Finite Group Databasebeta

← #11922881

Subgroups — #11922881

Order 1152.

616 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#10593655C212yesmaximal, complemented
3#10593628C212yesmaximal, complemented
4C4⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
5#10593625C211yesmaximal, characteristic, complemented
6#85332—33nomaximal
7#82444—31nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C2×C3⋊GL(2,𝔽3)C2212yes—
10#64139—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13C3⋊#859C2212yes—
14C2×C3×C4×Q8D311yescharacteristic, complemented
15#5477—31no—
16C2×C4×SL(2,𝔽3)D313yescomplemented
17#5421—36no—
18#5394—36no—
19C4⋊GL(2,𝔽3)—33no—
20#5391—33no—
21#5227—32no—
22#5027—31no—
23#5027—31no—
24#5026—32no—
25#5026—32no—
26#5021—31no—
27#5021—31no—
28#4967—32no—
29#4926—31no—
30C22×C32×C4—41no—
31C2×#638—42no—
32C2×C4⋊(C3⋊D3)—41no—
33C2×C3×SL(2,𝔽3)—21no—
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C3⋊GL(2,𝔽3)—22no—
36#3752—42no—
37C4⋊(C3⋊Dic3)—41no—
38#3112—91noSylow
39C22×C3×Q8C2×D311yescharacteristic
40C2×#235—61no—
41C22×SL(2,𝔽3)C2×D313yes—
42C2×GL(2,𝔽3)—36no—
43C3×C4×Q8—31no—
44C3×C4×Q8—31no—
45C3×C4×Q8C2×D311yescharacteristic
46C3×C4×Q8C2×D311yescharacteristic
47C2×C3×C4⋊C4—31no—
48C2×C3×C42—31no—
49C2×#215—32no—
50C2×#215—32no—
51#930—32no—
52#930—61no—
53#930—123no—
54#875—61no—
55#874—31no—
56C4×D12—32no—
57C4×D12—61no—
58#869—31no—
59C4×SL(2,𝔽3)C2×D313yes—
60C4×SL(2,𝔽3)C2×D313yes—
61#859—36no—
62#835—32no—
63#835—32no—
64#830—32no—
65#809—32no—
66#809—32no—
67#807—32no—
68#807—32no—
69C4⋊(C3⋊C8)—31no—
70C4⋊(C3⋊C8)—31no—
71#803—32no—
72C23×C32—41no—
73C22×C3⋊D3—42no—
74C2×C32×C4—41no—
75C2×C32×C4—41no—
76C2×C32×C4—82no—
77#638—84no—
78C2×C3⋊Dic3—42no—
79C4⋊(C3⋊D3)—82no—
80C3×SL(2,𝔽3)#4111yescharacteristic, complemented
81#532—91no—
82C2×C4×Q8C3⋊D311yescharacteristic, complemented
83#480—92no—
84#477—91no—
85#477—91no—
86#460—91no—
87#460—91no—
88#448—92no—
89#440—92no—
90#440—92no—
91#432—91no—
92C2×C3×Q8—21no—
93C2×C3×Q8—31no—
94C2×C3×Q8—61no—
95C2×C3×Q8C22×D311yescharacteristic
96C22×C3×C4—31no—
97C22×C3×C4—31no—
98C22×C3×C4—43no—
99C22×C3×C4S411yescharacteristic
100C2×#67—62no—
101C2×#67—126no—
102#235—124no—
103C2×D12—32no—
104C2×D12—61no—
105C2×D12—61no—
106C2×D12—123no—
107C2×C4×D3—62no—
108C2×Dic6—61no—
109C2×SL(2,𝔽3)—23no—
110C2×SL(2,𝔽3)C22×D313yes—
111GL(2,𝔽3)—66no—
112C3×C4⋊C4—31no—
113C3×C4⋊C4—31no—
114C3×C4⋊C4—61no—
115C3×C42—31no—
116C3×C42—31no—
117C3×C42—31no—
118C3×C42—31no—
119#215—62no—
120#215—64no—
121#215—62no—
122(C2×D3)⋊C4—62no—
123(C2×D3)⋊C4—62no—
124(C2×D3)⋊C4—126no—
125C4⋊Dic3—32no—
126C4⋊Dic3—61no—
127C4⋊Dic3—123no—
128C2×C3⋊C8—32no—
129C2×C3⋊C8—32no—
130C22×C32—42no—
131C22×C32—41no—
132C22×C32—81no—
133C22×C32—81no—
134C2×C3⋊D3—82no—
135C2×C3⋊D3—82no—
136C2×C3⋊D3—82no—
137C32×C4—42no—
138C32×C4—81no—
139C3⋊Dic3—82no—
140C2×#41—181no—
141C22×Q8C2×C3⋊D311yescharacteristic
142C2×SD16—92no—
143C2×SD16—92no—
144C4⋊Q8—91no—
145#127—91no—
146#124—181no—
147#121—92no—
148#121—181no—
149C4×Q8—31no—
150C4×Q8—31no—
151C4×Q8C2×C3⋊D311yescharacteristic
152C4×Q8C2×C3⋊D311yescharacteristic
153C4×D4—92no—
154C4×D4—181no—
155C2×C4⋊C4—31no—
156C2×C42—31no—
157#107—92no—
158#107—92no—
159C4⋊C8—91no—
160C4⋊C8—91no—
161#103—92no—
162#103—92no—
163#102—92no—
164#102—92no—
165C22⋊C8—92no—
166#97—92no—
167C23×C3—43no—
168C23×C3C2×S411yescharacteristic
169C22×D3—62no—
170C22×D3—126no—
171C3×Q8—31no—
172C3×Q8—32no—
173C3×Q8—61no—
174C3×Q8#23511yescharacteristic
175C2×C3×C4—22no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—43no—
181C2×C3×C4—43no—
182C2×C3×C4—61no—
183C2×C3×C4—61no—
184C2×C3×C4—62no—
185C2×C3×C4—86no—
186C2×C3×C4C2×S411yescharacteristic
187C2×C3×C4C2×S411yescharacteristic
188#67—124no—
189#67—2412no—
190C2×Dic3—62no—
191C2×Dic3—126no—
192D12—62no—
193D12—62no—
194D12—122no—
195D12—122no—
196D12—246no—
197C4×D3—124no—
198Dic6—122no—
199SL(2,𝔽3)#23513yescomplemented
200C3⋊C8—62no—
201C3⋊C8—62no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32—41no—
205C2×C32—82no—
206C3⋊D3—162no—
207#41—364no—
208C2×Q8—21no—
209C2×Q8—31no—
210C2×Q8—61no—
211C2×Q8—181no—
212C2×Q8C22×C3⋊D311yescharacteristic
213C2×D4—92no—
214C2×D4—181no—
215C2×D4—182no—
216C2×D4—181no—
217C22×C4—31no—
218C22×C4—31no—
219C22×C4—182no—
220C22×C4C3⋊S411yescharacteristic
221SD16—182no—
222SD16—184no—
223SD16—182no—
224C2×C8—92no—
225C2×C8—92no—
226C4⋊C4—31no—
227C4⋊C4—31no—
228C4⋊C4—61no—
229C4⋊C4—92no—
230C4⋊C4—181no—
231C22⋊C4—182no—
232C22⋊C4—182no—
233C42—31no—
234C42—31no—
235C42—31no—
236C42—31no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—46no—
240C22×C3—43no—
241C22×C3—83no—
242C22×C3—83no—
243C22×C3#98612yes—
244C22×C3C22×S411yescharacteristic
245C2×D3—122no—
246C2×D3—122no—
247C2×D3—122no—
248C2×D3—246no—
249C2×D3—246no—
250C2×D3—246no—
251C3×C4—21no—
252C3×C4—31no—
253C3×C4—31no—
254C3×C4—46no—
255C3×C4—61no—
256C3×C4—61no—
257C3×C4—61no—
258C3×C4—83no—
259C3×C4#98312yes—
260Dic3—122no—
261Dic3—246no—
262C32—41noSylow
263C23—182no—
264C23C2×C3⋊S411yescharacteristic
265Q8—31no—
266Q8—32no—
267Q8—61no—
268Q8—362no—
269Q8#382811yescharacteristic
270D4—182no—
271D4—182no—
272D4—362no—
273D4—364no—
274D4—362no—
275C2×C4—22no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—61no—
281C2×C4—61no—
282C2×C4—62no—
283C2×C4—182no—
284C2×C4—364no—
285C2×C4C2×C3⋊S411yescharacteristic
286C2×C4C2×C3⋊S411yescharacteristic
287C8—182no—
288C8—182no—
289C2×C3—22no—
290C2×C3—43no—
291C2×C3—43no—
292C2×C3—43no—
293C2×C3—86no—
294C2×C3#591211yescharacteristic
295C2×C3C2×#98311yescharacteristic
296C2×C3C2×#98611yescharacteristic
297D3—242no—
298D3—486no—
299C22—21no—
300C22—21no—
301C22—362no—
302C22—362no—
303C22—362no—
304C22#6426712yes—
305C22C22×C3⋊S411yescharacteristic
306C4—21no—
307C4—31no—
308C4—31no—
309C4—61no—
310C4—61no—
311C4—61no—
312C4—362no—
313C4#6426412yes—
314C3—43no—
315C3#8533211yescharacteristic, complemented
316C2—22no—
317C2—722no—
318C2#1059657511yescharacteristic
319C2C2×#6426411yescharacteristic
320C2C2×#6426711yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC2×C3×C4×Q8D3
socleC22×C3C22×S4
hypercenterC22×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)#41
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×SL(2,𝔽3)C2semidirect
2. #10593655C2semidirect
3. #10593628C2semidirect
4. C4⋊(C3⋊GL(2,𝔽3))C2semidirect
5. #10593625C2semidirect
14. C2×C3×C4×Q8D3semidirect
16. C2×C4×SL(2,𝔽3)D3semidirect
80. C3×SL(2,𝔽3)#41semidirect
82. C2×C4×Q8C3⋊D3semidirect
199. SL(2,𝔽3)#235semidirect
315. C3#85332semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×SL(2,𝔽3)2
#105936552
#105936282
C4⋊(C3⋊GL(2,𝔽3))2
#105936252
#853323
#824443
#641394