Finite Group Databasebeta

← C2×#10593207 #11922113

Subgroups — C2×#10593207 #11922113

Order 1152.

737 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 737 conjugacy classes of subgroups — too many to draw. Only its 83 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
2C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C2×C3×#859C211yesmaximal, characteristic, complemented
4#10593207C214yesmaximal, direct product factor
5C2×#5394—31nomaximal
6C2×#4817—31nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C2×C3⋊GL(2,𝔽3)C414yescomplemented
9C2×C3⋊GL(2,𝔽3)C2212yes—
10C2×#3722—41nomaximal
11Dic3×SL(2,𝔽3)C2212yescomplemented
12C3×#859C2212yescomplemented
13C22×GL(2,𝔽3)—31no—
14C22×GL(2,𝔽3)—61no—
15C2×Q8×Dic3D311yescharacteristic, complemented
16C22×#215—31no—
17C2×#893—31no—
18C2×C4×SL(2,𝔽3)—31no—
19C2×#859D311yescharacteristic, complemented
20#5394—34no—
21C2×C3×#103—31no—
22C2×#821—31no—
23C2×#815—31no—
24C2×#807—31no—
25#4817—34no—
26#4817—34no—
27C23×C3⋊D3—41no—
28C22×C3×Dic3—41no—
29C22×C3×Dic3—41no—
30C2×C3×SL(2,𝔽3)C2×C412yescomplemented
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C3⋊GL(2,𝔽3)C2×C414yescomplemented
33#3722—44no—
34C2×#440—91noSylow
35C22×C3×Q8C2×D311yescharacteristic
36C22×D12—31no—
37C22×C4×D3—61no—
38C22×SL(2,𝔽3)—21no—
39C22×SL(2,𝔽3)C2×D311yescharacteristic
40C2×GL(2,𝔽3)—34no—
41C2×GL(2,𝔽3)—32no—
42C2×GL(2,𝔽3)—64no—
43C2×GL(2,𝔽3)—62no—
44C22×C3×C8—31no—
45C2×C3×C4⋊C4—31no—
46Q8×Dic3—32no—
47Q8×Dic3C2×D312yescomplemented
48C2×#215—34no—
49C2×#215—32no—
50C2×#215—34no—
51C2×#215—32no—
52C2×(C2×D3)⋊C4—61no—
53C2×(C2×D3)⋊C4—121no—
54C2×C4⋊Dic3—31no—
55C2×C4×Dic3—31no—
56C22×C3⋊C8—31no—
57#893—34no—
58#893—62no—
59C4×SL(2,𝔽3)—32no—
60#859C2×D312yescomplemented
61C3×#103—32no—
62C3×#103—32no—
63#821—34no—
64#815—34no—
65#807—34no—
66C23×C32—41no—
67C22×C3⋊D3—44no—
68C22×C3⋊D3—42no—
69C22×C3⋊D3—82no—
70C22×C3⋊D3—82no—
71C2×C3×Dic3—42no—
72C2×C3×Dic3—42no—
73C2×C3×Dic3—82no—
74C2×C3×Dic3—82no—
75C3×SL(2,𝔽3)C22×C411yescharacteristic, complemented
76C22×SD16—91no—
77C2×C4×Q8—31no—
78C2×C4×D4—91no—
79#440—94no—
80#440—94no—
81C2×#107—91no—
82C2×#103—31no—
83C2×#102—91no—
84C2×#97—91no—
85C23×D3—61no—
86C23×D3—121no—
87C23×D3—241no—
88C2×C3×Q8—31no—
89C2×C3×Q8—32no—
90C2×C3×Q8C4×D312yes—
91C2×C3×Q8C22×D311yescharacteristic
92C22×C3×C4—31no—
93C22×C3×C4—61no—
94C22×C3×C4—121no—
95C22×Dic3—31no—
96C22×Dic3—41no—
97C22×Dic3S411yescharacteristic
98C2×D12—32no—
99C2×D12—34no—
100C2×D12—61no—
101C2×D12—62no—
102C2×C4×D3—64no—
103C2×C4×D3—124no—
104C2×SL(2,𝔽3)—22no—
105C2×SL(2,𝔽3)—21no—
106C2×SL(2,𝔽3)C4×D312yescomplemented
107C2×SL(2,𝔽3)C22×D311yescharacteristic
108GL(2,𝔽3)—34no—
109GL(2,𝔽3)—64no—
110C2×C3×C8—32no—
111C2×C3×C8—62no—
112C3×C4⋊C4—32no—
113C3×C4⋊C4—61no—
114#215—38no—
115#215—34no—
116#215—34no—
117(C2×D3)⋊C4—64no—
118(C2×D3)⋊C4—124no—
119C4⋊Dic3—32no—
120C4⋊Dic3—61no—
121C4×Dic3—32no—
122C4×Dic3—32no—
123C2×C3⋊C8—32no—
124C2×C3⋊C8—34no—
125C22×C32—42no—
126C22×C32—42no—
127C22×C32—42no—
128C22×C32—41no—
129C2×C3⋊D3—44no—
130C2×C3⋊D3—88no—
131C2×C3⋊D3—82no—
132C2×C3⋊D3—82no—
133C3×Dic3—82no—
134C3×Dic3—82no—
135C22×Q8D3211yescharacteristic
136C22×D4—91no—
137C23×C4—181no—
138C2×SD16—94no—
139C2×SD16—92no—
140C2×SD16—94no—
141C2×SD16—92no—
142C22×C8—31no—
143C22×C8—91no—
144C4×Q8—32no—
145C4×Q8—92no—
146C4×D4—94no—
147C4×D4—182no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4—91no—
150C2×C22⋊C4—181no—
151C2×C42—91no—
152#107—94no—
153#103—32no—
154#103—32no—
155#102—94no—
156#97—94no—
157C23×C3—41no—
158C23×C3—81no—
159C23×C3C2×S411yescharacteristic
160C22×D3—64no—
161C22×D3—62no—
162C22×D3—122no—
163C22×D3—124no—
164C22×D3—122no—
165C22×D3—122no—
166C22×D3—242no—
167C22×D3—244no—
168C22×D3—242no—
169C22×D3—242no—
170C22×D3—244no—
171C22×D3—244no—
172C3×Q8—32no—
173C3×Q8—31no—
174C3×Q8—61no—
175C3×Q8C2×C4×D311yescharacteristic
176C2×C3×C4—32no—
177C2×C3×C4—32no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—62no—
181C2×C3×C4—122no—
182C2×C3×C4—122no—
183C2×C3×C4—242no—
184C2×Dic3—22no—
185C2×Dic3—32no—
186C2×Dic3—42no—
187C2×Dic3—62no—
188C2×Dic3—82no—
189C2×Dic3C2×S412yes—
190D12—34no—
191D12—62no—
192D12—64no—
193C4×D3—128no—
194SL(2,𝔽3)—21no—
195SL(2,𝔽3)C2×C4×D311yescharacteristic, complemented
196C3×C8—62no—
197C3⋊C8—34no—
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#62412yes—
207C2×Q8C2×D3211yescharacteristic
208C2×D4—92no—
209C2×D4—94no—
210C2×D4—181no—
211C2×D4—182no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—61no—
215C22×C4—91no—
216C22×C4—184no—
217C22×C4—364no—
218SD16—98no—
219SD16—94no—
220SD16—94no—
221C2×C8—32no—
222C2×C8—62no—
223C2×C8—92no—
224C2×C8—94no—
225C4⋊C4—32no—
226C4⋊C4—61no—
227C4⋊C4—92no—
228C4⋊C4—181no—
229C22⋊C4—184no—
230C42—92no—
231C42—92no—
232C22×C3—41no—
233C22×C3—42no—
234C22×C3—42no—
235C22×C3—42no—
236C22×C3—81no—
237C22×C3—82no—
238C22×C3—82no—
239C22×C3—82no—
240C22×C3C4×S412yes—
241C22×C3#98312yes—
242C22×C3#98612yes—
243C22×C3C22×S411yescharacteristic
244C2×D3—64no—
245C2×D3—124no—
246C2×D3—128no—
247C2×D3—122no—
248C2×D3—122no—
249C2×D3—244no—
250C2×D3—242no—
251C2×D3—242no—
252C2×D3—248no—
253C2×D3—244no—
254C2×D3—244no—
255C2×D3—2416no—
256C3×C4—31no—
257C3×C4—31no—
258C3×C4—32no—
259C3×C4—122no—
260C3×C4—242no—
261Dic3—22no—
262Dic3—62no—
263Dic3—82no—
264C32—41noSylow
265C23—182no—
266C23—184no—
267C23—362no—
268C23—362no—
269C23D3×S411yescharacteristic
270Q8—32no—
271Q8—31no—
272Q8—61no—
273Q8C2×#62411yescharacteristic
274D4—94no—
275D4—182no—
276D4—184no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—32no—
280C2×C4—32no—
281C2×C4—32no—
282C2×C4—62no—
283C2×C4—62no—
284C2×C4—92no—
285C2×C4—122no—
286C2×C4—182no—
287C2×C4—368no—
288C8—62no—
289C8—94no—
290C2×C3—41no—
291C2×C3—41no—
292C2×C3—41no—
293C2×C3—44no—
294C2×C3—81no—
295C2×C3—81no—
296C2×C3—81no—
297C2×C3—84no—
298C2×C3#539414yescomplemented
299C2×C3C2×C4×S411yescharacteristic
300C2×C3C2×#98311yescharacteristic
301C2×C3C2×#98611yescharacteristic
302D3—124no—
303D3—244no—
304D3—248no—
305C22—184no—
306C22—362no—
307C22—362no—
308C22—368no—
309C22#6419912yes—
310C22#6420212yes—
311C22(C3⋊S4)⋊C412yes—
312C22C2×D3×S411yescharacteristic
313C4—31no—
314C4—31no—
315C4—32no—
316C4—62no—
317C4—122no—
318C4—182no—
319C3—41no—
320C3—81no—
321C3C2×#539411yescharacteristic, complemented
322C2—364no—
323C2#1059320714yesdirect product factor
324C2C2×#6419911yescharacteristic
325C2C2×#6420211yescharacteristic
326C2C2×(C3⋊S4)⋊C411yescharacteristic
subgroupG/H
center, hypercenterC23D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FrattiniC22C2×D3×S4
FittingC22×C3×Q8C2×D3
socleC23×C3C2×S4
prime radicalC22×C3⋊GL(2,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C23
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×SL(2,𝔽3)C2semidirect
3. C2×C3×#859C2semidirect
4. #10593207C2direct (N × H)
8. C2×C3⋊GL(2,𝔽3)C4semidirect
11. Dic3×SL(2,𝔽3)C22semidirect
12. C3×#859C22semidirect
15. C2×Q8×Dic3D3semidirect
19. C2×#859D3semidirect
30. C2×C3×SL(2,𝔽3)C2×C4semidirect
32. C3⋊GL(2,𝔽3)C2×C4semidirect
47. Q8×Dic3C2×D3semidirect
60. #859C2×D3semidirect
75. C3×SL(2,𝔽3)C22×C4semidirect
106. C2×SL(2,𝔽3)C4×D3semidirect
195. SL(2,𝔽3)C2×C4×D3semidirect
298. C2×C3#5394semidirect
321. C3C2×#5394semidirect
323. C2#10593207direct (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
C2×Dic3×SL(2,𝔽3)2
C2×C3×#8592
#105932072
C2×#53943
C2×#48173
C2×#37224