Finite Group Databasebeta

← #10708182

Subgroups — #10708182

Order 768.

558 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 59 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#74368C211yesmaximal, characteristic
2#69573C211yesmaximal, characteristic, complemented
3#68138C212yesmaximal, complemented
4C3×#1325C211yesmaximal, characteristic, complemented
5#67456C212yesmaximal, complemented
6#12246—31nomaximal, Sylow
7C2×#895—22no—
8#5500C2211yescharacteristic
9C2×C4×D12—21no—
10C3×#384C2211yescharacteristic
11#4991—21no—
12#4991C2212yes—
13#4987—22no—
14#4964—21no—
15#4731—22no—
16C3×#324C2212yes—
17#4530C2211yescharacteristic
18#4523—21no—
19#4467—22no—
20#2537—31no—
21#2060—31no—
22#1599—32no—
23#1345—32no—
24#1325D311yescharacteristic, complemented
25C22×D12—21no—
26C22×D12—21no—
27C22×D12D412yes—
28C22×C4×D3—22no—
29C2×C3×C4⋊C4—22no—
30C2×C3×C4⋊C4D411yescharacteristic
31C2×C3×C4⋊C4C2311yescharacteristic
32C2×C3×C42D411yescharacteristic
33C2×(C2×D3)⋊C4—22no—
34C2×(C2×D3)⋊C4—22no—
35C2×(C2×D3)⋊C4—22no—
36#926—21no—
37C2×C4⋊Dic3—21no—
38#924—22no—
39C2×#208—21no—
40C2×#204—42no—
41#895—48no—
42#890—41no—
43#888—41no—
44C4×D12—42no—
45C4×D12—42no—
46C3×C22⋊C8—22no—
47C3⋊#95—21no—
48C3⋊#95D412yes—
49#821—42no—
50#808—41no—
51#807—41no—
52#544—31no—
53C2×#121—62no—
54C2×C4×D4—61no—
55#450—62no—
56#428—61no—
57#394—32no—
58#394—61no—
59#384C2×D311yescharacteristic
60#380—62no—
61#343—31no—
62#339—61no—
63#328—62no—
64#324C2×D312yes—
65C23×D3—22no—
66C22×C3×C4—21no—
67C22×C3×C4—21no—
68C22×C3×C4C2×D411yescharacteristic
69C22×C3×C4C2×D412yes—
70C22×Dic3—22no—
71C2×D12—24no—
72C2×D12—42no—
73C2×D12—42no—
74C2×D12—42no—
75C2×D12—44no—
76C2×D12—44no—
77C2×D12—81no—
78C2×C4×D3—44no—
79C2×C4×D3—44no—
80C2×C4×D3—44no—
81C2×Dic6—22no—
82C2×C3×C8—42no—
83C3×C4⋊C4—41no—
84C3×C4⋊C4—41no—
85C3×C4⋊C4—44no—
86C3×C42—21no—
87C3×C42SD1612yescomplemented
88#217—41no—
89(C2×D3)⋊C4—44no—
90(C2×D3)⋊C4—44no—
91(C2×D3)⋊C4—42no—
92(C2×D3)⋊C4—42no—
93C4⋊Dic3—21no—
94C4⋊Dic3—21no—
95C4⋊Dic3—22no—
96C4⋊Dic3—42no—
97Dic3⋊C4—42no—
98C4×Dic3—41no—
99#208—22no—
100#208—41no—
101C2×C3⋊C8—41no—
102#204—84no—
103C22×D4—32no—
104C22×D4—61no—
105C22×D4—61no—
106C23×C4—62no—
107C2×SD16—122no—
108C2×M16—61no—
109C4⋊Q8—121no—
110#125—121no—
111C22⋊Q8—62no—
112#121—128no—
113C4×D4—122no—
114C4×D4—122no—
115#117—61no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—61no—
118C2×C4⋊C4D1211yescharacteristic
119C2×C4⋊C4C22×D311yescharacteristic
120C2×C22⋊C4—62no—
121C2×C22⋊C4—62no—
122C2×C22⋊C4—62no—
123C2×C42D1211yescharacteristic
124#107—121no—
125#106—121no—
126#102—122no—
127C22⋊C8—22no—
128#95—32no—
129#95—61no—
130C23×C3#12011yescharacteristic
131C22×D3—24no—
132C22×D3—42no—
133C22×D3—42no—
134C22×D3—42no—
135C22×D3—42no—
136C22×D3—42no—
137C22×D3—42no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—22no—
144C2×C3×C4—42no—
145C2×C3×C4—42no—
146C2×C3×C4—41no—
147C2×C3×C4C2×SD1611yescharacteristic
148C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
149C2×Dic3—42no—
150C2×Dic3—42no—
151C2×Dic3—42no—
152C2×Dic3—41no—
153C2×Dic3—41no—
154D12—48no—
155D12—44no—
156D12—82no—
157D12—84no—
158C4×D3—88no—
159Dic6—82no—
160C3×C8—82no—
161C3⋊C8—42no—
162C24—62no—
163C2×Q8—62no—
164C2×D4—64no—
165C2×D4—122no—
166C2×D4—122no—
167C2×D4—122no—
168C2×D4—124no—
169C2×D4—124no—
170C2×D4—241no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—62no—
174C22×C4—124no—
175C22×C4—124no—
176C22×C4—124no—
177C22×C4C2×D1211yescharacteristic
178C22×C4D3×D412yes—
179SD16—244no—
180M16—62no—
181M16—121no—
182C2×C8—42no—
183C2×C8—121no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—44no—
187C4⋊C4—62no—
188C4⋊C4—61no—
189C4⋊C4—61no—
190C4⋊C4—122no—
191C4⋊C4—122no—
192C22⋊C4—124no—
193C22⋊C4—124no—
194C22⋊C4—121no—
195C22⋊C4—122no—
196C22⋊C4—122no—
197C42—21no—
198C42—121no—
199C42#20412yescomplemented
200C22×C3—21no—
201C22×C3—21no—
202C22×C3#45011yescharacteristic
203C22×C3#45611yescharacteristic
204C22×C3B(4,𝔽2)11yescharacteristic
205C2×D3—44no—
206C2×D3—44no—
207C2×D3—44no—
208C2×D3—82no—
209C2×D3—82no—
210C2×D3—82no—
211C2×D3—82no—
212C3×C4—24no—
213C3×C4—21no—
214C3×C4—21no—
215C3×C4—42no—
216C3×C4—81no—
217Dic3—82no—
218Dic3—81no—
219Dic3—81no—
220C23—64no—
221C23—122no—
222C23—122no—
223C23—122no—
224C23—122no—
225C23—122no—
226C23—122no—
227C23#88211yescharacteristic
228Q8—242no—
229D4—128no—
230D4—124no—
231D4—242no—
232D4—244no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—122no—
243C2×C4—122no—
244C2×C4—122no—
245C2×C4—121no—
246C2×C4—121no—
247C2×C4—248no—
248C2×C4C2×#20411yescharacteristic
249C2×C4#90811yescharacteristic
250C8—82no—
251C8—122no—
252C2×C3—21no—
253C2×C3—21no—
254C2×C3#160411yescharacteristic
255C2×C3#219611yescharacteristic
256C2×C3#220411yescharacteristic
257D3—84no—
258C22—21no—
259C22—21no—
260C22—124no—
261C22—124no—
262C22—124no—
263C22—242no—
264C22—242no—
265C22—242no—
266C22—242no—
267C22#473111yescharacteristic
268C22#474111yescharacteristic
269C22#475211yescharacteristic
270C4—24no—
271C4—21no—
272C4—21no—
273C4—42no—
274C4—81no—
275C4—242no—
276C4—241no—
277C4—241no—
278C3#1224611yescharacteristic, complemented, Sylow
279C2—21no—
280C2—21no—
281C2—244no—
282C2#6813311yescharacteristic
283C2#6914611yescharacteristic
284C2#6916811yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1325C2
socleC22×C3B(4,𝔽2)
prime radical#74368C2
hypercenter#1325D3
hypocenterC3#12246
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
2. #69573C2semidirect
3. #68138C2semidirect
4. C3×#1325C2semidirect
5. #67456C2semidirect
24. #1325D3semidirect
87. C3×C42SD16semidirect
199. C42#204semidirect
278. C3#12246semidirect

Maximal subgroups

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

subgroupindex
#743682
#695732
#681382
C3×#13252
#674562
#122463