—61no—
112C2×C4⋊C4—122no—
113C2×C4⋊C4D1211yescharacteristic
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—32no—
117C2×C22⋊C4—62no—
118C2×C22⋊C4—62no—
119C2×C42D1211yescharacteristic
120#107—121no—
121#106—121no—
122#100—32no—
123#100—61no—
124#99—22no—
125#99Dic614yescomplemented
126#95—62no—
127#95—62no—
128#95—61no—
129#95—62no—
130#95—122no—
131C23×C3—22no—
132C23×C3—22no—
133C23×C3—42no—
134C23×C3—42no—
135C23×C3#12011yescharacteristic
136C23×C3C22⋊Q812yes—
137C23×C3C4⋊Q812yes—
138C3×D4—42no—
139C3×D4—41no—
140C3×D4—41no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—22no—
144C2×C3×C4—41no—
145C2×C3×C4—41no—
146C2×C3×C4—44no—
147C2×C3×C4—42no—
148C2×C3×C4—41no—
149C2×C3×C4—82no—
150C2×C3×C4—82no—
151C2×C3×C4C22⋊Q811yescharacteristic
152C2×C3×C4C4⋊Q811yescharacteristic
153C2×Dic3—42no—
154C2×Dic3—42no—
155C2×Dic3—42no—
156C2×Dic3—42no—
157C2×Dic3—44no—
158C2×Dic3—41no—
159C2×Dic3—41no—
160C2×Dic3—42no—
161C2×Dic3—42no—
162C2×Dic3—82no—
163C2×Dic3—82no—
164C2×Dic3—82no—
165Dic6—84no—
166C3⋊C8—42no—
167C24D3×D412yes—
168C2×Q8—122no—
169C2×Q8—241no—
170C2×Q8—241no—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4C2×Dic612yes—
174C2×D4#23711yescharacteristic
175C2×D4D3×Q811yescharacteristic
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—42no—
179C22×C4—62no—
180C22×C4—62no—
181C22×C4—62no—
182C22×C4—62no—
183C22×C4—122no—
184C22×C4—242no—
185C22×C4C2×D1211yescharacteristic
186M16—62no—
187M16—121no—
188C2×C8—121no—
189C4⋊C4—41no—
190C4⋊C4—61no—
191C4⋊C4—61no—
192C4⋊C4—241no—
193C4⋊C4—241no—
194C4⋊C4—244no—
195C22⋊C4—24no—
196C22⋊C4—44no—
197C22⋊C4—42no—
198C22⋊C4—122no—
199C22⋊C4—121no—
200C22⋊C4—122no—
201C22⋊C4—242no—
202C42—41no—
203C42—121no—
204C22×C3—21no—
205C22×C3—21no—
206C22×C3—22no—
207C22×C3—22no—
208C22×C3—42no—
209C22×C3—42no—
210C22×C3—42no—
211C22×C3—42no—
212C22×C3—82no—
213C22×C3#39511yescharacteristic
214C22×C3B(4,𝔽2)11yescharacteristic
215C22×C3#45811yescharacteristic
216C3×C4—21no—
217C3×C4—21no—
218C3×C4—42no—
219C3×C4—84no—
220C3×C4—81no—
221Dic3—82no—
222Dic3—81no—
223Dic3—82no—
224Dic3—81no—
225C23—22no—
226C23—22no—
227C23—42no—
228C23—42no—
229C23#87812yes—
230C23#88211yescharacteristic
231C23#88812yes—
232Q8—244no—
233D4—42no—
234D4—41no—
235D4—41no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—122no—
247C2×C4—122no—
248C2×C4—122no—
249C2×C4—122no—
250C2×C4—124no—
251C2×C4—121no—
252C2×C4—121no—
253C2×C4—122no—
254C2×C4—122no—
255C2×C4—242no—
256C2×C4—242no—
257C2×C4—242no—
258C2×C4#86911yescharacteristic
259C2×C4#89711yescharacteristic
260C8—122no—
261C2×C3—21no—
262C2×C3—21no—
263C2×C3—42no—
264C2×C3—42no—
265C2×C3#206211yescharacteristic
266C2×C3#219811yescharacteristic
267C2×C3#219911yescharacteristic
268C22—21no—
269C22—21no—
270C22—22no—
271C22—22no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—82no—
277C22#465611yescharacteristic
278C22#474111yescharacteristic
279C22#474211yescharacteristic
280C4—21no—
281C4—21no—
282C4—42no—
283C4—84no—
284C4—81no—
285C4—242no—
286C4—241no—
287C4—242no—
288C4—241no—
289C3#1346311yescharacteristic, complemented, Sylow
290C2—21no—
291C2—21no—
292C2—42no—
293C2—42no—
294C2#6895811yescharacteristic
295C2#6914811yescharacteristic
296C2#6915411yescharacteristic
subgroupG/H
centerC22#4741
commutator, prime radical, cosocleC22×C3×D4C23
FrattiniC22×D4C22×D3
FittingC3×#1393C2
socleC22×C3B(4,𝔽2)
hypercenter#1393D3
hypocenterC3#13463
LCSG ▷ C22×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1393

Complements

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

NH ≅ G/Ntype
43. C3×#99Q8semidirect
45. #833Q8semidirect
125. #99Dic6semidirect
289. C3#13463semidirect

Maximal subgroups

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

subgroupindex
#764212
#700212
#696422
C3×#13932
#676942
#134633