Finite Group Databasebeta

← #11922982

Subgroups — #11922982

Order 1152.

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

normal subgroups only

This group has 528 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, complemented
3#10593628C212yesmaximal, complemented
4C4⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic
5C4×C3⋊#226C211yesmaximal, characteristic, complemented
6#85433—33nomaximal
7#83708—31nomaximal
8C2×C3⋊GL(2,𝔽3)C2212yes—
9C2×C3⋊#226C2211yescharacteristic
10#64156—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2212yes—
13C3⋊#859C2211yescharacteristic
14C3×Q82D311yescharacteristic, complemented
15#5585—31no—
16Q8×SL(2,𝔽3)D313yescomplemented
17#5395—36no—
18#5394—36no—
19C4⋊GL(2,𝔽3)—33no—
20C4×#226—33no—
21#5091—31no—
22#5089—32no—
23#5088—31no—
24#5087—32no—
25C4×#216—31no—
26#5028—32no—
27#5027—31no—
28#5026—32no—
29#5024—31no—
30C2×C32×Q8—41no—
31C2×C4⋊(C3⋊D3)—41no—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C3⋊GL(2,𝔽3)—22no—
34C3⋊#226—21no—
35#3752—42no—
36#3752—42no—
37C4×C3⋊Dic3—41no—
38#3375—91noSylow
39C2×GL(2,𝔽3)—36no—
40C2×#226—33no—
41C3×C4⋊Q8—32no—
42C3×C4⋊Q8—31no—
43C3×C4×Q8—31no—
44C3×C4×Q8C2×D312yes—
45C3×C4×Q8C2×D311yescharacteristic
46#947—61no—
47#947—123no—
48C2×#216—31no—
49C2×#215—32no—
50#894—62no—
51#893—61no—
52#875—32no—
53#874—31no—
54C4×D12—32no—
55C4×Dic6—31no—
56C4×SL(2,𝔽3)C2×D313yes—
57C4×SL(2,𝔽3)C2×D316yes—
58#859—33no—
59#835—31no—
60#810—32no—
61#810—31no—
62#809—32no—
63#809—32no—
64#809—32no—
65#807—32no—
66#807—31no—
67C4⋊(C3⋊C8)—31no—
68C4⋊(C3⋊C8)—32no—
69#803—32no—
70C4×C3⋊C8—31no—
71C22×C3⋊D3—42no—
72C32×Q8—82no—
73C2×C32×C4—41no—
74C2×C32×C4—42no—
75C2×C3⋊Dic3—41no—
76C2×C3⋊Dic3—41no—
77C4⋊(C3⋊D3)—82no—
78C3×SL(2,𝔽3)#4111yescharacteristic
79#559—91no—
80Q82C3⋊D311yescharacteristic, complemented
81#489—92no—
82#486—91no—
83#477—91no—
84#474—92no—
85#464—92no—
86#460—91no—
87#446—91no—
88#440—92no—
89C4×Q16—91no—
90C2×C3×Q8—43no—
91C2×C3×Q8—61no—
92C2×C3×Q8S411yescharacteristic
93C2×C3×Q8C22×D311yescharacteristic
94C2×D12—32no—
95C2×D12—61no—
96C2×D12—61no—
97C2×D12—123no—
98C2×C4×D3—62no—
99C2×Dic6—31no—
100C2×SL(2,𝔽3)C22×D313yes—
101GL(2,𝔽3)—66no—
102#226—63no—
103C3×C4⋊C4—31no—
104C3×C4⋊C4—32no—
105C3×C4⋊C4—31no—
106C3×C4⋊C4—32no—
107C3×C42—31no—
108C3×C42—32no—
109#216—61no—
110#216—61no—
111#215—62no—
112#215—62no—
113(C2×D3)⋊C4—62no—
114(C2×D3)⋊C4—62no—
115(C2×D3)⋊C4—62no—
116(C2×D3)⋊C4—126no—
117(C2×D3)⋊C4—126no—
118C4⋊Dic3—31no—
119Dic3⋊C4—61no—
120C4×Dic3—61no—
121C4×Dic3—123no—
122C2×C3⋊C8—31no—
123C2×C3⋊C8—31no—
124C2×C3⋊C8—32no—
125C22×C32—41no—
126C2×C3⋊D3—82no—
127C2×C3⋊D3—82no—
128C2×C3⋊D3—82no—
129C32×C4—42no—
130C32×C4—82no—
131C3⋊Dic3—81no—
132C3⋊Dic3—81no—
133C2×Q16—91no—
134C2×SD16—92no—
135C4⋊Q8—32no—
136C4⋊Q8—31no—
137#127—91no—
138#124—92no—
139#124—181no—
140#123—182no—
141C4×Q8—31no—
142C4×Q8—91no—
143C4×Q8C2×C3⋊D311yescharacteristic
144C4×Q8C2×C3⋊D312yes—
145C4×D4—92no—
146C4×D4—181no—
147#107—91no—
148#107—92no—
149C4⋊C8—91no—
150C4⋊C8—92no—
151#103—91no—
152#103—92no—
153#103—91no—
154#102—92no—
155#102—92no—
156#102—92no—
157#97—92no—
158C4×C8—91no—
159C22×D3—62no—
160C22×D3—126no—
161C3×Q8—22no—
162C3×Q8—31no—
163C3×Q8—86no—
164C3×Q8—241no—
165C3×Q8#23911yescharacteristic
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—32no—
169C2×C3×C4—43no—
170C2×C3×C4—46no—
171C2×C3×C4C2×S412yes—
172C2×C3×C4C2×S411yescharacteristic
173C2×Dic3—61no—
174C2×Dic3—61no—
175C2×Dic3—123no—
176C2×Dic3—123no—
177D12—62no—
178D12—62no—
179D12—122no—
180D12—246no—
181D12—241no—
182C4×D3—242no—
183Dic6—61no—
184Dic6—61no—
185SL(2,𝔽3)#23913yes—
186C3⋊C8—61no—
187C3⋊C8—62no—
188C3⋊C8—61no—
189C2×C32—41no—
190C2×C32—41no—
191C2×C32—41no—
192C3⋊D3—162no—
193C2×Q8—61no—
194C2×Q8—91no—
195C2×Q8C3⋊S411yescharacteristic
196C2×Q8C22×C3⋊D311yescharacteristic
197C2×D4—92no—
198C2×D4—181no—
199C2×D4—181no—
200C22×C4—182no—
201Q16—181no—
202Q16—181no—
203SD16—182no—
204SD16—182no—
205C2×C8—91no—
206C2×C8—91no—
207C2×C8—92no—
208C4⋊C4—32no—
209C4⋊C4—31no—
210C4⋊C4—31no—
211C4⋊C4—32no—
212C4⋊C4—91no—
213C4⋊C4—181no—
214C22⋊C4—182no—
215C22⋊C4—182no—
216C22⋊C4—182no—
217C42—31no—
218C42—32no—
219C42—181no—
220C22×C3—43no—
221C22×C3C22×S411yescharacteristic
222C2×D3—122no—
223C2×D3—122no—
224C2×D3—122no—
225C2×D3—246no—
226C2×D3—246no—
227C2×D3—246no—
228C3×C4—22no—
229C3×C4—31no—
230C3×C4—31no—
231C3×C4—46no—
232C3×C4—62no—
233C3×C4—61no—
234C3×C4—86no—
235C3×C4#98312yes—
236Dic3—121no—
237Dic3—121no—
238Dic3—243no—
239Dic3—243no—
240C32—41noSylow
241C23—182no—
242Q8—22no—
243Q8—31no—
244Q8—181no—
245Q8—181no—
246Q8—241no—
247Q8#383211yescharacteristic
248D4—182no—
249D4—182no—
250D4—362no—
251D4—721no—
252C2×C4—31no—
253C2×C4—32no—
254C2×C4—31no—
255C2×C4—181no—
256C2×C4—181no—
257C2×C4—722no—
258C2×C4C2×C3⋊S412yes—
259C2×C4C2×C3⋊S411yescharacteristic
260C8—182no—
261C8—181no—
262C8—181no—
263C2×C3—43no—
264C2×C3—43no—
265C2×C3—43no—
266C2×C3C2×#98311yescharacteristic
267C2×C3#591911yescharacteristic
268C2×C3#592511yescharacteristic
269D3—242no—
270D3—486no—
271C22—362no—
272C22—362no—
273C22—362no—
274C22C22×C3⋊S411yescharacteristic
275C4—22no—
276C4—31no—
277C4—31no—
278C4—62no—
279C4—61no—
280C4—361no—
281C4—361no—
282C4#6426412yes—
283C3—43no—
284C3#8543311yescharacteristic, complemented
285C2—722no—
286C2C2×#6426411yescharacteristic
287C2#1059658211yescharacteristic
288C2#1059658811yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×Q82D3
socleC22×C3C22×S4
prime radicalC4⋊(C3⋊GL(2,𝔽3))C2
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
2. #10593629C2semidirect
3. #10593628C2semidirect
5. C4×C3⋊#226C2semidirect
14. C3×Q82D3semidirect
16. Q8×SL(2,𝔽3)D3semidirect
80. Q82C3⋊D3semidirect
284. C3#85433semidirect

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
#105936282
C4⋊(C3⋊GL(2,𝔽3))2
C4×C3⋊#2262
#854333
#837083
#641564