Finite Group Databasebeta

← #11922967

Subgroups — #11922967

Order 1152.

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

normal subgroups only

This group has 464 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
2C4⋊(C3⋊#226)C212yesmaximal
3C4⋊(C3⋊#226)C211yesmaximal, characteristic
4#10593622C212yesmaximal
5C4×C3⋊#226C211yesmaximal, characteristic
6#85417—33nomaximal
7#83698—31nomaximal
8C2×C3⋊#226C2211yescharacteristic
9C2×C3⋊#226C2212yes—
10#64099—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2212yes—
13C3⋊#859C2211yescharacteristic
14C3×#557D311yescharacteristic
15#5570—31no—
16#557⋊C3D313yes—
17C4⋊#226—33no—
18C4⋊#226—36no—
19#5388—36no—
20C4×#226—33no—
21#5094—31no—
22#5093—31no—
23#5069—32no—
24#5063—32no—
25#5031—32no—
26#5031—31no—
27#5030—32no—
28C4×#216—31no—
29#5024—31no—
30C2×C3×SL(2,𝔽3)C2311yescharacteristic
31C3⋊#226—21no—
32C3⋊#226—22no—
33C32×C4⋊C4—41no—
34C4⋊(C3⋊Dic3)—42no—
35C4⋊(C3⋊Dic3)—41no—
36#3750—42no—
37C4×C3⋊Dic3—41no—
38#3374—91noSylow
39C2×#226—33no—
40C2×#226—36no—
41C3×C4⋊Q8—31no—
42C3×#125—32no—
43C3×C4×Q8—31no—
44C3×C4×Q8C2×D311yescharacteristic
45C3×C4×Q8C2×D312yes—
46Q8×Dic3—61no—
47C2×#216—31no—
48C2×#216—32no—
49#890—62no—
50#890—61no—
51#890—123no—
52#869—32no—
53#869—31no—
54C4×Dic6—31no—
55C4×Dic6—32no—
56C4×SL(2,𝔽3)C2×D313yes—
57C4×SL(2,𝔽3)C2×D316yes—
58#859—33no—
59#835—31no—
60#810—31no—
61#810—32no—
62#810—32no—
63#810—32no—
64#810—32no—
65#808—32no—
66#807—31no—
67C4⋊(C3⋊C8)—32no—
68C4⋊(C3⋊C8)—31no—
69#803—32no—
70C4×C3⋊C8—31no—
71C2×C32×C4—41no—
72C2×C32×C4—42no—
73C2×C3⋊Dic3—41no—
74C2×C3⋊Dic3—42no—
75C2×C3⋊Dic3—41no—
76C3×SL(2,𝔽3)#4111yescharacteristic
77#557—91no—
78#557C3⋊D311yescharacteristic
79#491—92no—
80#487—91no—
81#479—92no—
82#475—91no—
83#462—92no—
84#462—91no—
85#446—91no—
86#441—92no—
87C4×Q16—91no—
88C2×C3×Q8—31no—
89C2×C3×Q8C22×D311yescharacteristic
90C2×Dic6—31no—
91C2×Dic6—32no—
92C2×SL(2,𝔽3)C22×D313yes—
93#226—63no—
94#226—66no—
95C3×C4⋊C4—31no—
96C3×C4⋊C4—32no—
97C3×C4⋊C4—32no—
98C3×C4⋊C4—43no—
99C3×C4⋊C4—61no—
100C3×C4⋊C4S411yescharacteristic
101C3×C42—31no—
102C3×C42—32no—
103#216—61no—
104#216—62no—
105#216—61no—
106#216—62no—
107C4⋊Dic3—31no—
108C4⋊Dic3—61no—
109C4⋊Dic3—62no—
110C4⋊Dic3—62no—
111C4⋊Dic3—61no—
112C4⋊Dic3—123no—
113C4⋊Dic3—126no—
114Dic3⋊C4—61no—
115Dic3⋊C4—62no—
116Dic3⋊C4—126no—
117C4×Dic3—62no—
118C4×Dic3—61no—
119C4×Dic3—123no—
120C2×C3⋊C8—31no—
121C2×C3⋊C8—31no—
122C2×C3⋊C8—32no—
123C22×C32—41no—
124C32×C4—42no—
125C32×C4—82no—
126C3⋊Dic3—81no—
127C3⋊Dic3—81no—
128C3⋊Dic3—82no—
129C2×Q16—91no—
130C2×Q16—92no—
131C4⋊Q8—31no—
132C4⋊Q8—91no—
133C4⋊Q8—92no—
134#125—32no—
135#125—182no—
136#125—181no—
137C4×Q8—31no—
138C4×Q8—91no—
139C4×Q8—92no—
140C4×Q8—181no—
141C4×Q8C2×C3⋊D311yescharacteristic
142C4×Q8C2×C3⋊D312yes—
143#107—91no—
144#106—92no—
145C4⋊C8—91no—
146C4⋊C8—92no—
147#103—91no—
148#103—92no—
149#103—92no—
150#103—92no—
151#103—92no—
152#103—91no—
153#97—92no—
154C4×C8—91no—
155C3×Q8—31no—
156C3×Q8—62no—
157C3×Q8#23911yescharacteristic
158C2×C3×C4—31no—
159C2×C3×C4—32no—
160C2×C3×C4—31no—
161C2×C3×C4—43no—
162C2×C3×C4—46no—
163C2×C3×C4C2×S412yes—
164C2×C3×C4C2×S411yescharacteristic
165C2×Dic3—61no—
166C2×Dic3—61no—
167C2×Dic3—62no—
168C2×Dic3—123no—
169C2×Dic3—123no—
170C2×Dic3—126no—
171Dic6—61no—
172Dic6—62no—
173Dic6—61no—
174Dic6—62no—
175SL(2,𝔽3)#23913yes—
176C3⋊C8—61no—
177C3⋊C8—61no—
178C3⋊C8—62no—
179C2×C32—41no—
180C2×C32—41no—
181C2×C32—41no—
182C2×Q8—31no—
183C2×Q8—91no—
184C2×Q8—92no—
185C2×Q8C22×C3⋊D311yescharacteristic
186Q16—181no—
187Q16—182no—
188Q16—181no—
189Q16—182no—
190C2×C8—91no—
191C2×C8—91no—
192C2×C8—92no—
193C4⋊C4—32no—
194C4⋊C4—31no—
195C4⋊C4—32no—
196C4⋊C4—61no—
197C4⋊C4—91no—
198C4⋊C4—181no—
199C4⋊C4—182no—
200C4⋊C4—181no—
201C4⋊C4—182no—
202C4⋊C4—181no—
203C4⋊C4—182no—
204C4⋊C4C3⋊S411yescharacteristic
205C42—31no—
206C42—32no—
207C42—182no—
208C42—181no—
209C22×C3—43no—
210C22×C3C22×S411yescharacteristic
211C3×C4—22no—
212C3×C4—31no—
213C3×C4—31no—
214C3×C4—46no—
215C3×C4—62no—
216C3×C4—61no—
217C3×C4—86no—
218C3×C4#98412yes—
219Dic3—121no—
220Dic3—122no—
221Dic3—121no—
222Dic3—243no—
223Dic3—243no—
224Dic3—246no—
225C32—41noSylow
226Q8—31no—
227Q8—62no—
228Q8—181no—
229Q8—182no—
230Q8—181no—
231Q8—182no—
232Q8#383211yescharacteristic
233C2×C4—31no—
234C2×C4—32no—
235C2×C4—31no—
236C2×C4—181no—
237C2×C4—182no—
238C2×C4—181no—
239C2×C4C2×C3⋊S412yes—
240C2×C4C2×C3⋊S411yescharacteristic
241C8—181no—
242C8—182no—
243C8—181no—
244C2×C3—43no—
245C2×C3—43no—
246C2×C3—43no—
247C2×C3#591911yescharacteristic
248C2×C3C2×#98411yescharacteristic
249C2×C3#592711yescharacteristic
250C22C22×C3⋊S411yescharacteristic
251C4—22no—
252C4—31no—
253C4—31no—
254C4—62no—
255C4—61no—
256C4—362no—
257C4—361no—
258C4—361no—
259C4#6426512yes—
260C3—43no—
261C3#8541711yescharacteristic, complemented
262C2#1059658211yescharacteristic
263C2C2×#6426511yescharacteristic
264C2#1059659011yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, prime radical, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×#557D3
socleC22×C3C22×S4
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
261. C3#85417semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
C4⋊(C3⋊#226)2
#105936222
C4×C3⋊#2262
#854173
#836983
#640994