Finite Group Databasebeta

← #11922969

Subgroups — #11922969

Order 1152.

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

normal subgroups only

This group has 514 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×#557⋊C3C211yesmaximal, characteristic, complemented
2#10593629C212yesmaximal
3C4×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic
4C4⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
5C4×C3⋊#226C212yesmaximal, complemented
6#85419—33nomaximal
7#83689—31nomaximal
8C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
9C2×C3⋊#226C2212yes—
10#64101—41nomaximal
11C3×C4×SL(2,𝔽3)C2212yes—
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13C3⋊#859C2211yescharacteristic
14C3×#557D311yescharacteristic, complemented
15#5584—31no—
16#557⋊C3D313yescomplemented
17#5395—36no—
18C4×GL(2,𝔽3)—33no—
19C4⋊#226—33no—
20C4×#226—36no—
21#5093—31no—
22#5087—31no—
23#5068—32no—
24#5063—32no—
25#5031—31no—
26C4×#216—32no—
27#5028—32no—
28C4×#215—31no—
29Q8×C3⋊C8—31no—
30C2×C4×C3⋊D3—41no—
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C3⋊GL(2,𝔽3)—21no—
33C3⋊#226—22no—
34C32×C4⋊C4—41no—
35#3752—42no—
36C4⋊(C3⋊Dic3)—41no—
37C4×C3⋊Dic3—42no—
38#3364—91noSylow
39C2×GL(2,𝔽3)—33no—
40C2×#226—36no—
41C3×C4⋊Q8—31no—
42C3×#125—32no—
43C3×C4×Q8—31no—
44C3×C4×Q8C2×D311yescharacteristic
45C3×C4×Q8C2×D312yes—
46#946—61no—
47C2×#216—32no—
48C2×#215—31no—
49#898—62no—
50#892—61no—
51#892—123no—
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#807—32no—
67C4⋊(C3⋊C8)—32no—
68C4⋊(C3⋊C8)—31no—
69C4×C3⋊C8—31no—
70C4×C3⋊C8—32no—
71C22×C3⋊D3—41no—
72C2×C32×C4—41no—
73C2×C32×C4—42no—
74C2×C3⋊Dic3—41no—
75C2×C3⋊Dic3—42no—
76C4×C3⋊D3—44no—
77C3×SL(2,𝔽3)#4111yescharacteristic
78#557C3⋊D311yescharacteristic, complemented
79#556—91no—
80#488—92no—
81#487—91no—
82#479—92no—
83#474—91no—
84#464—92no—
85#462—91no—
86C8×Q8—91no—
87C4×Q16—92no—
88C4×SD16—91no—
89C2×C3×Q8—31no—
90C2×C3×Q8C22×D311yescharacteristic
91C2×D12—31no—
92C2×C4×D3—61no—
93C2×C4×D3—123no—
94C2×Dic6—32no—
95C2×SL(2,𝔽3)C22×D313yes—
96GL(2,𝔽3)—63no—
97#226—66no—
98C3×C4⋊C4—32no—
99C3×C4⋊C4—31no—
100C3×C4⋊C4—32no—
101C3×C4⋊C4—43no—
102C3×C4⋊C4—61no—
103C3×C4⋊C4S411yescharacteristic
104C3×C42—32no—
105C3×C42—31no—
106#216—62no—
107#216—62no—
108#215—61no—
109#215—61no—
110(C2×D3)⋊C4—61no—
111(C2×D3)⋊C4—62no—
112(C2×D3)⋊C4—62no—
113(C2×D3)⋊C4—126no—
114C4⋊Dic3—31no—
115C4⋊Dic3—61no—
116C4⋊Dic3—61no—
117C4⋊Dic3—123no—
118Dic3⋊C4—62no—
119C4×Dic3—62no—
120C4×Dic3—126no—
121C2×C3⋊C8—31no—
122C2×C3⋊C8—32no—
123C2×C3⋊C8—31no—
124C22×C32—41no—
125C2×C3⋊D3—42no—
126C2×C3⋊D3—81no—
127C2×C3⋊D3—81no—
128C32×C4—42no—
129C32×C4—82no—
130C3⋊Dic3—42no—
131C3⋊Dic3—82no—
132C2×Q16—92no—
133C2×SD16—91no—
134C4⋊Q8—31no—
135C4⋊Q8—91no—
136#126—182no—
137#125—32no—
138#124—92no—
139C22⋊Q8—181no—
140C4×Q8—31no—
141C4×Q8—92no—
142C4×Q8C2×C3⋊D311yescharacteristic
143C4×Q8C2×C3⋊D312yes—
144C4×D4—91no—
145#117—181no—
146#107—92no—
147#106—91no—
148C4⋊C8—91no—
149C4⋊C8—92no—
150#103—92no—
151#103—92no—
152#103—92no—
153#103—91no—
154#102—91no—
155#102—92no—
156C4×C8—91no—
157C4×C8—92no—
158C22×D3—61no—
159C22×D3—123no—
160C3×Q8—31no—
161C3×Q8—62no—
162C3×Q8#23911yescharacteristic
163C2×C3×C4—31no—
164C2×C3×C4—32no—
165C2×C3×C4—31no—
166C2×C3×C4—43no—
167C2×C3×C4—46no—
168C2×C3×C4C2×S412yes—
169C2×C3×C4C2×S411yescharacteristic
170C2×Dic3—61no—
171C2×Dic3—62no—
172C2×Dic3—123no—
173C2×Dic3—126no—
174D12—61no—
175D12—61no—
176C4×D3—1212no—
177C4×D3—122no—
178Dic6—62no—
179Dic6—62no—
180SL(2,𝔽3)#23913yes—
181C3⋊C8—61no—
182C3⋊C8—61no—
183C3⋊C8—62no—
184C2×C32—41no—
185C2×C32—41no—
186C2×C32—41no—
187C3⋊D3—82no—
188C2×Q8—31no—
189C2×Q8—92no—
190C2×Q8C22×C3⋊D311yescharacteristic
191C2×D4—91no—
192C22×C4—181no—
193Q16—182no—
194Q16—182no—
195SD16—181no—
196SD16—181no—
197C2×C8—91no—
198C2×C8—92no—
199C2×C8—91no—
200C4⋊C4—31no—
201C4⋊C4—32no—
202C4⋊C4—32no—
203C4⋊C4—61no—
204C4⋊C4—91no—
205C4⋊C4—182no—
206C4⋊C4—181no—
207C4⋊C4—181no—
208C4⋊C4C3⋊S411yescharacteristic
209C22⋊C4—181no—
210C22⋊C4—182no—
211C22⋊C4—182no—
212C42—31no—
213C42—32no—
214C42—182no—
215C22×C3—43no—
216C22×C3C22×S411yescharacteristic
217C2×D3—126no—
218C2×D3—121no—
219C2×D3—121no—
220C2×D3—121no—
221C2×D3—243no—
222C2×D3—243no—
223C3×C4—22no—
224C3×C4—31no—
225C3×C4—31no—
226C3×C4—46no—
227C3×C4—62no—
228C3×C4—61no—
229C3×C4—86no—
230C3×C4#98512yes—
231Dic3—126no—
232Dic3—122no—
233Dic3—121no—
234Dic3—246no—
235C32—41noSylow
236C23—181no—
237Q8—31no—
238Q8—62no—
239Q8—182no—
240Q8—182no—
241Q8#383211yescharacteristic
242D4—181no—
243D4—181no—
244C2×C4—31no—
245C2×C4—31no—
246C2×C4—32no—
247C2×C4—181no—
248C2×C4—182no—
249C2×C4—362no—
250C2×C4C2×C3⋊S412yes—
251C2×C4C2×C3⋊S411yescharacteristic
252C8—181no—
253C8—182no—
254C8—181no—
255C2×C3—43no—
256C2×C3—43no—
257C2×C3—43no—
258C2×C3#591911yescharacteristic
259C2×C3C2×#98511yescharacteristic
260C2×C3#592711yescharacteristic
261D3—246no—
262D3—241no—
263C22—361no—
264C22—361no—
265C22—361no—
266C22C22×C3⋊S411yescharacteristic
267C4—22no—
268C4—31no—
269C4—31no—
270C4—62no—
271C4—61no—
272C4—362no—
273C4—361no—
274C4#6426612yes—
275C3—43no—
276C3#8541911yescharacteristic, complemented
277C2—721no—
278C2#1059658211yescharacteristic
279C2C2×#6426611yescharacteristic
280C2#1059659011yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×#557D3
socleC22×C3C22×S4
prime radicalC2×C3⋊GL(2,𝔽3)C22
hypercenterC4⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)#41
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#557⋊C3C2semidirect
4. C4⋊(C3⋊#226)C2semidirect
5. C4×C3⋊#226C2semidirect
14. C3×#557D3semidirect
16. #557⋊C3D3semidirect
78. #557C3⋊D3semidirect
276. C3#85419semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
#105936292
C4×C3⋊GL(2,𝔽3)2
C4⋊(C3⋊#226)2
C4×C3⋊#2262
#854193
#836893
#641014