Finite Group Databasebeta

← #10707449

Subgroups — #10707449

Order 768.

398 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 63 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×#2809C211yesmaximal, characteristic, complemented
2#69718C212yesmaximal
3#69694C211yesmaximal, characteristic
4#67652C212yesmaximal, complemented
5#67643C211yesmaximal, characteristic, complemented
6#12131—31nomaximal, Sylow
7C2×#942—21no—
8#5510C2211yescharacteristic
9C3×#400C2212yes—
10C3×#398—22no—
11C3×#391—21no—
12C3×#391C2211yescharacteristic
13C3×#389—21no—
14C3×#387—22no—
15#4968—22no—
16C2×#805—21no—
17#4530C2211yescharacteristic
18#4528C2212yes—
19#4451—22no—
20#2809D311yescharacteristic, complemented
21#2007—31no—
22#1616—32no—
23#1353—32no—
24#1319—31no—
25C22×C3×Q8D411yescharacteristic
26C2×#235—21no—
27C2×C3×C4⋊C4—22no—
28C2×C3×C4⋊C4—21no—
29C2×C3×C4⋊C4—21no—
30C2×C3×C4⋊C4—21no—
31C2×C3×C4⋊C4D412yes—
32C2×C3×C4⋊C4C2311yescharacteristic
33C2×C3×C42—21no—
34C2×C3×C42D411yescharacteristic
35C2×#216—41no—
36#942—42no—
37#942—42no—
38C2×#215—41no—
39#930—22no—
40C2×#208—21no—
41#897—41no—
42#895—41no—
43C3×#95—22no—
44C3×#95—22no—
45C3×#95—22no—
46C3×#95—22no—
47C3×#95D412yes—
48#830—22no—
49#809—42no—
50#808—42no—
51#805—42no—
52C2×#137—61no—
53#537—31no—
54#481—62no—
55C2×#104—61no—
56#400C2×D312yes—
57#398—22no—
58#391—21no—
59#391C2×D311yescharacteristic
60#389—21no—
61#387—22no—
62#343—31no—
63#327—62no—
64#324—32no—
65C2×C3×Q8—21no—
66C2×C3×Q8—41no—
67C2×C3×Q8—41no—
68C2×C3×Q8SD1612yes—
69C22×C3×C4—21no—
70C22×C3×C4—21no—
71C22×C3×C4—22no—
72C22×C3×C4—22no—
73C22×C3×C4C2×D411yescharacteristic
74C22×C3×C4C2×D412yes—
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—41no—
85C3×C4⋊C4—42no—
86C3×C4⋊C4—44no—
87C3×C4⋊C4—42no—
88C3×C4⋊C4—81no—
89C3×C4⋊C4—81no—
90C3×C42—41no—
91C3×C42—81no—
92#216—44no—
93#215—44no—
94(C2×D3)⋊C4—42no—
95C4⋊Dic3—22no—
96#208—22no—
97#208—41no—
98C2×C3⋊C8—41no—
99C2×C3⋊C8—42no—
100C2×#41—61no—
101C22×Q8#6711yescharacteristic
102#137—122no—
103#137—122no—
104C2×Q16—121no—
105C2×SD16—121no—
106C2×M16—61no—
107C22⋊Q8—121no—
108#121—62no—
109#121—121no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4#6712yes—
115C2×C4⋊C4C22×D311yescharacteristic
116C2×C42—21no—
117C2×C42#6711yescharacteristic
118#106—122no—
119#104—122no—
120#102—122no—
121C22⋊C8—62no—
122#95—22no—
123#95—22no—
124#95—22no—
125#95—22no—
126#95#6712yes—
127C23×C3#12011yescharacteristic
128C22×D3—41no—
129C3×Q8—42no—
130C3×Q8—81no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—41no—
137C2×C3×C4—41no—
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×C4C2×SD1611yescharacteristic
148C2×C3×C4#13711yescharacteristic
149#67—81no—
150#67—81no—
151C2×Dic3—41no—
152D12—41no—
153D12—41no—
154D12—82no—
155C4×D3—81no—
156C4×D3—81no—
157Dic6—41no—
158Dic6—41no—
159C3⋊C8—42no—
160C3⋊C8—82no—
161#41—121no—
162#41—121no—
163#41—121no—
164#41—121no—
165C2×Q8—21no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—61no—
169C2×Q8#21512yes—
170C2×D4—61no—
171C2×D4—62no—
172C2×D4—121no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—121no—
178C22×C4C2×#6711yescharacteristic
179C22×C4C2×#6712yes—
180Q16—124no—
181SD16—124no—
182M16—62no—
183M16—121no—
184C2×C8—121no—
185C2×C8—122no—
186C4⋊C4—41no—
187C4⋊C4—42no—
188C4⋊C4—44no—
189C4⋊C4—42no—
190C4⋊C4—62no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C22⋊C4—122no—
194C42—41no—
195C42—81no—
196C22×C3—21no—
197C22×C3—21no—
198C22×C3#44811yescharacteristic
199C22×C3#45311yescharacteristic
200C22×C3B(4,𝔽2)11yescharacteristic
201C2×D3—81no—
202C2×D3—81no—
203C2×D3—81no—
204C3×C4—21no—
205C3×C4—21no—
206C3×C4—42no—
207C3×C4—42no—
208C3×C4—42no—
209C3×C4—82no—
210C3×C4—82no—
211Dic3—81no—
212C23—121no—
213C23#95311yescharacteristic
214Q8—42no—
215Q8—81no—
216Q8—121no—
217Q8—121no—
218D4—121no—
219D4—121no—
220D4—242no—
221D4—241no—
222D4—241no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—121no—
240C2×C4—241no—
241C2×C4—241no—
242C2×C4C2×#21511yescharacteristic
243C2×C4#94211yescharacteristic
244C8—122no—
245C8—242no—
246C2×C3—21no—
247C2×C3—21no—
248C2×C3#159611yescharacteristic
249C2×C3#220111yescharacteristic
250C2×C3#220411yescharacteristic
251D3—161no—
252C22—21no—
253C22—21no—
254C22—241no—
255C22—241no—
256C22—241no—
257C22#522711yescharacteristic
258C22#524111yescharacteristic
259C22#524411yescharacteristic
260C4—21no—
261C4—21no—
262C4—42no—
263C4—42no—
264C4—42no—
265C4—82no—
266C4—82no—
267C4—241no—
268C3#1213111yescharacteristic, complemented, Sylow
269C2—21no—
270C2—21no—
271C2—481no—
272C2#7189211yescharacteristic
273C2#7193011yescharacteristic
274C2#7195811yescharacteristic
subgroupG/H
centerC22#5244
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2809C2
socleC22×C3B(4,𝔽2)
prime radical#5510C22
hypercenter#2809D3
hypocenterC3#12131
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2809

Complements

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

NH ≅ G/Ntype
1. C3×#2809C2semidirect
4. #67652C2semidirect
5. #67643C2semidirect
20. #2809D3semidirect
268. C3#12131semidirect

Maximal subgroups

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

subgroupindex
C3×#28092
#697182
#696942
#676522
#676432
#121313