Finite Group Databasebeta

← #10698426

Subgroups — #10698426

Order 768.

594 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C3×#2608C211yesmaximal, characteristic, complemented
2#69537C211yesmaximal, characteristic
3#69454C212yesmaximal
4#67629C211yesmaximal, characteristic, complemented
5#67616C212yesmaximal, complemented
6#10388—31nomaximal, Sylow
7C22×#235C2211yescharacteristic
8C3×#396—22no—
9C3×#389—22no—
10C3×#387—22no—
11C3×#381C2211yescharacteristic
12C3×C4×C22⋊C4C2212yes—
13#5211C2211yescharacteristic
14#5207C2212yes—
15C2×#806—22no—
16C2×#805—22no—
17C2×#805—22no—
18#2608D311yescharacteristic, complemented
19#1789—32no—
20#1782—31no—
21#1306—31no—
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×C4⋊C4—22no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C22⋊C4—22no—
35C2×C3×C22⋊C4D412yes—
36C2×C3×C42D412yes—
37C2×C3×C42D412yes—
38C2×#217—21no—
39C2×#208—22no—
40C3×#95—22no—
41C3×#95—22no—
42C3×#95—22no—
43C3⋊#95—22no—
44#830—22no—
45#830—22no—
46#806—44no—
47#805—44no—
48#805—44no—
49C22×#41—31no—
50C2×#104—62no—
51C2×#104—62no—
52C2×#99—62no—
53#407—32no—
54#396—22no—
55#389—22no—
56#387—22no—
57#381—31no—
58#381C2×D311yescharacteristic
59C4×C22⋊C4C2×D312yes—
60C24×C3C2×D411yescharacteristic
61C23×D3—21no—
62C22×C3×C4—22no—
63C22×C3×C4—22no—
64C22×C3×C4—22no—
65C22×C3×C4—42no—
66C22×C3×C4C2×D412yes—
67C22×C3×C4C2×Q811yescharacteristic
68C22×C3×C4#4112yes—
69C22×C3×C4#4111yescharacteristic
70C2×#67—24no—
71C2×#67—42no—
72C2×#67—41no—
73C2×#67—41no—
74C22×Dic3—21no—
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×C4⋊C4—44no—
90C3×C4⋊C4—44no—
91C3×C4⋊C4—82no—
92C3×C22⋊C4—44no—
93C3×C22⋊C4—42no—
94C3×C42—42no—
95C3×C42—42no—
96#217—42no—
97#208—42no—
98#208—42no—
99C2×C3⋊C8—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—62no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—61no—
114C2×C22⋊C4#6712yes—
115C2×C42#6712yes—
116C2×C42#6712yes—
117#104—124no—
118#104—124no—
119#99—124no—
120C22⋊C8—62no—
121C22⋊C8—62no—
122#95—22no—
123#95—22no—
124#95—22no—
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—24no—
138C2×C3×C4—41no—
139C2×C3×C4—41no—
140C2×C3×C4—42no—
141C2×C3×C4—42no—
142C2×C3×C4—42no—
143C2×C3×C4—42no—
144C2×C3×C4—42no—
145C2×C3×C4—42no—
146C2×C3×C4—42no—
147C2×C3×C4—42no—
148C2×C3×C4—42no—
149C2×C3×C4C22⋊Q812yes—
150C2×C3×C4#12412yes—
151#67—44no—
152#67—44no—
153C2×Dic3—22no—
154C2×Dic3—41no—
155C2×Dic3—41no—
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×#6711yescharacteristic
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—22no—
183C22×C4—22no—
184C22×C4—42no—
185C22×C4—61no—
186C22×C4—122no—
187C22×C4—122no—
188C22×C4—122no—
189C22×C4—121no—
190C22×C4D3×Q811yescharacteristic
191C22×C4#23912yes—
192C22×C4#23911yescharacteristic
193C22×C4C2×#6712yes—
194M16—122no—
195M16—122no—
196C2×C8—122no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—82no—
200C22⋊C4—44no—
201C22⋊C4—42no—
202C22⋊C4—122no—
203C42—42no—
204C42—42no—
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—44no—
220C3×C4—82no—
221C3×C4—82no—
222Dic3—42no—
223Dic3—161no—
224C23—22no—
225C23—42no—
226C23—62no—
227C23—121no—
228C23—121no—
229C23—242no—
230C23#94611yescharacteristic
231C23#94711yescharacteristic
232C23#95311yescharacteristic
233Q8—64no—
234Q8—122no—
235D4—64no—
236D4—122no—
237D4—124no—
238D4—124no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—62no—
254C2×C4—124no—
255C2×C4—124no—
256C2×C4—121no—
257C2×C4—121no—
258C2×C4—241no—
259C2×C4—242no—
260C2×C4#94612yes—
261C2×C4#94712yes—
262C8—242no—
263C2×C3—22no—
264C2×C3—41no—
265C2×C3—41no—
266C2×C3#202911yescharacteristic
267C2×C3#203112yes—
268D3—82no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—21no—
273C22—21no—
274C22—82no—
275C22—122no—
276C22—242no—
277C22—242no—
278C22—242no—
279C22#523111yescharacteristic
280C22#524412yes—
281C4—22no—
282C4—22no—
283C4—44no—
284C4—82no—
285C4—82no—
286C4—122no—
287C4—481no—
288C3#1038811yescharacteristic, complemented, Sylow
289C2—22no—
290C2—41no—
291C2—41no—
292C2—242no—
293C2#7184212yes—
294C2#7189711yescharacteristic
subgroupG/H
centerC22#5231
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2608C2
socleC22×C3#393
prime radicalC22×#235C22
hypercenter#2608D3
hypocenterC3#10388
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2608

Complements

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

NH ≅ G/Ntype
1. C3×#2608C2semidirect
4. #67629C2semidirect
5. #67616C2semidirect
18. #2608D3semidirect
288. C3#10388semidirect

Maximal subgroups

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

subgroupindex
C3×#26082
#695372
#694542
#676292
#676162
#103883