Finite Group Databasebeta

← C2×#10593629 #11922877

Subgroups — C2×#10593629 #11922877

Order 1152.

796 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C22×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic
2C22×C3⋊#226C211yesmaximal, characteristic, complemented
3C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593629C214yesmaximal, direct product factor
5C2×#5395—33nomaximal
6C2×#5028—31nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C2×C3⋊GL(2,𝔽3)—22no—
9C2×C3⋊GL(2,𝔽3)C2212yes—
10C2×C3⋊#226—22no—
11C2×C3⋊#226C2212yescomplemented
12C2×#3752—41nomaximal
13C3×C4×SL(2,𝔽3)C2212yescomplemented
14C22×GL(2,𝔽3)—33no—
15C22×#226—33no—
16C2×C3×C4×Q8D311yescharacteristic, complemented
17C22×#216—31no—
18C22×#215—31no—
19C2×#875—31no—
20C2×C4×SL(2,𝔽3)D313yescomplemented
21#5395—312no—
22#5028—34no—
23#5028—34no—
24C2×#810—31no—
25C2×#809—31no—
26C2×C4⋊(C3⋊C8)—31no—
27C23×C3⋊D3—41no—
28C22×C32×C4—41no—
29C22×C3⋊Dic3—41no—
30C2×C3×SL(2,𝔽3)D412yes—
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C3⋊GL(2,𝔽3)—22no—
33C3⋊#226—22no—
34#3752—44no—
35C2×#464—91noSylow
36C22×C3×Q8C2×D311yescharacteristic
37C22×D12—31no—
38C22×Dic6—31no—
39C22×SL(2,𝔽3)C2×D313yes—
40C2×GL(2,𝔽3)—36no—
41C2×GL(2,𝔽3)—66no—
42C2×#226—36no—
43C2×#226—66no—
44C3×C4×Q8—32no—
45C3×C4×Q8C2×D312yescomplemented
46C2×C3×C4⋊C4—31no—
47C2×C3×C42—31no—
48C2×#216—32no—
49C2×#216—32no—
50C2×#216—62no—
51C2×#216—62no—
52C2×#215—32no—
53C2×#215—32no—
54C2×#215—62no—
55C2×#215—62no—
56C2×(C2×D3)⋊C4—61no—
57C2×(C2×D3)⋊C4—61no—
58C2×(C2×D3)⋊C4—123no—
59C22×C3⋊C8—31no—
60C22×C3⋊C8—31no—
61#875—34no—
62#875—62no—
63C4×SL(2,𝔽3)C2×D316yescomplemented
64#810—34no—
65#809—34no—
66C4⋊(C3⋊C8)—34no—
67C23×C32—41no—
68C22×C3⋊D3—42no—
69C22×C3⋊D3—44no—
70C22×C3⋊D3—82no—
71C22×C3⋊D3—82no—
72C2×C32×C4—42no—
73C2×C32×C4—82no—
74C2×C3⋊Dic3—42no—
75C2×C3⋊Dic3—82no—
76C3×SL(2,𝔽3)C2×D411yescharacteristic
77C22×Q16—91no—
78C22×SD16—91no—
79C2×#124—91no—
80C2×C4×Q8C3⋊D311yescharacteristic, complemented
81#464—94no—
82#464—94no—
83C2×C4⋊C8—91no—
84C2×#103—91no—
85C2×#102—91no—
86C23×D3—61no—
87C23×D3—123no—
88C2×C3×Q8—31no—
89C2×C3×Q8—32no—
90C2×C3×Q8D1212yes—
91C2×C3×Q8C22×D311yescharacteristic
92C22×C3×C4—31no—
93C22×C3×C4—31no—
94C22×C3×C4—43no—
95C22×C3×C4S411yescharacteristic
96C22×Dic3—61no—
97C22×Dic3—123no—
98C2×D12—32no—
99C2×D12—62no—
100C2×D12—62no—
101C2×D12—61no—
102C2×Dic6—32no—
103C2×Dic6—62no—
104C2×Dic6—62no—
105C2×Dic6—61no—
106C2×SL(2,𝔽3)D1216yes—
107C2×SL(2,𝔽3)C22×D313yes—
108GL(2,𝔽3)—66no—
109#226—66no—
110C3×C4⋊C4—32no—
111C3×C4⋊C4—61no—
112C3×C42—32no—
113C3×C42—32no—
114#216—64no—
115#216—62no—
116#216—62no—
117#215—64no—
118#215—62no—
119#215—62no—
120(C2×D3)⋊C4—64no—
121(C2×D3)⋊C4—64no—
122(C2×D3)⋊C4—1212no—
123C2×C3⋊C8—32no—
124C2×C3⋊C8—32no—
125C2×C3⋊C8—62no—
126C2×C3⋊C8—62no—
127C22×C32—42no—
128C22×C32—42no—
129C22×C32—42no—
130C22×C32—41no—
131C2×C3⋊D3—44no—
132C2×C3⋊D3—82no—
133C2×C3⋊D3—82no—
134C2×C3⋊D3—88no—
135C32×C4—82no—
136C3⋊Dic3—82no—
137C22×Q8—91no—
138C22×Q8C2×C3⋊D311yescharacteristic
139C22×D4—91no—
140C2×Q16—92no—
141C2×Q16—92no—
142C2×Q16—182no—
143C2×Q16—182no—
144C2×SD16—92no—
145C2×SD16—92no—
146C2×SD16—182no—
147C2×SD16—182no—
148C22×C8—91no—
149C22×C8—91no—
150#124—94no—
151#124—182no—
152C4×Q8—32no—
153C4×Q8C2×C3⋊D312yescomplemented
154C2×C4⋊C4—31no—
155C2×C22⋊C4—181no—
156C2×C22⋊C4—181no—
157C2×C42—31no—
158C4⋊C8—94no—
159#103—94no—
160#102—94no—
161C23×C3—43no—
162C23×C3C2×S411yescharacteristic
163C22×D3—62no—
164C22×D3—126no—
165C22×D3—1212no—
166C22×D3—122no—
167C22×D3—122no—
168C22×D3—122no—
169C22×D3—246no—
170C22×D3—246no—
171C3×Q8—32no—
172C3×Q8—31no—
173C3×Q8—61no—
174C3×Q8C2×D1211yescharacteristic
175C2×C3×C4—22no—
176C2×C3×C4—32no—
177C2×C3×C4—32no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—32no—
181C2×C3×C4—46no—
182C2×C3×C4—62no—
183C2×C3×C4—86no—
184C2×C3×C4C2×S412yes—
185C2×Dic3—62no—
186C2×Dic3—126no—
187C2×Dic3—122no—
188C2×Dic3—246no—
189D12—62no—
190D12—62no—
191D12—122no—
192Dic6—62no—
193Dic6—62no—
194Dic6—122no—
195SL(2,𝔽3)C2×D1213yes—
196C3⋊C8—62no—
197C3⋊C8—62no—
198C2×C32—44no—
199C2×C32—41no—
200C2×C32—41no—
201C2×C32—41no—
202C3⋊D3—84no—
203C24—181no—
204C2×Q8—31no—
205C2×Q8—32no—
206C2×Q8—92no—
207C2×Q8—181no—
208C2×Q8—182no—
209C2×Q8—182no—
210C2×Q8C4⋊(C3⋊D3)12yes—
211C2×Q8C22×C3⋊D311yescharacteristic
212C2×D4—92no—
213C2×D4—182no—
214C2×D4—182no—
215C2×D4—181no—
216C22×C4—31no—
217C22×C4—31no—
218C22×C4—181no—
219C22×C4C3⋊S411yescharacteristic
220Q16—184no—
221Q16—182no—
222Q16—182no—
223SD16—184no—
224SD16—182no—
225SD16—182no—
226C2×C8—92no—
227C2×C8—92no—
228C2×C8—182no—
229C2×C8—182no—
230C4⋊C4—32no—
231C4⋊C4—61no—
232C22⋊C4—184no—
233C22⋊C4—184no—
234C42—32no—
235C42—32no—
236C22×C3—46no—
237C22×C3—46no—
238C22×C3—46no—
239C22×C3—43no—
240C22×C3C4⋊S412yes—
241C22×C3#98312yes—
242C22×C3#98512yes—
243C22×C3C22×S411yescharacteristic
244C2×D3—1212no—
245C2×D3—122no—
246C2×D3—122no—
247C2×D3—122no—
248C2×D3—246no—
249C2×D3—246no—
250C2×D3—2424no—
251C2×D3—244no—
252C3×C4—22no—
253C3×C4—31no—
254C3×C4—31no—
255C3×C4—32no—
256C3×C4—62no—
257C3×C4—86no—
258Dic3—122no—
259Dic3—246no—
260C32—41noSylow
261C23—182no—
262C23—362no—
263C23—362no—
264C23—362no—
265C23C2×C3⋊S411yescharacteristic
266Q8—32no—
267Q8—31no—
268Q8—61no—
269Q8—182no—
270Q8—182no—
271Q8—362no—
272Q8C2×C4⋊(C3⋊D3)11yescharacteristic
273D4—182no—
274D4—182no—
275D4—362no—
276C2×C4—22no—
277C2×C4—32no—
278C2×C4—32no—
279C2×C4—31no—
280C2×C4—31no—
281C2×C4—32no—
282C2×C4—62no—
283C2×C4—182no—
284C2×C4—362no—
285C2×C4C2×C3⋊S412yes—
286C8—182no—
287C8—182no—
288C2×C3—412no—
289C2×C3—43no—
290C2×C3—43no—
291C2×C3—43no—
292C2×C3#539514yescomplemented
293C2×C3C2×C4⋊S411yescharacteristic
294C2×C3C2×#98311yescharacteristic
295C2×C3C2×#98511yescharacteristic
296D3—2412no—
297D3—242no—
298C22—362no—
299C22—362no—
300C22—362no—
301C22—724no—
302C22C4⋊(C3⋊S4)12yes—
303C22#6426412yes—
304C22#6426612yes—
305C22C22×C3⋊S411yescharacteristic
306C4—22no—
307C4—31no—
308C4—31no—
309C4—32no—
310C4—62no—
311C4—362no—
312C3—43no—
313C3C2×#539511yescharacteristic, complemented
314C2—722no—
315C2#1059362914yesdirect product factor
316C2C2×C4⋊(C3⋊S4)11yescharacteristic
317C2C2×#6426411yescharacteristic
318C2C2×#6426611yescharacteristic
subgroupG/H
centerC23C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FrattiniC22C22×C3⋊S4
FittingC2×C3×C4×Q8D3
socleC23×C3C2×S4
prime radicalC22×C3⋊GL(2,𝔽3)C2
hypercenterC22×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊#226C2semidirect
3. C2×C3×C4×SL(2,𝔽3)C2semidirect
4. #10593629C2direct (N × H)
11. C2×C3⋊#226C22semidirect
13. C3×C4×SL(2,𝔽3)C22semidirect
16. C2×C3×C4×Q8D3semidirect
20. C2×C4×SL(2,𝔽3)D3semidirect
45. C3×C4×Q8C2×D3semidirect
63. C4×SL(2,𝔽3)C2×D3semidirect
80. C2×C4×Q8C3⋊D3semidirect
153. C4×Q8C2×C3⋊D3semidirect
292. C2×C3#5395semidirect
313. C3C2×#5395semidirect
315. C2#10593629direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊GL(2,𝔽3)2
C22×C3⋊#2262
C2×C3×C4×SL(2,𝔽3)2
#105936292
C2×#53953
C2×#50283
C2×#37524