Finite Group Databasebeta

← #11780686

Subgroups — #11780686

Order 1152.

502 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 72 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#10596171C211yesmaximal, characteristic
2C3×#5069C211yesmaximal, characteristic, complemented
3#10590294C212yesmaximal, complemented
4C3×#4543C211yesmaximal, characteristic, complemented
5#10589087C212yesmaximal, complemented
6#71251—31nomaximal
7#68374—31nomaximal
8C32×#125C2211yescharacteristic
9#64105—22no—
10#64104—22no—
11#64103—22no—
12#64084C2212yes—
13C3×#869C2211yescharacteristic
14C3×#810—22no—
15C3×#803C2212yes—
16C3×#803C2211yescharacteristic
17#63567—22no—
18#63564—22no—
19#5700—31no—
20#5700—31no—
21#5700—61no—
22C3×#491—31no—
23#5089—32no—
24#5069D311yescharacteristic, complemented
25#4714—32no—
26C3×#330—31no—
27#4543D311yescharacteristic, complemented
28#4464—32no—
29#4464—32no—
30C2×C4⋊(C3⋊D3)—22no—
31C2×C4⋊(C3⋊D3)D412yes—
32C2×C4×C3⋊D3—22no—
33C2×#634D412yes—
34C2×C3×Dic6—21no—
35C32×C4⋊C4—22no—
36C32×C4⋊C4D412yes—
37C32×C42C2311yescharacteristic
38#3752—22no—
39#3752—22no—
40#3752—22no—
41#3750—22no—
42C3×C4⋊Dic3—22no—
43C2×C3×C3⋊C8—22no—
44C2×C3×C3⋊C8—21no—
45#1662—91noSylow
46C3×C4⋊Q8—31no—
47C3×#125—21no—
48C3×#125C2×D311yescharacteristic
49C3×#125C2×D311yescharacteristic
50#896—62no—
51#896—62no—
52#896—64no—
53#895—62no—
54#895—62no—
55#895—64no—
56#894—64no—
57#894—62no—
58#894—62no—
59#875—32no—
60#875—32no—
61#875—62no—
62#869C2×D311yescharacteristic
63C3×#103—62no—
64C3×#97—32no—
65C3×#97—31no—
66#821—62no—
67#816—62no—
68#810—22no—
69#810—62no—
70#809—62no—
71#803C2×D311yescharacteristic
72#803C2×D312yes—
73C22×C3⋊D3—22no—
74C2×C32×C4—22no—
75C2×C32×C4C2×D411yescharacteristic
76C2×C32×C4C2×D412yes—
77C2×C3⋊Dic3—22no—
78C4⋊(C3⋊D3)—42no—
79C4⋊(C3⋊D3)—82no—
80C4×C3⋊D3—82no—
81#634—42no—
82C2×C3×Dic3—41no—
83C3×Dic6—42no—
84C3×C3⋊C8—41no—
85C3×C3⋊C8—42no—
86#562—91no—
87#491—31no—
88#489—92no—
89#330—31no—
90#329—92no—
91C2×C3×Q8—61no—
92C2×D12—32no—
93C2×D12—32no—
94C2×D12—62no—
95C2×D12—64no—
96C2×D12—62no—
97C2×D12—62no—
98C2×C4×D3—64no—
99C2×C4×D3—62no—
100C2×C4×D3—62no—
101C2×Dic6—21no—
102C2×Dic6—32no—
103C2×Dic6—32no—
104C2×Dic6—62no—
105C2×C3×C8—61no—
106C2×C3×C8—62no—
107C3×C4⋊C4—24no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—62no—
112C3×C4⋊C4D1212yes—
113C3×C4⋊C4#6712yes—
114C3×C42—21no—
115C3×C42C22×D311yescharacteristic
116C3×C42C22×D311yescharacteristic
117(C2×D3)⋊C4—64no—
118(C2×D3)⋊C4—64no—
119(C2×D3)⋊C4—64no—
120(C2×D3)⋊C4—62no—
121(C2×D3)⋊C4—62no—
122(C2×D3)⋊C4—62no—
123(C2×D3)⋊C4—62no—
124(C2×D3)⋊C4—62no—
125(C2×D3)⋊C4—62no—
126C4⋊Dic3—22no—
127Dic3⋊C4—64no—
128Dic3⋊C4—62no—
129Dic3⋊C4—62no—
130C2×C3⋊C8—22no—
131C2×C3⋊C8—21no—
132C22×C32#12011yescharacteristic
133C2×C3⋊D3—42no—
134C2×C3⋊D3—42no—
135C2×C3⋊D3—42no—
136C32×C4—21no—
137C32×C4—22no—
138C32×C4—42no—
139C3⋊Dic3—42no—
140C3×Dic3—81no—
141C4⋊Q8—31no—
142#125D3211yescharacteristic
143#124—92no—
144#123—182no—
145C22⋊Q8—182no—
146#121—182no—
147#103—62no—
148#103—182no—
149#102—182no—
150#97—31no—
151#97—32no—
152C22×D3—64no—
153C22×D3—62no—
154C22×D3—62no—
155C3×Q8—122no—
156C2×C3×C4—24no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4—21no—
160C2×C3×C4—22no—
161C2×C3×C4—121no—
162C2×C3×C4C2×D1211yescharacteristic
163C2×C3×C4D3×D412yes—
164C2×C3×C4D3×D412yes—
165C2×C3×C4C2×#6711yescharacteristic
166C2×Dic3—41no—
167C2×Dic3—64no—
168C2×Dic3—62no—
169C2×Dic3—62no—
170D12—122no—
171D12—122no—
172D12—242no—
173D12—244no—
174D12—242no—
175D12—242no—
176C4×D3—244no—
177C4×D3—242no—
178C4×D3—242no—
179Dic6—42no—
180Dic6—122no—
181Dic6—122no—
182Dic6—242no—
183C3×C8—122no—
184C3×C8—121no—
185C3⋊C8—41no—
186C3⋊C8—42no—
187C2×C32#45411yescharacteristic
188C2×C32#45512yes—
189C3⋊D3—82no—
190C2×Q8—61no—
191C2×Q8—92no—
192C2×D4—92no—
193C2×D4—182no—
194C22×C4—182no—
195C2×C8—61no—
196C2×C8—62no—
197C4⋊C4—22no—
198C4⋊C4—62no—
199C4⋊C4—182no—
200C4⋊C4#62612yes—
201C22⋊C4—182no—
202C22⋊C4—182no—
203C22⋊C4—182no—
204C42C2×D3211yescharacteristic
205C22×C3—21no—
206C22×C3#88211yescharacteristic
207C22×C3#93711yescharacteristic
208C2×D3—124no—
209C2×D3—124no—
210C2×D3—124no—
211C2×D3—122no—
212C2×D3—122no—
213C2×D3—122no—
214C2×D3—122no—
215C2×D3—122no—
216C2×D3—122no—
217C3×C4—22no—
218C3×C4—21no—
219C3×C4—21no—
220C3×C4—22no—
221C3×C4—42no—
222C3×C4—41no—
223C3×C4—44no—
224C3×C4—42no—
225C3×C4—42no—
226C3×C4—241no—
227Dic3—81no—
228Dic3—124no—
229Dic3—122no—
230Dic3—122no—
231C32#166211yescharacteristic, complemented, Sylow
232C23—182no—
233Q8—122no—
234Q8—362no—
235D4—362no—
236D4—722no—
237C2×C4—22no—
238C2×C4—121no—
239C2×C4—182no—
240C2×C4—722no—
241C2×C4C2×#62611yescharacteristic
242C2×C4#381112yes—
243C8—121no—
244C8—122no—
245C2×C3—21no—
246C2×C3—22no—
247C2×C3#475311yescharacteristic
248C2×C3#482512yes—
249C2×C3#506211yescharacteristic
250C2×C3#520212yes—
251D3—244no—
252D3—242no—
253D3—242no—
254C22—362no—
255C22—362no—
256C22—362no—
257C22#6398111yescharacteristic
258C4—21no—
259C4—22no—
260C4—42no—
261C4—241no—
262C4—362no—
263C3—21no—
264C3#6837411yescharacteristic, complemented
265C3#7125111yescharacteristic, complemented
266C2—722no—
267C2#1059164711yescharacteristic
268C2#1059169012yes—
subgroupG/H
centerC22#63981
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×#125C22
socleC22×C32#120
prime radical#10596171C2
hypercenter#125D32
hypocenterC32#1662
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
2. C3×#5069C2semidirect
3. #10590294C2semidirect
4. C3×#4543C2semidirect
5. #10589087C2semidirect
24. #5069D3semidirect
27. #4543D3semidirect
231. C32#1662semidirect
264. C3#68374semidirect
265. C3#71251semidirect

Maximal subgroups

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

subgroupindex
#105961712
C3×#50692
#105902942
C3×#45432
#105890872
#712513
#683743