Finite Group Databasebeta

← #11922968

Subgroups — #11922968

Order 1152.

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

normal subgroups only

This group has 544 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×#557⋊C3C211yesmaximal, characteristic, complemented
2C4×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic
3C4×C3⋊GL(2,𝔽3)C212yesmaximal
4#10593625C211yesmaximal, characteristic, complemented
5#10593623C212yesmaximal, complemented
6#85418—33nomaximal
7#83693—31nomaximal
8C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
9C4⋊C4×C3⋊D3—41nomaximal
10C3×C4×SL(2,𝔽3)C2212yes—
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3⋊#859C2211yescharacteristic
13C3⋊#859C2212yes—
14C3×#557D311yescharacteristic, complemented
15#5579—31no—
16#557⋊C3D313yescomplemented
17C4×GL(2,𝔽3)—33no—
18C4×GL(2,𝔽3)—36no—
19#5391—33no—
20#5389—36no—
21#5087—31no—
22#5080—31no—
23#5066—32no—
24#5065—32no—
25C4×#215—32no—
26C4×#215—31no—
27Q8×C3⋊C8—31no—
28#5022—32no—
29#5021—31no—
30C2×C4×C3⋊D3—41no—
31C2×C4×C3⋊D3—42no—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C3⋊GL(2,𝔽3)Q812yescomplemented
34C32×C4⋊C4—41no—
35C4⋊(C3⋊Dic3)—41no—
36#3750—42no—
37#3381—91noSylow
38C2×GL(2,𝔽3)—33no—
39C3×C4⋊Q8—31no—
40C3×#125—32no—
41C3×C4×Q8—31no—
42C3×C4×Q8C2×D311yescharacteristic
43C3×C4×Q8C2×D312yes—
44#946—61no—
45C2×#215—31no—
46#896—62no—
47D3×C4⋊C4—61no—
48D3×C4⋊C4—123no—
49C4×D12—32no—
50C4×D12—31no—
51#870—32no—
52#869—31no—
53C4×SL(2,𝔽3)C2×D316yes—
54C4×SL(2,𝔽3)C2×D313yes—
55#859—33no—
56#859—36no—
57#835—32no—
58#835—31no—
59#809—31no—
60#809—32no—
61#808—32no—
62#808—31no—
63#808—32no—
64#807—32no—
65#807—32no—
66C4⋊(C3⋊C8)—31no—
67C4⋊(C3⋊C8)—32no—
68C4×C3⋊C8—32no—
69C4×C3⋊C8—31no—
70C22×C3⋊D3—41no—
71C2×C32×C4—41no—
72C2×C32×C4—42no—
73C2×C3⋊Dic3—41no—
74C2×C3⋊Dic3—42no—
75C4×C3⋊D3—44no—
76C4×C3⋊D3—84no—
77C3×SL(2,𝔽3)C2×Q811yescharacteristic
78#557C3⋊D311yescharacteristic, complemented
79#554—91no—
80#499—92no—
81#498—91no—
82#479—92no—
83#478—92no—
84#477—91no—
85#474—91no—
86C8×Q8—91no—
87C4×SD16—91no—
88C4×SD16—92no—
89C2×C3×Q8—31no—
90C2×C3×Q8C22×D311yescharacteristic
91C2×D12—31no—
92C2×C4×D3—61no—
93C2×C4×D3—62no—
94C2×C4×D3—123no—
95C2×C4×D3—126no—
96C2×Dic6—61no—
97C2×SL(2,𝔽3)C22×D313yes—
98GL(2,𝔽3)—36no—
99C3×C4⋊C4—31no—
100C3×C4⋊C4—32no—
101C3×C4⋊C4—32no—
102C3×C4⋊C4—43no—
103C3×C4⋊C4—61no—
104C3×C4⋊C4S411yescharacteristic
105C3×C42—31no—
106C3×C42—32no—
107#215—32no—
108#215—32no—
109(C2×D3)⋊C4—61no—
110(C2×D3)⋊C4—62no—
111C4⋊Dic3—31no—
112C4⋊Dic3—32no—
113C4⋊Dic3—61no—
114C4⋊Dic3—123no—
115Dic3⋊C4—62no—
116Dic3⋊C4—62no—
117Dic3⋊C4—126no—
118C2×C3⋊C8—31no—
119C2×C3⋊C8—32no—
120C2×C3⋊C8—31no—
121C22×C32—41no—
122C2×C3⋊D3—42no—
123C2×C3⋊D3—42no—
124C2×C3⋊D3—42no—
125C32×C4—42no—
126C32×C4—82no—
127C3⋊Dic3—42no—
128C3⋊Dic3—82no—
129C2×SD16—91no—
130C4⋊Q8—31no—
131C4⋊Q8—91no—
132#125—32no—
133#125—92no—
134C22⋊Q8—182no—
135C22⋊Q8—181no—
136C4×Q8—31no—
137C4×Q8C2×C3⋊D311yescharacteristic
138C4×Q8C2×C3⋊D312yes—
139C4×D4—91no—
140C4×D4—92no—
141C2×C4⋊C4—181no—
142#107—92no—
143#107—92no—
144#106—92no—
145#106—91no—
146#106—92no—
147C4⋊C8—91no—
148C4⋊C8—92no—
149#103—92no—
150#103—91no—
151#102—92no—
152#102—91no—
153C4×C8—91no—
154C4×C8—92no—
155C22×D3—61no—
156C22×D3—123no—
157C3×Q8—31no—
158C3×Q8—62no—
159C3×Q8D3×Q811yescharacteristic
160C2×C3×C4—31no—
161C2×C3×C4—32no—
162C2×C3×C4—31no—
163C2×C3×C4—43no—
164C2×C3×C4—46no—
165C2×C3×C4C2×S412yes—
166C2×C3×C4C2×S411yescharacteristic
167C2×Dic3—61no—
168C2×Dic3—62no—
169C2×Dic3—123no—
170C2×Dic3—126no—
171D12—32no—
172D12—61no—
173C4×D3—1212no—
174C4×D3—122no—
175C4×D3—242no—
176C4×D3—2412no—
177Dic6—241no—
178SL(2,𝔽3)D3×Q813yes—
179C3⋊C8—32no—
180C3⋊C8—62no—
181C3⋊C8—61no—
182C2×C32—41no—
183C2×C32—41no—
184C2×C32—41no—
185C3⋊D3—44no—
186C2×Q8—31no—
187C2×Q8—181no—
188C2×Q8C22×C3⋊D311yescharacteristic
189C2×D4—91no—
190C22×C4—181no—
191C22×C4—182no—
192SD16—92no—
193SD16—92no—
194C2×C8—91no—
195C2×C8—91no—
196C2×C8—92no—
197C4⋊C4—32no—
198C4⋊C4—31no—
199C4⋊C4—32no—
200C4⋊C4—61no—
201C4⋊C4—92no—
202C4⋊C4—91no—
203C4⋊C4—182no—
204C4⋊C4—182no—
205C4⋊C4—181no—
206C4⋊C4C3⋊S411yescharacteristic
207C22⋊C4—181no—
208C22⋊C4—182no—
209C42—32no—
210C42—31no—
211C22×C3—43no—
212C22×C3C22×S411yescharacteristic
213C2×D3—62no—
214C2×D3—126no—
215C2×D3—126no—
216C2×D3—126no—
217C2×D3—121no—
218C2×D3—121no—
219C3×C4—22no—
220C3×C4—31no—
221C3×C4—31no—
222C3×C4—46no—
223C3×C4—62no—
224C3×C4—61no—
225C3×C4—86no—
226C3×C4#98512yes—
227Dic3—126no—
228Dic3—121no—
229Dic3—122no—
230Dic3—246no—
231C32—41noSylow
232C23—181no—
233Q8—31no—
234Q8—62no—
235Q8—721no—
236Q8Q8×C3⋊D311yescharacteristic
237D4—92no—
238D4—181no—
239C2×C4—31no—
240C2×C4—31no—
241C2×C4—32no—
242C2×C4—181no—
243C2×C4—182no—
244C2×C4—362no—
245C2×C4—722no—
246C2×C4C2×C3⋊S412yes—
247C2×C4C2×C3⋊S411yescharacteristic
248C8—92no—
249C8—181no—
250C8—182no—
251C2×C3—43no—
252C2×C3—43no—
253C2×C3—43no—
254C2×C3Q8×S411yescharacteristic
255C2×C3C2×#98511yescharacteristic
256C2×C3#592611yescharacteristic
257D3—1212no—
258D3—122no—
259C22—182no—
260C22—361no—
261C22—361no—
262C22C22×C3⋊S411yescharacteristic
263C4—22no—
264C4—31no—
265C4—31no—
266C4—62no—
267C4—61no—
268C4—361no—
269C4—362no—
270C4#6426612yes—
271C3—43no—
272C3#8541811yescharacteristic, complemented
273C2—362no—
274C2Q8×C3⋊S411yescharacteristic
275C2C2×#6426611yescharacteristic
276C2#1059658911yescharacteristic
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)C2×Q8
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. #10593625C2semidirect
5. #10593623C2semidirect
14. C3×#557D3semidirect
16. #557⋊C3D3semidirect
33. C3⋊GL(2,𝔽3)Q8semidirect
78. #557C3⋊D3semidirect
272. C3#85418semidirect

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⋊GL(2,𝔽3)2
#105936252
#105936232
#854183
#836933
C4⋊C4×C3⋊D34