Finite Group Databasebeta

← #10698430

Subgroups — #10698430

Order 768.

594 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 73 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
2#69991C212yesmaximal
3#69537C211yesmaximal, characteristic
4C3×#1306C211yesmaximal, characteristic, complemented
5#67682C212yesmaximal, complemented
6#10388—31nomaximal, Sylow
7C22×#235C2211yescharacteristic
8C3×#407C2212yes—
9C3×#381C2211yescharacteristic
10#5211C2211yescharacteristic
11C4×#217C2212yes—
12C2×#825—22no—
13C2×#825—22no—
14C2×#806—22no—
15#4948—22no—
16#4654—22no—
17#4647—22no—
18#2608—31no—
19#1789—32no—
20#1782—31no—
21#1306D311yescharacteristic, complemented
22#1284—32no—
23C23×C3×C4C2311yescharacteristic
24C22×#67—21no—
25C2×#235—22no—
26C2×#235—22no—
27C2×#235—44no—
28C22×D12Q811yescharacteristic
29C22×C4×D3—21no—
30C22×Dic6Q811yescharacteristic
31C2×C3×M16—22no—
32C2×C3×C22⋊C4D412yes—
33C2×#217—22no—
34C2×#217—21no—
35C2×C4⋊Dic3—22no—
36C2×Dic3⋊C4—22no—
37C2×Dic3⋊C4—22no—
38C2×C4×Dic3D412yes—
39C2×C4×Dic3D412yes—
40C3×C22⋊C8—22no—
41C3×C22⋊C8—22no—
42C3×#95—22no—
43C3⋊#95—22no—
44C3⋊#95—22no—
45C3⋊#95—22no—
46#825—44no—
47#825—44no—
48#806—44no—
49C22×#41—31no—
50C2×#104—62no—
51C2×#104—62no—
52C2×#99—62no—
53#407C2×D312yes—
54#396—62no—
55#389—62no—
56#387—62no—
57#381—31no—
58#381C2×D311yescharacteristic
59C4×C22⋊C4—32no—
60C24×C3C2×D411yescharacteristic
61C23×D3—21no—
62C22×C3×C4—22no—
63C22×C3×C4—42no—
64C22×C3×C4C2×D412yes—
65C22×C3×C4C2×Q811yescharacteristic
66C22×C3×C4#4112yes—
67C22×C3×C4#4111yescharacteristic
68C2×#67—24no—
69C2×#67—42no—
70C2×#67—41no—
71C2×#67—41no—
72C22×Dic3—22no—
73C22×Dic3—21no—
74C22×Dic3—22no—
75C22×Dic3—41no—
76#235—24no—
77#235—24no—
78#235—44no—
79#235—44no—
80#235—42no—
81#235—42no—
82C2×D12—22no—
83C2×D12—42no—
84C2×C4×D3—42no—
85C2×C4×D3—42no—
86C2×C4×D3—42no—
87C2×Dic6—22no—
88C2×Dic6—42no—
89C3×M16—42no—
90C3×M16—42no—
91C2×C3×C8—42no—
92C3×C22⋊C4—42no—
93#217—44no—
94#217—42no—
95C4⋊Dic3—44no—
96Dic3⋊C4—44no—
97Dic3⋊C4—82no—
98C4×Dic3—42no—
99C4×Dic3—42no—
100C2×#41—62no—
101C2×#41—62no—
102C2×#41—124no—
103C22×Q8—31no—
104C22×D4—31no—
105C22×D4—61no—
106C23×C4—61no—
107C23×C4C22×D311yescharacteristic
108C2×M16—22no—
109C2×C4⋊C4—62no—
110C2×C4⋊C4—62no—
111C2×C4⋊C4—62no—
112C2×C22⋊C4—62no—
113C2×C22⋊C4—61no—
114C2×C22⋊C4D1212yes—
115C2×C42—32no—
116C2×C42—32no—
117#104—124no—
118#104—124no—
119#99—124no—
120C22⋊C8—22no—
121C22⋊C8—22no—
122#95—22no—
123#95—62no—
124#95—62no—
125#95—62no—
126C23×C3—22no—
127C23×C3—42no—
128C23×C3#12011yescharacteristic
129C23×C3C22⋊Q811yescharacteristic
130C23×C3#12411yescharacteristic
131C22×D3—22no—
132C22×D3—41no—
133C22×D3—41no—
134C22×D3—82no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—41no—
138C2×C3×C4—41no—
139C2×C3×C4—42no—
140C2×C3×C4—42no—
141C2×C3×C4—42no—
142C2×C3×C4—42no—
143C2×C3×C4C22⋊Q812yes—
144C2×C3×C4#12412yes—
145#67—44no—
146#67—44no—
147C2×Dic3—24no—
148C2×Dic3—22no—
149C2×Dic3—42no—
150C2×Dic3—41no—
151C2×Dic3—41no—
152C2×Dic3—42no—
153C2×Dic3—42no—
154C2×Dic3—42no—
155C2×Dic3—42no—
156C2×Dic3—81no—
157C2×Dic3—82no—
158D12—24no—
159D12—42no—
160C4×D3—44no—
161C4×D3—44no—
162Dic6—24no—
163Dic6—42no—
164C3×C8—82no—
165C24—61no—
166C24C2×D1211yescharacteristic
167#41—64no—
168#41—64no—
169#41—122no—
170#41—122no—
171#41—124no—
172#41—124no—
173C2×Q8—62no—
174C2×Q8—122no—
175C2×D4—64no—
176C2×D4—62no—
177C2×D4—122no—
178C2×D4—122no—
179C2×D4—121no—
180C2×D4—121no—
181C22×C4—22no—
182C22×C4—42no—
183C22×C4—62no—
184C22×C4—61no—
185C22×C4—62no—
186C22×C4—122no—
187C22×C4—122no—
188C22×C4—122no—
189C22×C4—121no—
190C22×C4#23512yes—
191C22×C4D3×D412yes—
192C22×C4#23711yescharacteristic
193C22×C4D3×Q811yescharacteristic
194M16—42no—
195M16—42no—
196C2×C8—42no—
197C4⋊C4—124no—
198C4⋊C4—124no—
199C4⋊C4—242no—
200C22⋊C4—42no—
201C22⋊C4—124no—
202C22⋊C4—122no—
203C42—122no—
204C42—122no—
205C22×C3—22no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—82no—
211C22×C3#39311yescharacteristic
212C22×C3B(4,𝔽2)12yes—
213C2×D3—42no—
214C2×D3—82no—
215C2×D3—82no—
216C2×D3—82no—
217C3×C4—22no—
218C3×C4—22no—
219C3×C4—82no—
220Dic3—44no—
221Dic3—42no—
222Dic3—82no—
223Dic3—161no—
224C23—22no—
225C23—42no—
226C23—62no—
227C23—121no—
228C23—121no—
229C23—242no—
230C23#87511yescharacteristic
231C23#88211yescharacteristic
232C23#89711yescharacteristic
233Q8—64no—
234Q8—122no—
235D4—64no—
236D4—122no—
237D4—124no—
238D4—124no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—64no—
248C2×C4—62no—
249C2×C4—124no—
250C2×C4—124no—
251C2×C4—122no—
252C2×C4—121no—
253C2×C4—121no—
254C2×C4—122no—
255C2×C4—122no—
256C2×C4—122no—
257C2×C4—122no—
258C2×C4—241no—
259C2×C4—242no—
260C2×C4#88512yes—
261C2×C4#89612yes—
262C8—82no—
263C2×C3—22no—
264C2×C3—41no—
265C2×C3—41no—
266C2×C3#202911yescharacteristic
267C2×C3#203112yes—
268D3—82no—
269C22—22no—
270C22—21no—
271C22—21no—
272C22—22no—
273C22—22no—
274C22—82no—
275C22—122no—
276C22—242no—
277C22—242no—
278C22—242no—
279C22#467311yescharacteristic
280C22#474112yes—
281C4—22no—
282C4—22no—
283C4—82no—
284C4—124no—
285C4—122no—
286C4—242no—
287C4—481no—
288C3#1038811yescharacteristic, complemented, Sylow
289C2—22no—
290C2—41no—
291C2—41no—
292C2—242no—
293C2#6897111yescharacteristic
294C2#6904212yes—
subgroupG/H
centerC22#4673
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1306C2
socleC22×C3#393
prime radicalC22×#235C22
hypercenter#1306D3
hypocenterC3#10388
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1306

Complements

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

NH ≅ G/Ntype
1. #74089C2semidirect
4. C3×#1306C2semidirect
5. #67682C2semidirect
21. #1306D3semidirect
288. C3#10388semidirect

Maximal subgroups

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

subgroupindex
#740892
#699912
#695372
C3×#13062
#676822
#103883