Finite Group Databasebeta

← #11922981

Subgroups — #11922981

Order 1152.

512 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 69 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×Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#10593629C212yesmaximal
3C4×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic
4C4⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
5#10593622C212yesmaximal, complemented
6#85432—33nomaximal
7#83704—31nomaximal
8C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
9C2×C3⋊#226C2212yes—
10#64155—41nomaximal
11C3×C4×SL(2,𝔽3)C2212yes—
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13C3⋊#859C2211yescharacteristic
14C3×Q82D311yescharacteristic, complemented
15#5584—31no—
16Q8×SL(2,𝔽3)D313yescomplemented
17#5395—36no—
18C4×GL(2,𝔽3)—33no—
19C4⋊#226—33no—
20#5388—36no—
21#5094—32no—
22#5093—31no—
23#5089—32no—
24#5084—31no—
25#5031—31no—
26#5030—32no—
27#5028—32no—
28C4×#215—31no—
29#5024—31no—
30C2×C32×Q8—41no—
31C2×C4×C3⋊D3—41no—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C3⋊GL(2,𝔽3)—21no—
34C3⋊#226—22no—
35#3752—42no—
36C4⋊(C3⋊Dic3)—41no—
37#3750—42no—
38#3369—91noSylow
39C2×GL(2,𝔽3)—33no—
40C2×#226—36no—
41C3×C4⋊Q8—31no—
42C3×C4⋊Q8—32no—
43C3×C4×Q8—31no—
44C3×C4×Q8C2×D311yescharacteristic
45C3×C4×Q8C2×D312yes—
46#946—61no—
47#946—123no—
48C2×#216—32no—
49C2×#215—31no—
50#898—62no—
51#892—61no—
52#875—32no—
53C4×D12—31no—
54#869—31no—
55C4×Dic6—32no—
56C4×SL(2,𝔽3)C2×D316yes—
57C4×SL(2,𝔽3)C2×D313yes—
58#859—33no—
59#835—31no—
60#810—32no—
61#810—32no—
62#810—32no—
63#809—31no—
64#809—32no—
65#808—31no—
66#808—32no—
67C4⋊(C3⋊C8)—32no—
68C4⋊(C3⋊C8)—31no—
69#803—32no—
70C4×C3⋊C8—31no—
71C22×C3⋊D3—41no—
72C32×Q8—82no—
73C2×C32×C4—41no—
74C2×C32×C4—42no—
75C2×C3⋊Dic3—42no—
76C2×C3⋊Dic3—41no—
77C4×C3⋊D3—82no—
78C3×SL(2,𝔽3)#4111yescharacteristic
79Q82C3⋊D311yescharacteristic, complemented
80#556—91no—
81#489—92no—
82#487—91no—
83#476—91no—
84#475—92no—
85#464—92no—
86#462—91no—
87#446—91no—
88#441—92no—
89C4×SD16—91no—
90C2×C3×Q8—43no—
91C2×C3×Q8—61no—
92C2×C3×Q8S411yescharacteristic
93C2×C3×Q8C22×D311yescharacteristic
94C2×D12—31no—
95C2×C4×D3—61no—
96C2×C4×D3—123no—
97C2×Dic6—32no—
98C2×SL(2,𝔽3)C22×D313yes—
99GL(2,𝔽3)—63no—
100#226—66no—
101C3×C4⋊C4—32no—
102C3×C4⋊C4—31no—
103C3×C4⋊C4—31no—
104C3×C4⋊C4—32no—
105C3×C42—31no—
106C3×C42—32no—
107#216—62no—
108#216—62no—
109#215—61no—
110#215—61no—
111(C2×D3)⋊C4—61no—
112(C2×D3)⋊C4—62no—
113(C2×D3)⋊C4—62no—
114(C2×D3)⋊C4—126no—
115C4⋊Dic3—31no—
116C4⋊Dic3—61no—
117C4⋊Dic3—61no—
118C4⋊Dic3—123no—
119Dic3⋊C4—62no—
120Dic3⋊C4—126no—
121C4×Dic3—62no—
122C2×C3⋊C8—31no—
123C2×C3⋊C8—31no—
124C2×C3⋊C8—32no—
125C22×C32—41no—
126C2×C3⋊D3—42no—
127C2×C3⋊D3—81no—
128C2×C3⋊D3—81no—
129C32×C4—42no—
130C32×C4—82no—
131C3⋊Dic3—81no—
132C3⋊Dic3—82no—
133C2×Q16—92no—
134C2×SD16—91no—
135C4⋊Q8—32no—
136C4⋊Q8—31no—
137C4⋊Q8—91no—
138#126—182no—
139#124—92no—
140C22⋊Q8—181no—
141C4×Q8—31no—
142C4×Q8—92no—
143C4×Q8C2×C3⋊D311yescharacteristic
144C4×Q8C2×C3⋊D312yes—
145C4×D4—91no—
146#117—181no—
147#106—91no—
148#106—92no—
149C4⋊C8—91no—
150C4⋊C8—92no—
151#103—92no—
152#103—92no—
153#103—92no—
154#103—91no—
155#102—91no—
156#102—92no—
157#97—92no—
158C4×C8—91no—
159C22×D3—61no—
160C22×D3—123no—
161C3×Q8—22no—
162C3×Q8—31no—
163C3×Q8—86no—
164C3×Q8—241no—
165C3×Q8#23911yescharacteristic
166C2×C3×C4—31no—
167C2×C3×C4—32no—
168C2×C3×C4—31no—
169C2×C3×C4—43no—
170C2×C3×C4—46no—
171C2×C3×C4C2×S412yes—
172C2×C3×C4C2×S411yescharacteristic
173C2×Dic3—61no—
174C2×Dic3—62no—
175C2×Dic3—123no—
176C2×Dic3—126no—
177D12—61no—
178D12—61no—
179C4×D3—122no—
180C4×D3—246no—
181Dic6—62no—
182Dic6—62no—
183SL(2,𝔽3)#23913yes—
184C3⋊C8—61no—
185C3⋊C8—61no—
186C3⋊C8—62no—
187C2×C32—41no—
188C2×C32—41no—
189C2×C32—41no—
190C3⋊D3—82no—
191C2×Q8—61no—
192C2×Q8—92no—
193C2×Q8C3⋊S411yescharacteristic
194C2×Q8C22×C3⋊D311yescharacteristic
195C2×D4—91no—
196C22×C4—181no—
197Q16—182no—
198Q16—182no—
199SD16—181no—
200SD16—181no—
201C2×C8—91no—
202C2×C8—91no—
203C2×C8—92no—
204C4⋊C4—32no—
205C4⋊C4—31no—
206C4⋊C4—31no—
207C4⋊C4—32no—
208C4⋊C4—91no—
209C4⋊C4—181no—
210C4⋊C4—181no—
211C4⋊C4—182no—
212C22⋊C4—182no—
213C22⋊C4—181no—
214C22⋊C4—182no—
215C42—31no—
216C42—32no—
217C42—182no—
218C22×C3—43no—
219C22×C3C22×S411yescharacteristic
220C2×D3—126no—
221C2×D3—121no—
222C2×D3—121no—
223C2×D3—121no—
224C2×D3—243no—
225C2×D3—243no—
226C3×C4—22no—
227C3×C4—31no—
228C3×C4—31no—
229C3×C4—46no—
230C3×C4—62no—
231C3×C4—61no—
232C3×C4—86no—
233C3×C4#98312yes—
234Dic3—121no—
235Dic3—122no—
236Dic3—243no—
237Dic3—246no—
238C32—41noSylow
239C23—181no—
240Q8—22no—
241Q8—31no—
242Q8—182no—
243Q8—182no—
244Q8—241no—
245Q8#383211yescharacteristic
246D4—181no—
247D4—181no—
248C2×C4—31no—
249C2×C4—32no—
250C2×C4—31no—
251C2×C4—181no—
252C2×C4—182no—
253C2×C4—362no—
254C2×C4C2×C3⋊S412yes—
255C2×C4C2×C3⋊S411yescharacteristic
256C8—181no—
257C8—182no—
258C8—181no—
259C2×C3—43no—
260C2×C3—43no—
261C2×C3—43no—
262C2×C3C2×#98311yescharacteristic
263C2×C3#591911yescharacteristic
264C2×C3#592411yescharacteristic
265D3—246no—
266D3—241no—
267C22—361no—
268C22—361no—
269C22—361no—
270C22C22×C3⋊S411yescharacteristic
271C4—22no—
272C4—31no—
273C4—31no—
274C4—62no—
275C4—61no—
276C4—361no—
277C4—362no—
278C4#6426412yes—
279C3—43no—
280C3#8543211yescharacteristic, complemented
281C2—721no—
282C2C2×#6426411yescharacteristic
283C2#1059658211yescharacteristic
284C2#1059658711yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×Q82D3
socleC22×C3C22×S4
prime radicalC2×C3⋊GL(2,𝔽3)C22
hypercenterC2×Q8C3⋊S4
hypocenterC3×SL(2,𝔽3)#41
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×SL(2,𝔽3)C2semidirect
4. C4⋊(C3⋊#226)C2semidirect
5. #10593622C2semidirect
14. C3×Q82D3semidirect
16. Q8×SL(2,𝔽3)D3semidirect
79. Q82C3⋊D3semidirect
280. C3#85432semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×SL(2,𝔽3)2
#105936292
C4×C3⋊GL(2,𝔽3)2
C4⋊(C3⋊#226)2
#105936222
#854323
#837043
#641554