Finite Group Databasebeta

← #11922975

Subgroups — #11922975

Order 1152.

484 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 71 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
2#10593623C212yesmaximal
3#10593622C212yesmaximal
4C4×C3⋊#226C211yesmaximal, characteristic
5#10593620C211yesmaximal, characteristic
6#85426—33nomaximal
7#83709—31nomaximal
8C2×C3⋊#226C2211yescharacteristic
9#64153—41nomaximal
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11C3×C4×SL(2,𝔽3)C2212yes—
12C3⋊#859C2212yes—
13C3⋊#859C2211yescharacteristic
14C3×Q82D311yescharacteristic
15#5567—31no—
16Q8×SL(2,𝔽3)D313yes—
17#5389—36no—
18#5388—36no—
19C4×#226—33no—
20#5386—33no—
21#5094—32no—
22#5091—31no—
23#5081—32no—
24#5080—31no—
25#5030—32no—
26C4×#216—31no—
27#5024—31no—
28#5022—32no—
29#5020—31no—
30C2×C32×Q8—41no—
31C2×#634—41no—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C3⋊#226Q812yescomplemented
34#3750—42no—
35#3750—42no—
36C4×C3⋊Dic3—41no—
37#3387—91noSylow
38C2×#226—33no—
39C3×C4⋊Q8—31no—
40C3×C4⋊Q8—32no—
41C3×C4×Q8—31no—
42C3×C4×Q8C2×D311yescharacteristic
43C3×C4×Q8C2×D312yes—
44#944—61no—
45#944—123no—
46C2×#216—31no—
47#889—62no—
48#887—61no—
49#870—32no—
50#869—31no—
51C4×Dic6—31no—
52C4×Dic6—32no—
53C4×SL(2,𝔽3)C2×D313yes—
54C4×SL(2,𝔽3)C2×D316yes—
55#859—36no—
56#859—33no—
57#835—32no—
58#835—31no—
59#810—31no—
60#810—32no—
61#808—32no—
62#808—32no—
63#808—32no—
64#807—32no—
65#807—31no—
66C4⋊(C3⋊C8)—31no—
67C4⋊(C3⋊C8)—32no—
68#803—32no—
69C4×C3⋊C8—31no—
70C32×Q8—82no—
71C2×C32×C4—41no—
72C2×C32×C4—42no—
73C2×C3⋊Dic3—41no—
74C2×C3⋊Dic3—41no—
75C2×C3⋊Dic3—42no—
76#634—82no—
77C3×SL(2,𝔽3)C2×Q811yescharacteristic
78Q82C3⋊D311yescharacteristic
79#557—91no—
80#501—92no—
81#498—91no—
82#479—92no—
83#477—91no—
84#475—92no—
85#475—91no—
86#446—91no—
87#441—92no—
88C4×Q16—91no—
89C2×C3×Q8—43no—
90C2×C3×Q8—61no—
91C2×C3×Q8S411yescharacteristic
92C2×C3×Q8C22×D311yescharacteristic
93C2×Dic6—31no—
94C2×Dic6—61no—
95C2×Dic6—123no—
96C2×SL(2,𝔽3)C22×D313yes—
97#226—36no—
98C3×C4⋊C4—32no—
99C3×C4⋊C4—31no—
100C3×C4⋊C4—31no—
101C3×C4⋊C4—32no—
102C3×C42—31no—
103C3×C42—32no—
104#216—32no—
105#216—32no—
106C4⋊Dic3—31no—
107C4⋊Dic3—32no—
108C4⋊Dic3—61no—
109Dic3⋊C4—61no—
110Dic3⋊C4—62no—
111Dic3⋊C4—62no—
112Dic3⋊C4—62no—
113Dic3⋊C4—126no—
114Dic3⋊C4—126no—
115C4×Dic3—62no—
116C4×Dic3—61no—
117C4×Dic3—123no—
118C2×C3⋊C8—31no—
119C2×C3⋊C8—31no—
120C2×C3⋊C8—32no—
121C22×C32—41no—
122C32×C4—42no—
123C32×C4—82no—
124C3⋊Dic3—42no—
125C3⋊Dic3—42no—
126C3⋊Dic3—82no—
127C2×Q16—91no—
128C4⋊Q8—32no—
129C4⋊Q8—31no—
130C4⋊Q8—91no—
131C4⋊Q8—181no—
132#125—92no—
133#125—182no—
134C4×Q8—31no—
135C4×Q8—91no—
136C4×Q8—92no—
137C4×Q8—181no—
138C4×Q8C2×C3⋊D311yescharacteristic
139C4×Q8C2×C3⋊D312yes—
140#107—91no—
141#107—92no—
142#106—92no—
143#106—92no—
144#106—92no—
145C4⋊C8—91no—
146C4⋊C8—92no—
147#103—91no—
148#103—92no—
149#103—91no—
150#103—92no—
151#97—92no—
152C4×C8—91no—
153C3×Q8—22no—
154C3×Q8—31no—
155C3×Q8—86no—
156C3×Q8—241no—
157C3×Q8D3×Q811yescharacteristic
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—32no—
172Dic6—61no—
173Dic6—122no—
174Dic6—246no—
175SL(2,𝔽3)D3×Q813yes—
176C3⋊C8—32no—
177C3⋊C8—61no—
178C3⋊C8—62no—
179C2×C32—41no—
180C2×C32—41no—
181C2×C32—41no—
182C2×Q8—61no—
183C2×Q8—91no—
184C2×Q8—181no—
185C2×Q8C3⋊S411yescharacteristic
186C2×Q8C22×C3⋊D311yescharacteristic
187Q16—92no—
188Q16—92no—
189C2×C8—91no—
190C2×C8—91no—
191C2×C8—92no—
192C4⋊C4—32no—
193C4⋊C4—31no—
194C4⋊C4—31no—
195C4⋊C4—32no—
196C4⋊C4—92no—
197C4⋊C4—91no—
198C4⋊C4—181no—
199C4⋊C4—182no—
200C4⋊C4—181no—
201C4⋊C4—182no—
202C4⋊C4—182no—
203C42—31no—
204C42—32no—
205C42—182no—
206C42—181no—
207C22×C3—43no—
208C22×C3C22×S411yescharacteristic
209C3×C4—22no—
210C3×C4—31no—
211C3×C4—31no—
212C3×C4—46no—
213C3×C4—62no—
214C3×C4—61no—
215C3×C4—86no—
216C3×C4#98312yes—
217Dic3—62no—
218Dic3—126no—
219Dic3—126no—
220Dic3—122no—
221Dic3—121no—
222Dic3—246no—
223C32—41noSylow
224Q8—22no—
225Q8—31no—
226Q8—92no—
227Q8—181no—
228Q8—241no—
229Q8—362no—
230Q8Q8×C3⋊D311yescharacteristic
231C2×C4—31no—
232C2×C4—32no—
233C2×C4—31no—
234C2×C4—181no—
235C2×C4—181no—
236C2×C4—182no—
237C2×C4C2×C3⋊S412yes—
238C2×C4C2×C3⋊S411yescharacteristic
239C8—92no—
240C8—181no—
241C8—182no—
242C2×C3—43no—
243C2×C3—43no—
244C2×C3—43no—
245C2×C3C2×#98311yescharacteristic
246C2×C3Q8×S411yescharacteristic
247C2×C3#592411yescharacteristic
248C22C22×C3⋊S411yescharacteristic
249C4—22no—
250C4—31no—
251C4—31no—
252C4—62no—
253C4—61no—
254C4—182no—
255C4—362no—
256C4—361no—
257C4#6426412yes—
258C3—43no—
259C3#8542611yescharacteristic, complemented
260C2C2×#6426411yescharacteristic
261C2Q8×C3⋊S411yescharacteristic
262C2#1059658711yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, prime radical, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×Q82D3
socleC22×C3C22×S4
hypercenterC2×Q8C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×Q8
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
33. C3⋊#226Q8semidirect
259. C3#85426semidirect

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
#105936232
#105936222
C4×C3⋊#2262
#105936202
#854263
#837093
#641534