Finite Group Databasebeta

← #85827⋊C3 #11923204

Subgroups — #85827⋊C3 #11923204

Order 1152.

785 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 785 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#1010⋊C3C212yesmaximal, complemented
2C3×C4×#41⋊C3C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×D12C214yesmaximal, complemented
4#85827C311yesmaximal, characteristic, complemented
5#3499⋊C3—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7#1010⋊C3—24no—
8C2×D3×SL(2,𝔽3)C2214yes—
9C3×#874—41nomaximal
10C3×C4×SL(2,𝔽3)C2212yes—
11C2×#1010C2×C312yescomplemented
12C2×2−1+4⋊C3—32no—
13C3×C4×#41C2×C311yescharacteristic, complemented
14Q8×D12C2×C314yescomplemented
15#5555—34no—
16#5526—32no—
17#5477—31no—
18C2×#869—31no—
19D4×SL(2,𝔽3)—34no—
20C4×#41⋊C3—21no—
21C4×#41⋊C3D311yescharacteristic, complemented
22C2×C3×D12—42no—
23C2×C3×D12—44no—
24C3×#41⋊C3D412yescomplemented
25C2×C3×SL(2,𝔽3)C2311yescharacteristic
26D3×SL(2,𝔽3)—24no—
27C32×C42—41no—
28#3499—31noSylow
29C2×C3×#41C22×C311yescharacteristic
30#1010—24no—
31#1010—64no—
32C2×D3×Q8C22×C314yes—
33C2×#237—34no—
34C2×#235—32no—
35C22×Dic6—32no—
362−1+4⋊C3—64no—
37C2×#41⋊C3—21no—
38C2×#41⋊C3C2×D311yescharacteristic
39C22×SL(2,𝔽3)—34no—
40C3×#127—121no—
41C3×C4×Q8C22×C312yes—
42C3×C4×D4—32no—
43C3×#117—31no—
44C2×C3×C42—31no—
45#930—34no—
46C2×C4⋊Dic3—34no—
47#897—38no—
48#886—38no—
49#875—32no—
50#874A411yescharacteristic
51C4×D12—34no—
52#869—32no—
53#869—31no—
54#869—34no—
55C4×SL(2,𝔽3)—22no—
56C4×SL(2,𝔽3)C2×D312yes—
57C22×C3×D3—44no—
58C2×C32×C4—41no—
59C2×C32×C4—42no—
60C3×D12—84no—
61C3×D12—88no—
62C3×SL(2,𝔽3)C2×D411yescharacteristic
63C2×2−1+4—32no—
64D4×Q8—34no—
65#547—94no—
66#536—92no—
67#532—91no—
68C2×C4⋊Q8—91no—
69C4×#41C3×D311yescharacteristic, complemented
70C3×#41—32no—
71C3×#41C3×D412yescomplemented
72C2×C3×Q8C23×C311yescharacteristic
73C2×C3×D4—31no—
74C2×C3×D4—122no—
75C2×C3×D4—124no—
76C22×C3×C4—31no—
77C22×C3×C4—32no—
78C2×#67—34no—
79C22×Dic3—34no—
80D3×Q8—24no—
81D3×Q8—64no—
82#237—68no—
83#237—68no—
84#235—64no—
85#235—64no—
86C2×D12C2×A414yes—
87C2×D12C2×A412yes—
88C2×C4×D3—34no—
89C2×Dic6—32no—
90C2×Dic6—34no—
91C2×Dic6—64no—
92C2×Dic6—64no—
93#41⋊C3—22no—
94#41⋊C3D1212yescomplemented
95C2×SL(2,𝔽3)—21no—
96C2×SL(2,𝔽3)—64no—
97C2×SL(2,𝔽3)C22×D311yescharacteristic
98C3×C4⋊C4—32no—
99C3×C22⋊C4—32no—
100C3×C42—32no—
101C3×C42—31no—
102C3×C42—41no—
103C3×C42—81no—
104C3×C42C2×A411yescharacteristic
105(C2×D3)⋊C4—38no—
106C4⋊Dic3—38no—
107C4⋊Dic3—34no—
108C22×C32—41no—
109C2×C3×D3—88no—
110C2×C3×D3—84no—
111C32×C4—42no—
112C32×C4—44no—
1132−1+4—64no—
1142−1+4—184no—
115C2×#41—92no—
116C2×#41—94no—
117C2×#41C2×C3×D311yescharacteristic
118C22×Q8—34no—
119C22×Q8—92no—
120C4⋊Q8—92no—
121C4⋊Q8—94no—
122C4⋊Q8—91no—
123#127—31no—
124#124—92no—
125#123—98no—
126C22⋊Q8—98no—
127#121—94no—
128C4×Q8C2×C3×D312yes—
129C4×D4—32no—
130C4×D4—94no—
131#117—31no—
132C2×C4⋊C4—94no—
133C2×C42—31no—
134C23×C3—31no—
135C23×C3—124no—
136C22×D3C22×A414yes—
137C3×Q8—31no—
138C3×Q8C2×C3×D411yescharacteristic
139C3×D4—32no—
140C3×D4—32no—
141C3×D4—244no—
142C3×D4—248no—
143C2×C3×C4—31no—
144C2×C3×C4—32no—
145C2×C3×C4—32no—
146C2×C3×C4—32no—
147C2×C3×C4—42no—
148C2×C3×C4—41no—
149C2×C3×C4—64no—
150C2×C3×C4—82no—
151C2×C3×C4—81no—
152C2×C3×C4C22×A411yescharacteristic
153C2×C3×C4C22×A412yes—
154#67—68no—
155C2×Dic3—34no—
156C2×Dic3—68no—
157D12—24no—
158D12—28no—
159C4×D3—64no—
160C4×D3—64no—
161Dic6—64no—
162Dic6—64no—
163Dic6—64no—
164SL(2,𝔽3)—21no—
165SL(2,𝔽3)C2×D1211yescharacteristic
166C2×C32—41no—
167C2×C32—42no—
168C3×D3—164no—
169#41—32no—
170#41—184no—
171#41—184no—
172#41—188no—
173#41—188no—
174#41C3×D1212yescomplemented
175C2×Q8—64no—
176C2×Q8—94no—
177C2×Q8—92no—
178C2×Q8—184no—
179C2×Q8—184no—
180C2×Q8—184no—
181C2×Q8C22×C3×D311yescharacteristic
182C2×D4—31no—
183C2×D4—34no—
184C2×D4—32no—
185C2×D4—94no—
186C22×C4—31no—
187C22×C4—32no—
188C22×C4—94no—
189C22×C4—94no—
190C4⋊C4—32no—
191C4⋊C4—98no—
192C4⋊C4—94no—
193C22⋊C4—32no—
194C22⋊C4—98no—
195C42—31no—
196C42—32no—
197C42D3×A411yescharacteristic
198C22×C3—32no—
199C22×C3—41no—
200C22×C3—62no—
201C22×C3—81no—
202C22×C3—248no—
203C22×C3—244no—
204C22×C3C23×A411yescharacteristic
205C2×D3—24no—
206C2×D3—28no—
207C3×C4—31no—
208C3×C4—31no—
209C3×C4—42no—
210C3×C4—44no—
211C3×C4—62no—
212C3×C4—82no—
213C3×C4—84no—
214C3×C4D4×A412yes—
215C3×C42−1+4⋊C314yescomplemented
216Dic3—64no—
217C32—41noSylow
218C23—31no—
219C23—34no—
220Q8—31no—
221Q8—184no—
222Q8—184no—
223Q8—184no—
224Q8C2×C3×D1211yescharacteristic
225D4—32no—
226D4—32no—
227D4—64no—
228D4—68no—
229D4—188no—
230C2×C4—31no—
231C2×C4—32no—
232C2×C4—32no—
233C2×C4—32no—
234C2×C4—64no—
235C2×C4—94no—
236C2×C4—184no—
237C2×C4—184no—
238C2×C4—188no—
239C2×C4C2×D3×A411yescharacteristic
240C2×C4C2×D3×A412yes—
241C2×C3—41no—
242C2×C3—42no—
243C2×C3—62no—
244C2×C3—81no—
245C2×C3—82no—
246C2×C3—484no—
247C2×C3C2×D4×A411yescharacteristic
248C2×C3C2×2−1+4⋊C312yes—
249D3—44no—
250C22—32no—
251C22—62no—
252C22—64no—
253C22—68no—
254C22C22×D3×A411yescharacteristic
255C4—31no—
256C4—31no—
257C4—62no—
258C4—184no—
259C4A4×D1212yes—
260C4#1010⋊C314yescomplemented
261C3—41no—
262C3—81no—
263C3#3499⋊C311yescharacteristic, complemented
264C2—62no—
265C2—124no—
266C2C2×A4×D1211yescharacteristic
267C2C2×#1010⋊C312yes—
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×C4×#41C2×C3
socleC22×C3C23×A4
hypercenterC42D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#1010⋊C3C2semidirect
2. C3×C4×#41⋊C3C2semidirect
3. SL(2,𝔽3)×D12C2semidirect
4. #85827C3semidirect
11. C2×#1010C2×C3semidirect
13. C3×C4×#41C2×C3semidirect
14. Q8×D12C2×C3semidirect
21. C4×#41⋊C3D3semidirect
24. C3×#41⋊C3D4semidirect
69. C4×#41C3×D3semidirect
71. C3×#41C3×D4semidirect
94. #41⋊C3D12semidirect
174. #41C3×D12semidirect
215. C3×C42−1+4⋊C3semidirect
260. C4#1010⋊C3semidirect
263. C3#3499⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#1010⋊C32
C3×C4×#41⋊C32
SL(2,𝔽3)×D122
#858273
#3499⋊C33
C3×#8744