Finite Group Databasebeta

← C22×C3×#985 #11923696

Subgroups — C22×C3×#985 #11923696

Order 1152.

1649 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 1649 conjugacy classes of subgroups — too many to draw. Only its 220 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×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×C3×#985C2112yesmaximal, direct product factor
3C22×C3×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C22×C3×#226C211yesmaximal, characteristic, complemented
5C22×#985C311yesmaximal, characteristic, direct product factor
6C22×C3×#135—31nomaximal
7C22×C3×C4×D3—41nomaximal
8C2×C3×#41⋊C3C2216yescomplemented
9C22×C3×SL(2,𝔽3)C2211yescharacteristic
10C3×#985C22116yesdirect product factor
11C2×C3×GL(2,𝔽3)C2216yescomplemented
12C2×C3×#226C2216yescomplemented
13C22×C3×#41—31no—
14C22×C3×#41D311yescharacteristic, complemented
15C22×#41⋊C3—21no—
16C22×#41⋊C3C2×C311yescharacteristic, complemented
17C2×#985C2×C3112yesdirect product factor
18C22×GL(2,𝔽3)C2×C311yescharacteristic, complemented
19C22×#226C2×C311yescharacteristic, complemented
20C2×C3×#135—312no—
21C2×C3×#135—312no—
22C22×C3×Q16—31no—
23C22×C3×SD16—31no—
24C22×C3×SD16—31no—
25C22×C3×D8—31no—
26C23×C3×C8—31no—
27C23×C3×D3—41no—
28C22×C32×C4—41no—
29C22×C3×Dic3—41no—
30C2×C3×C4×D3—412no—
31C3×#41⋊C3C2314yescomplemented
32C2×C3×SL(2,𝔽3)C2313yes—
33C3×GL(2,𝔽3)C2314yescomplemented
34C3×#226C2314yescomplemented
35C22×#135—31noSylow
36C2×C3×#41—36no—
37C2×C3×#41—312no—
38C2×C3×#41—66no—
39C2×C3×#41C2×D316yescomplemented
40C22×C3×Q8—31no—
41C22×C3×Q8C2×D311yescharacteristic
42C22×C3×D4—31no—
43C22×C3×D4—31no—
44C22×C3×D4—61no—
45C23×C3×C4—31no—
46C23×C3×C4—61no—
47C22×C4×D3—41no—
48C2×#41⋊C3—26no—
49C2×#41⋊C3C22×C316yescomplemented
50C22×SL(2,𝔽3)—21no—
51C22×SL(2,𝔽3)C22×C311yescharacteristic
52#985C22×C3116yesdirect product factor
53C2×GL(2,𝔽3)C22×C316yescomplemented
54C2×#226C22×C316yescomplemented
55C3×#135—348no—
56C3×#135—316no—
57C2×C3×Q16—36no—
58C2×C3×Q16—36no—
59C2×C3×SD16—312no—
60C2×C3×SD16—36no—
61C2×C3×SD16—36no—
62C2×C3×D8—312no—
63C22×C3×C8—31no—
64C22×C3×C8—31no—
65C22×C3×C8—312no—
66C23×C32—41no—
67C22×C3×D3—46no—
68C22×C3×D3—48no—
69C2×C32×C4—46no—
70C2×C3×Dic3—46no—
71C3×C4×D3—416no—
72C3×SL(2,𝔽3)C2411yescharacteristic
73C22×#41—31no—
74C22×#41C3×D311yescharacteristic, complemented
75C2×#135—312no—
76C2×#135—312no—
77C22×Q16—31no—
78C22×SD16—31no—
79C22×SD16—31no—
80C22×D8—31no—
81C23×C8—31no—
82C24×C3—31no—
83C24×C3—61no—
84C23×D3—41no—
85C3×#41—312no—
86C3×#41—316no—
87C3×#41—64no—
88C3×#41—624no—
89C3×#41C22×D314yescomplemented
90C2×C3×Q8—33no—
91C2×C3×Q8—36no—
92C2×C3×Q8—63no—
93C2×C3×Q8C22×D313yes—
94C2×C3×D4—36no—
95C2×C3×D4—36no—
96C2×C3×D4—63no—
97C2×C3×D4—63no—
98C2×C3×D4—612no—
99C22×C3×C4—31no—
100C22×C3×C4—36no—
101C22×C3×C4—36no—
102C22×C3×C4—41no—
103C22×C3×C4—61no—
104C22×C3×C4—612no—
105C22×C3×C4—81no—
106C22×C3×C4S411yescharacteristic
107C22×Dic3—41no—
108C2×C4×D3—412no—
109#41⋊C3—24no—
110#41⋊C3C23×C314yescomplemented
111C2×SL(2,𝔽3)—23no—
112C2×SL(2,𝔽3)C23×C313yes—
113GL(2,𝔽3)C23×C314yescomplemented
114#226C23×C314yescomplemented
115C3×Q16—312no—
116C3×Q16—34no—
117C3×SD16—316no—
118C3×SD16—312no—
119C3×SD16—34no—
120C3×D8—316no—
121C2×C3×C8—36no—
122C2×C3×C8—36no—
123C2×C3×C8—316no—
124C22×C32—44no—
125C22×C32—43no—
126C2×C3×D3—44no—
127C2×C3×D3—424no—
128C32×C4—44no—
129C3×Dic3—44no—
130C2×#41—36no—
131C2×#41—312no—
132C2×#41—66no—
133C2×#41C2×C3×D316yescomplemented
134C22×Q8—31no—
135C22×Q8C2×C3×D311yescharacteristic
136C22×D4—31no—
137C22×D4—31no—
138C22×D4—61no—
139C23×C4—31no—
140C23×C4—61no—
141#135—348no—
142#135—316no—
143C2×Q16—36no—
144C2×Q16—36no—
145C2×SD16—312no—
146C2×SD16—36no—
147C2×SD16—36no—
148C2×D8—312no—
149C22×C8—31no—
150C22×C8—31no—
151C22×C8—312no—
152C23×C3—36no—
153C23×C3—41no—
154C23×C3—64no—
155C23×C3—66no—
156C23×C3—81no—
157C23×C3—124no—
158C23×C3C2×S411yescharacteristic
159C22×D3—46no—
160C22×D3—48no—
161C3×Q8—33no—
162C3×Q8—34no—
163C3×Q8—61no—
164C3×Q8—66no—
165C3×Q8C23×D311yescharacteristic
166C3×D4—34no—
167C3×D4—34no—
168C3×D4—66no—
169C3×D4—66no—
170C3×D4—616no—
171C2×C3×C4—33no—
172C2×C3×C4—33no—
173C2×C3×C4—34no—
174C2×C3×C4—312no—
175C2×C3×C4—46no—
176C2×C3×C4—66no—
177C2×C3×C4—616no—
178C2×C3×C4—86no—
179C2×C3×C4C2×S416yes—
180C2×Dic3—46no—
181C4×D3—416no—
182SL(2,𝔽3)—21no—
183SL(2,𝔽3)C24×C311yescharacteristic
184C3×C8—34no—
185C3×C8—34no—
186C2×C32—46no—
187C2×C32—41no—
188C3×D3—48no—
189C24—31no—
190C24—61no—
191#41—312no—
192#41—316no—
193#41—64no—
194#41—624no—
195#41C22×C3×D314yescomplemented
196C2×Q8—33no—
197C2×Q8—36no—
198C2×Q8—63no—
199C2×Q8C22×C3×D313yes—
200C2×D4—36no—
201C2×D4—36no—
202C2×D4—63no—
203C2×D4—63no—
204C2×D4—612no—
205C22×C4—31no—
206C22×C4—36no—
207C22×C4—36no—
208C22×C4—61no—
209C22×C4—612no—
210C22×C4C3×S411yescharacteristic
211Q16—312no—
212Q16—34no—
213SD16—316no—
214SD16—312no—
215SD16—34no—
216D8—316no—
217C2×C8—36no—
218C2×C8—36no—
219C2×C8—316no—
220C22×C3—34no—
221C22×C3—43no—
222C22×C3—44no—
223C22×C3—612no—
224C22×C3—64no—
225C22×C3—83no—
226C22×C3—84no—
227C22×C3—1212no—
228C22×C3#98514yesdirect product factor
229C22×C3C22×S413yes—
230C2×D3—44no—
231C2×D3—424no—
232C3×C4—31no—
233C3×C4—33no—
234C3×C4—44no—
235C3×C4—64no—
236C3×C4—84no—
237C3×C4C22×S414yes—
238Dic3—44no—
239C32—41noSylow
240C23—36no—
241C23—64no—
242C23—66no—
243C23—124no—
244C23C2×C3×S411yescharacteristic
245Q8—33no—
246Q8—34no—
247Q8—61no—
248Q8—66no—
249Q8C23×C3×D311yescharacteristic
250D4—34no—
251D4—34no—
252D4—66no—
253D4—66no—
254D4—616no—
255C2×C4—33no—
256C2×C4—33no—
257C2×C4—34no—
258C2×C4—312no—
259C2×C4—66no—
260C2×C4—616no—
261C2×C4C2×C3×S416yes—
262C8—34no—
263C8—34no—
264C2×C3—41no—
265C2×C3—46no—
266C2×C3—64no—
267C2×C3—81no—
268C2×C3—86no—
269C2×C3—124no—
270C2×C3C2×#98516yesdirect product factor
271C2×C3C23×S411yescharacteristic
272D3—48no—
273C22—34no—
274C22—612no—
275C22—64no—
276C22—1212no—
277C22C3×#98514yesdirect product factor
278C22C22×C3×S413yes—
279C4—31no—
280C4—33no—
281C4—64no—
282C4C22×C3×S414yes—
283C3—41no—
284C3—81no—
285C3C22×#98511yescharacteristic, direct product factor
286C2—64no—
287C2—124no—
288C2C2×C3×#98516yesdirect product factor
289C2C23×C3×S411yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C4S4
commutator, cosocle, hypocenterSL(2,𝔽3)C24×C3
FrattiniC2C23×C3×S4
FittingC22×C3×#41D3
socleC23×C3C2×S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#41⋊C3C2semidirect
2. C2×C3×#985C2direct (N × H)
3. C22×C3×GL(2,𝔽3)C2semidirect
4. C22×C3×#226C2semidirect
5. C22×#985C3direct (N × H)
8. C2×C3×#41⋊C3C22semidirect
10. C3×#985C22direct (N × H)
11. C2×C3×GL(2,𝔽3)C22semidirect
12. C2×C3×#226C22semidirect
14. C22×C3×#41D3semidirect
16. C22×#41⋊C3C2×C3semidirect
17. C2×#985C2×C3direct (N × H)
18. C22×GL(2,𝔽3)C2×C3semidirect
19. C22×#226C2×C3semidirect
31. C3×#41⋊C3C23semidirect
33. C3×GL(2,𝔽3)C23semidirect
34. C3×#226C23semidirect
39. C2×C3×#41C2×D3semidirect
49. C2×#41⋊C3C22×C3semidirect
52. #985C22×C3direct (N × H)
53. C2×GL(2,𝔽3)C22×C3semidirect
54. C2×#226C22×C3semidirect
74. C22×#41C3×D3semidirect
89. C3×#41C22×D3semidirect
110. #41⋊C3C23×C3semidirect
113. GL(2,𝔽3)C23×C3semidirect
114. #226C23×C3semidirect
133. C2×#41C2×C3×D3semidirect
195. #41C22×C3×D3semidirect
228. C22×C3#985direct (N × H)
270. C2×C3C2×#985direct (N × H)
277. C22C3×#985direct (N × H)
285. C3C22×#985direct (N × H)
288. C2C2×C3×#985direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×C3×#9852
C22×C3×GL(2,𝔽3)2
C22×C3×#2262
C22×#9853
C22×C3×#1353
C22×C3×C4×D34