Finite Group Databasebeta

← #10698312

Subgroups — #10698312

Order 768.

454 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 101 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×#2399C211yesmaximal, characteristic, complemented
2#69469C212yesmaximal
3#69440C211yesmaximal, characteristic
4#67620C212yesmaximal, complemented
5#67614C211yesmaximal, characteristic, complemented
6#10370—31nomaximal, Sylow
7C2×C3×C4×Q8—21no—
8#5477C2211yescharacteristic
9C3×#398—22no—
10C3×#395—22no—
11C3×C4×C4⋊C4—21no—
12C3×C4×C4⋊C4C2212yes—
13C3×#376C2211yescharacteristic
14C2×#805—21no—
15C2×#805—21no—
16#4925C2212yes—
17#4924C2211yescharacteristic
18#4451—22no—
19#4451—22no—
20#2399D311yescharacteristic, complemented
21#1963—31no—
22#1502—32no—
23#1303—32no—
24#1274—31no—
25C22×C3×Q8—21no—
26C2×#235—21no—
27C3×C4×Q8—42no—
28C3×C4×Q8—42no—
29C2×C3×C4⋊C4—22no—
30C2×C3×C4⋊C4—22no—
31C2×C3×C4⋊C4—21no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C42—22no—
34C2×C3×C42D411yescharacteristic
35C2×C3×C42D411yescharacteristic
36C2×C3×C42C2311yescharacteristic
37#930—22no—
38C2×#208—21no—
39#875—21no—
40#874Q811yescharacteristic, complemented
41C4×D12—22no—
42#869Q811yescharacteristic, complemented
43C3×#95—21no—
44C3×#95—21no—
45C3×#95—22no—
46C3×#95D412yes—
47C3×#95D412yes—
48#830—22no—
49#805—42no—
50#805—42no—
51C4⋊(C3⋊C8)—22no—
52C4⋊(C3⋊C8)—22no—
53C4×C3⋊C8—22no—
54#532—31no—
55C2×C4×Q8—21no—
56#431—32no—
57#423—31no—
58C2×#104—61no—
59C2×#104—61no—
60#398—22no—
61#395—22no—
62C4×C4⋊C4—21no—
63C4×C4⋊C4C2×D312yes—
64#376C2×D311yescharacteristic
65#327—62no—
66#327—62no—
67C2×C3×Q8—42no—
68C2×C3×Q8—44no—
69C22×C3×C4—21no—
70C22×C3×C4—21no—
71C22×C3×C4—22no—
72C22×C3×C4—22no—
73C22×C3×C4C2×D412yes—
74C22×C3×C4C2×D411yescharacteristic
75C2×#67—41no—
76#235—41no—
77#235—41no—
78#235—41no—
79#235—41no—
80C2×D12—21no—
81C2×D12—22no—
82C2×C4×D3—41no—
83C2×Dic6—21no—
84C3×C4⋊C4—44no—
85C3×C4⋊C4—44no—
86C3×C4⋊C4—44no—
87C3×C4⋊C4—42no—
88C3×C42—21no—
89C3×C42—21no—
90C3×C42—42no—
91C3×C42—42no—
92C3×C42D812yes—
93C3×C42SD1612yes—
94C3×C42C2×Q811yescharacteristic
95C3×C42#4112yes—
96C3×C42#4111yescharacteristic
97(C2×D3)⋊C4—42no—
98C4⋊Dic3—22no—
99#208—41no—
100#208—41no—
101C2×C3⋊C8—42no—
102C2×C3⋊C8—41no—
103C2×#41—61no—
104C22×Q8—21no—
105C2×M16—61no—
106C4⋊Q8—31no—
107#127—31no—
108#124—61no—
109#121—62no—
110C4×Q8—42no—
111C4×Q8—42no—
112C4×D4—62no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C42—22no—
118C2×C42#6711yescharacteristic
119C2×C42#6711yescharacteristic
120C2×C42C22×D311yescharacteristic
121C4⋊C8—62no—
122C4⋊C8—62no—
123#104—122no—
124#104—122no—
125C22⋊C8—62no—
126C4×C8—62no—
127#95—21no—
128#95—21no—
129#95—22no—
130#95#6712yes—
131#95#6712yes—
132C23×C3#12011yescharacteristic
133C22×D3—41no—
134C3×Q8—48no—
135C2×C3×C4—24no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—42no—
139C2×C3×C4—42no—
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×C4C22⋊Q812yes—
147C2×C3×C4C22⋊Q811yescharacteristic
148C2×C3×C4#12412yes—
149C2×C3×C4#12411yescharacteristic
150C2×C3×C4C2×D811yescharacteristic
151C2×C3×C4C2×SD1611yescharacteristic
152C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
153C2×C3×C4#13711yescharacteristic
154#67—81no—
155#67—81no—
156C2×Dic3—41no—
157D12—41no—
158D12—41no—
159D12—82no—
160C4×D3—81no—
161C4×D3—81no—
162Dic6—41no—
163Dic6—41no—
164C3⋊C8—82no—
165C3⋊C8—81no—
166#41—121no—
167#41—121no—
168#41—121no—
169#41—121no—
170C2×Q8—42no—
171C2×Q8—44no—
172C2×Q8—61no—
173C2×D4—61no—
174C2×D4—62no—
175C2×D4—121no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—121no—
181C22×C4C2×#6712yes—
182C22×C4C2×#6711yescharacteristic
183M16—121no—
184M16—121no—
185C2×C8—122no—
186C2×C8—121no—
187C4⋊C4—44no—
188C4⋊C4—42no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—62no—
192C22⋊C4—122no—
193C42—21no—
194C42—21no—
195C42—42no—
196C42—42no—
197C42#21312yes—
198C42#21512yes—
199C42D3×Q811yescharacteristic
200C42#23911yescharacteristic
201C42#23912yes—
202C22×C3—22no—
203C22×C3#39311yescharacteristic
204C22×C3#44711yescharacteristic
205C22×C3#44811yescharacteristic
206C2×D3—81no—
207C2×D3—82no—
208C3×C4—22no—
209C3×C4—22no—
210C3×C4—42no—
211C3×C4—44no—
212C3×C4—42no—
213C3×C4—82no—
214C3×C4#47411yescharacteristic
215C3×C4#47611yescharacteristic
216C3×C4#48611yescharacteristic
217C3×C4#48911yescharacteristic
218Dic3—81no—
219C23—121no—
220C23#95311yescharacteristic
221Q8—48no—
222Q8—121no—
223Q8—121no—
224D4—121no—
225D4—121no—
226D4—242no—
227D4—241no—
228D4—241no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—121no—
241C2×C4—241no—
242C2×C4—241no—
243C2×C4C2×#21311yescharacteristic
244C2×C4#93211yescharacteristic
245C2×C4C2×#21511yescharacteristic
246C2×C4#94211yescharacteristic
247C2×C4#94612yes—
248C2×C4#94611yescharacteristic
249C2×C4#94712yes—
250C2×C4#94711yescharacteristic
251C8—242no—
252C8—241no—
253C2×C3—21no—
254C2×C3—21no—
255C2×C3#202311yescharacteristic
256C2×C3#203112yes—
257D3—161no—
258C22—22no—
259C22—241no—
260C22—242no—
261C22#521711yescharacteristic
262C22#522711yescharacteristic
263C22#523111yescharacteristic
264C4—22no—
265C4—22no—
266C4—44no—
267C4—42no—
268C4—42no—
269C4—82no—
270C4—241no—
271C4#508411yescharacteristic
272C4#508711yescharacteristic
273C4#508811yescharacteristic
274C4#508911yescharacteristic
275C3#1037011yescharacteristic, complemented, Sylow
276C2—21no—
277C2—21no—
278C2—481no—
279C2#7179211yescharacteristic
280C2#7184212yes—
subgroupG/H
centerC2×C4#947
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2399C2
socleC22×C3#393
prime radical#5477C22
hypercenter#2399D3
hypocenterC3#10370
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
1. C3×#2399C2semidirect
4. #67620C2semidirect
5. #67614C2semidirect
20. #2399D3semidirect
40. #874Q8semidirect
42. #869Q8semidirect
275. C3#10370semidirect

Maximal subgroups

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

subgroupindex
C3×#23992
#694692
#694402
#676202
#676142
#103703