Finite Group Databasebeta

← #10698429

Subgroups — #10698429

Order 768.

594 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 75 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#74089C211yesmaximal, characteristic, complemented
2C3×#1782C211yesmaximal, characteristic
3#72138C212yesmaximal
4#67629C211yesmaximal, characteristic, complemented
5#67628C212yesmaximal, complemented
6#10388—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C2×C3×#99—22no—
9C3×#381C2211yescharacteristic
10C3×#381C2211yescharacteristic
11C2×#837—22no—
12C2×#837—22no—
13C4×#217C2212yes—
14#5207C2212yes—
15#4948—22no—
16#4654—22no—
17#4647—22no—
18#2608—31no—
19#1789—32no—
20#1782D311yescharacteristic
21#1306—31no—
22#1284—32no—
23C2×C3×#41—22no—
24C2×C3×#41—22no—
25C2×C3×#41—44no—
26C22×C3×Q8Q811yescharacteristic
27C22×C3×D4—21no—
28C22×C3×D4Q811yescharacteristic
29C23×C3×C4—21no—
30C23×C3×C4C2311yescharacteristic
31C2×C3×C22⋊C4—21no—
32C2×C3×C22⋊C4D412yes—
33C2×#217—22no—
34C2×C4⋊Dic3—22no—
35C2×Dic3⋊C4—22no—
36C2×Dic3⋊C4—22no—
37C2×C4×Dic3D412yes—
38C2×C4×Dic3D412yes—
39C2×#208—22no—
40C3×#99—44no—
41C3×#95—22no—
42C3×#95—22no—
43#837—44no—
44#837—44no—
45C3⋊#95—22no—
46C3⋊#95—22no—
47#830—22no—
48#830—22no—
49C22×#41C2×D311yescharacteristic
50C2×#104—62no—
51C2×#104—62no—
52C2×#99—22no—
53#407—32no—
54#396—62no—
55#389—62no—
56#387—62no—
57#381C2×D311yescharacteristic
58#381C2×D311yescharacteristic
59C4×C22⋊C4—32no—
60C24×C3—21no—
61C24×C3C2×D411yescharacteristic
62C3×#41—24no—
63C3×#41—24no—
64C3×#41—42no—
65C3×#41—42no—
66C3×#41—44no—
67C3×#41—44no—
68C2×C3×Q8—22no—
69C2×C3×Q8—42no—
70C2×C3×D4—24no—
71C2×C3×D4—22no—
72C2×C3×D4—42no—
73C2×C3×D4—42no—
74C2×C3×D4—41no—
75C2×C3×D4—41no—
76C22×C3×C4—21no—
77C22×C3×C4—22no—
78C22×C3×C4—42no—
79C22×C3×C4—42no—
80C22×C3×C4—42no—
81C22×C3×C4—42no—
82C22×C3×C4—41no—
83C22×C3×C4C2×D412yes—
84C22×C3×C4C2×Q811yescharacteristic
85C22×C3×C4#4111yescharacteristic
86C22×C3×C4#4112yes—
87C22×Dic3—22no—
88C22×Dic3—22no—
89C3×C22⋊C4—42no—
90C3×C22⋊C4—42no—
91#217—44no—
92C4⋊Dic3—44no—
93Dic3⋊C4—44no—
94Dic3⋊C4—82no—
95C4×Dic3—42no—
96C4×Dic3—42no—
97#208—42no—
98#208—42no—
99C2×C3⋊C8—42no—
100C2×#41—22no—
101C2×#41—22no—
102C2×#41—44no—
103C22×Q8Dic611yescharacteristic
104C22×D4—21no—
105C22×D4Dic611yescharacteristic
106C23×C4—21no—
107C23×C4C22×D311yescharacteristic
108C2×M16—62no—
109C2×C4⋊C4—62no—
110C2×C4⋊C4—62no—
111C2×C4⋊C4—62no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—62no—
114C2×C22⋊C4#6712yes—
115C2×C42—32no—
116C2×C42—32no—
117#104—124no—
118#104—124no—
119#99—44no—
120C22⋊C8—62no—
121C22⋊C8—62no—
122#95—22no—
123#95—22no—
124#95—62no—
125#95—62no—
126C23×C3—22no—
127C23×C3—22no—
128C23×C3—42no—
129C23×C3—41no—
130C23×C3—41no—
131C23×C3—82no—
132C23×C3#12011yescharacteristic
133C23×C3C22⋊Q811yescharacteristic
134C23×C3#12411yescharacteristic
135C3×Q8—24no—
136C3×Q8—42no—
137C3×D4—24no—
138C3×D4—42no—
139C3×D4—44no—
140C3×D4—44no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—41no—
145C2×C3×C4—41no—
146C2×C3×C4—42no—
147C2×C3×C4—44no—
148C2×C3×C4—44no—
149C2×C3×C4—41no—
150C2×C3×C4—41no—
151C2×C3×C4—42no—
152C2×C3×C4—42no—
153C2×C3×C4—42no—
154C2×C3×C4—81no—
155C2×C3×C4—82no—
156C2×C3×C4C22⋊Q812yes—
157C2×C3×C4#12412yes—
158C2×Dic3—24no—
159C2×Dic3—42no—
160C2×Dic3—42no—
161C2×Dic3—42no—
162C2×Dic3—42no—
163C2×Dic3—42no—
164C3⋊C8—82no—
165C24—21no—
166C24C2×#6711yescharacteristic
167#41—24no—
168#41—24no—
169#41—42no—
170#41—42no—
171#41—44no—
172#41—44no—
173C2×Q8—22no—
174C2×Q8—42no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—41no—
180C2×D4—41no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—62no—
189C22×C4—62no—
190C22×C4C2×Dic611yescharacteristic
191C22×C4#23511yescharacteristic
192C22×C4D3×D412yes—
193C22×C4#23712yes—
194M16—122no—
195M16—122no—
196C2×C8—122no—
197C4⋊C4—124no—
198C4⋊C4—124no—
199C4⋊C4—242no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—124no—
203C42—122no—
204C42—122no—
205C22×C3—22no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—42no—
211C22×C3—82no—
212C22×C3—82no—
213C22×C3—82no—
214C22×C3—82no—
215C22×C3#39311yescharacteristic
216C22×C3B(4,𝔽2)12yes—
217C3×C4—22no—
218C3×C4—22no—
219C3×C4—42no—
220C3×C4—82no—
221C3×C4—161no—
222Dic3—44no—
223Dic3—82no—
224C23—22no—
225C23—22no—
226C23—42no—
227C23—41no—
228C23—41no—
229C23—82no—
230C23#92411yescharacteristic
231C23#93611yescharacteristic
232C23#93711yescharacteristic
233Q8—24no—
234Q8—42no—
235D4—24no—
236D4—42no—
237D4—44no—
238D4—44no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—64no—
253C2×C4—81no—
254C2×C4—82no—
255C2×C4—122no—
256C2×C4—122no—
257C2×C4—122no—
258C2×C4—122no—
259C2×C4—122no—
260C2×C4#87812yes—
261C2×C4#88512yes—
262C8—242no—
263C2×C3—22no—
264C2×C3—41no—
265C2×C3—41no—
266C2×C3—82no—
267C2×C3#202911yescharacteristic
268C2×C3#203112yes—
269C22—22no—
270C22—21no—
271C22—21no—
272C22—22no—
273C22—22no—
274C22—42no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22#494711yescharacteristic
280C22#503212yes—
281C4—22no—
282C4—22no—
283C4—42no—
284C4—82no—
285C4—124no—
286C4—161no—
287C4—242no—
288C3#1038811yescharacteristic, complemented, Sylow
289C2—22no—
290C2—41no—
291C2—41no—
292C2—82no—
293C2#6949711yescharacteristic
294C2#7003812yes—
subgroupG/H
centerC22#4947
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1782C2
socleC22×C3#393
prime radicalC22×C3×#41C22
hypercenter#1782D3
hypocenterC3#10388
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
1. #74089C2semidirect
4. #67629C2semidirect
5. #67628C2semidirect
288. C3#10388semidirect

Maximal subgroups

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

subgroupindex
#740892
C3×#17822
#721382
#676292
#676282
#103883