Finite Group Databasebeta

← C2×#11724211 #13180859

Subgroups — C2×#11724211 #13180859

Order 1728.

1011 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 1011 conjugacy classes of subgroups — too many to draw. Only its 70 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⋊D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C3×C3⋊GL(2,𝔽3)C212yesmaximal, complemented
3#11724211C214yesmaximal, direct product factor
4C2×D3×GL(2,𝔽3)—32nomaximal
5C2×#63940—31nomaximal
6C22×#1113—41nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
8C3⋊D3×SL(2,𝔽3)C2212yescomplemented
9C3×C3⋊GL(2,𝔽3)C2214yescomplemented
10C2×Q8×C3⋊D3D311yescharacteristic, complemented
11C2×#3802—31no—
12C2×D3×SL(2,𝔽3)—32no—
13C2×D3×SL(2,𝔽3)—61no—
14C2×C3⋊GL(2,𝔽3)D312yescomplemented
15C2×C3×GL(2,𝔽3)—32no—
16C2×C3×GL(2,𝔽3)—62no—
17D3×GL(2,𝔽3)—38no—
18C2×C3×#215—32no—
19#63940—34no—
20#63940—34no—
21C2×#3716—32no—
22C2×#3710—31no—
23C22×C3×C3⋊D3—42no—
24C22×C3×C3⋊D3—41no—
25C2×#1113—44no—
26C2×#1113—48no—
27C32×SL(2,𝔽3)C2311yescharacteristic, complemented
28C22×GL(2,𝔽3)—91no—
29C2×D3×SD16—92no—
30C22×D32—61no—
31C22×D32—122no—
32C2×C32×Q8C2×D311yescharacteristic, complemented
33Q8×C3⋊D3—32no—
34Q8×C3⋊D3C2×D312yescomplemented
35C2×C4×C3⋊D3—31no—
36C2×#634—31no—
37C2×C3×D12—32no—
38C2×C3×SL(2,𝔽3)—22no—
39C2×C3×SL(2,𝔽3)—21no—
40C2×C3×SL(2,𝔽3)—41no—
41C2×C3×SL(2,𝔽3)C2×D312yescomplemented
42C2×#625—61no—
43#3802—34no—
44#3802—62no—
45D3×SL(2,𝔽3)—34no—
46D3×SL(2,𝔽3)—62no—
47C3⋊GL(2,𝔽3)C2×D314yescomplemented
48C3×GL(2,𝔽3)—34no—
49C3×GL(2,𝔽3)—64no—
50C3×#215—34no—
51C3×#215—34no—
52C2×C3×C3⋊C8—32no—
53#3716—38no—
54#3710—34no—
55C22×C33—41no—
56C2×C3×C3⋊D3—44no—
57C2×C3×C3⋊D3—48no—
58C2×C3×C3⋊D3—44no—
59C2×C3×C3⋊D3—42no—
60#1113—416no—
61C2×D3×Q8—32no—
62C2×D3×Q8—61no—
63C2×D3×D4—92no—
64C22×SL(2,𝔽3)—91no—
65C2×GL(2,𝔽3)—32no—
66C2×GL(2,𝔽3)—62no—
67C2×GL(2,𝔽3)—94no—
68C2×C3×SD16—92no—
69C2×#215—32no—
70C2×#214—92no—
71D3×SD16—98no—
72D3×SD16—98no—
73C2×#204—92no—
74C2×D3×C8—92no—
75C22×C3⋊D3—42no—
76C22×C3⋊D3S411yescharacteristic
77C22×C3×D3—62no—
78C22×C3×D3—122no—
79C22×C3×D3—122no—
80C22×C3×D3—241no—
81C22×C3×D3—242no—
82C2×D32—64no—
83C2×D32—128no—
84C2×D32—1216no—
85C2×D32—124no—
86C32×Q8—31no—
87C32×Q8C22×D311yescharacteristic, complemented
88C2×C32×C4—31no—
89C2×C3⋊Dic3—31no—
90C4×C3⋊D3—32no—
91C4×C3⋊D3—32no—
92#634—32no—
93#634—61no—
94C3×D12—34no—
95C3×D12—62no—
96C3×SL(2,𝔽3)—22no—
97C3×SL(2,𝔽3)—21no—
98C3×SL(2,𝔽3)—41no—
99C3×SL(2,𝔽3)C22×D312yescomplemented
100#625—64no—
101C3×C3⋊C8—34no—
102C22×SD16—271noSylow
103C2×C33—42no—
104C2×C33—41no—
105C3×C3⋊D3—48no—
106C3×C3⋊D3—44no—
107C23×D3—182no—
108C23×D3—361no—
109C2×C3×Q8—21no—
110C2×C3×Q8D3212yescomplemented
111C2×C3×D4—92no—
112C2×#67—182no—
113D3×Q8—34no—
114D3×Q8—62no—
115D3×Q8—94no—
116D3×Q8—182no—
117D3×D4—98no—
118D3×D4—184no—
119C2×D12—32no—
120C2×C4×D3—92no—
121C2×C4×D3—181no—
122C2×Dic6—92no—
123C2×Dic6—181no—
124C2×SL(2,𝔽3)—22no—
125C2×SL(2,𝔽3)—41no—
126C2×SL(2,𝔽3)—92no—
127C2×SL(2,𝔽3)D3211yescharacteristic, complemented
128GL(2,𝔽3)—34no—
129GL(2,𝔽3)—64no—
130C3×SD16—94no—
131C3×SD16—94no—
132C2×C3×C8—92no—
133#215—34no—
134#215—34no—
135#214—98no—
136C2×C3⋊C8—32no—
137#204—98no—
138D3×C8—98no—
139C22×C32—42no—
140C22×C32—81no—
141C22×C32—82no—
142C22×C32—161no—
143C22×C32C2×S411yescharacteristic
144C2×C3⋊D3—44no—
145C2×C3⋊D3—48no—
146C2×C3⋊D3GL(2,𝔽3)14yescomplemented
147C2×C3⋊D3C2×S412yes—
148C2×C3×D3—64no—
149C2×C3×D3—128no—
150C2×C3×D3—124no—
151C2×C3×D3—124no—
152C2×C3×D3—128no—
153C2×C3×D3—124no—
154C2×C3×D3—244no—
155C2×C3×D3—242no—
156C2×C3×D3—244no—
157C2×C3×D3—248no—
158D32—1232no—
159D32—128no—
160C32×C4—31no—
161C32×C4—31no—
162C3⋊Dic3—32no—
163C22×Q8—91no—
164C22×D4—271no—
165C2×SD16—92no—
166C2×SD16—272no—
167C2×SD16—274no—
168C2×SD16—274no—
169C22×C8—271no—
170C33—41noSylow
171C23×C3—182no—
172C23×C3—361no—
173C22×D3—32no—
174C22×D3—61no—
175C22×D3—62no—
176C22×D3—122no—
177C22×D3—188no—
178C22×D3—242no—
179C22×D3—364no—
180C22×D3—368no—
181C22×D3—368no—
182C3×Q8—21no—
183C3×Q8—32no—
184C3×Q8—61no—
185C3×Q8C2×D3212yescomplemented
186C3×D4—94no—
187C3×D4—182no—
188C2×C3×C4—32no—
189C2×C3×C4—61no—
190#67—188no—
191C2×Dic3—92no—
192C2×Dic3—181no—
193D12—34no—
194D12—62no—
195C4×D3—94no—
196C4×D3—94no—
197C4×D3—182no—
198C4×D3—182no—
199Dic6—94no—
200Dic6—182no—
201Dic6—182no—
202Dic6—182no—
203SL(2,𝔽3)—22no—
204SL(2,𝔽3)—41no—
205SL(2,𝔽3)C2×D3211yescharacteristic, complemented
206C3×C8—94no—
207C3⋊C8—34no—
208C2×C32—42no—
209C2×C32—44no—
210C2×C32—81no—
211C2×C32—82no—
212C2×C32—82no—
213C2×C32—84no—
214C2×C32—161no—
215C2×C32—162no—
216C2×C32C2×GL(2,𝔽3)12yescomplemented
217C2×C32C22×S411yescharacteristic
218C3⋊D3—48no—
219C3⋊D3C2×GL(2,𝔽3)14yescomplemented
220C3×D3—128no—
221C3×D3—128no—
222C3×D3—124no—
223C3×D3—244no—
224C3×D3—248no—
225C24—541no—
226C2×Q8—92no—
227C2×Q8—271no—
228C2×Q8—272no—
229C2×Q8#111311yescharacteristic, complemented
230C2×D4—92no—
231C2×D4—274no—
232C2×D4—542no—
233C2×D4—541no—
234C22×C4—271no—
235SD16—94no—
236SD16—94no—
237SD16—278no—
238C2×C8—92no—
239C2×C8—274no—
240C22×C3—21no—
241C22×C3—41no—
242C22×C3—82no—
243C22×C3—161no—
244C22×C3—184no—
245C22×C3—364no—
246C22×C3—362no—
247C22×C3—364no—
248C22×C3D3×S412yes—
249C2×D3—34no—
250C2×D3—38no—
251C2×D3—62no—
252C2×D3—64no—
253C2×D3—64no—
254C2×D3—124no—
255C2×D3—128no—
256C2×D3—124no—
257C2×D3—244no—
258C2×D3—248no—
259C2×D3—3616no—
260C2×D3—3616no—
261C3×C4—32no—
262C3×C4—32no—
263C3×C4—61no—
264C3×C4—61no—
265Dic3—94no—
266Dic3—182no—
267C32—42no—
268C32—81no—
269C32—82no—
270C32—161no—
271C32C22×GL(2,𝔽3)11yescharacteristic, complemented
272C23—91no—
273C23—182no—
274C23—544no—
275C23—1084no—
276Q8—31no—
277Q8—272no—
278Q8—541no—
279Q8C2×#111311yescharacteristic, complemented
280D4—94no—
281D4—182no—
282D4—544no—
283C2×C4—31no—
284C2×C4—272no—
285C2×C4—272no—
286C2×C4—271no—
287C8—94no—
288C2×C3—22no—
289C2×C3—21no—
290C2×C3—42no—
291C2×C3—41no—
292C2×C3—84no—
293C2×C3—82no—
294C2×C3—162no—
295C2×C3—161no—
296C2×C3—364no—
297C2×C3—364no—
298C2×C3D3×GL(2,𝔽3)14yescomplemented
299C2×C3C2×D3×S412yes—
300D3—38no—
301D3—64no—
302D3—128no—
303D3—124no—
304D3—248no—
305C22—94no—
306C22—92no—
307C22—184no—
308C22—364no—
309C22—1088no—
310C22#8910111yescharacteristic
311C4—31no—
312C4—31no—
313C4—272no—
314C3—21no—
315C3—41no—
316C3—82no—
317C3—161no—
318C3C2×D3×GL(2,𝔽3)12yescomplemented
319C2—94no—
320C2—364no—
321C2#1172421112yesdirect product factor
322C2C2×#8910111yescharacteristic
subgroupG/H
center, hypercenterC22#89101
commutator, cosocle, hypocenterC32×SL(2,𝔽3)C23
FrattiniC2C2×#89101
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊D3×SL(2,𝔽3)C2semidirect
2. C2×C3×C3⋊GL(2,𝔽3)C2semidirect
3. #11724211C2direct (N × H)
7. C2×C32×SL(2,𝔽3)C22semidirect
8. C3⋊D3×SL(2,𝔽3)C22semidirect
9. C3×C3⋊GL(2,𝔽3)C22semidirect
10. C2×Q8×C3⋊D3D3semidirect
14. C2×C3⋊GL(2,𝔽3)D3semidirect
27. C32×SL(2,𝔽3)C23semidirect
32. C2×C32×Q8C2×D3semidirect
34. Q8×C3⋊D3C2×D3semidirect
41. C2×C3×SL(2,𝔽3)C2×D3semidirect
47. C3⋊GL(2,𝔽3)C2×D3semidirect
87. C32×Q8C22×D3semidirect
99. C3×SL(2,𝔽3)C22×D3semidirect
110. C2×C3×Q8D32semidirect
127. C2×SL(2,𝔽3)D32semidirect
146. C2×C3⋊D3GL(2,𝔽3)semidirect
185. C3×Q8C2×D32semidirect
205. SL(2,𝔽3)C2×D32semidirect
216. C2×C32C2×GL(2,𝔽3)semidirect
219. C3⋊D3C2×GL(2,𝔽3)semidirect
229. C2×Q8#1113semidirect
271. C32C22×GL(2,𝔽3)semidirect
279. Q8C2×#1113semidirect
298. C2×C3D3×GL(2,𝔽3)semidirect
318. C3C2×D3×GL(2,𝔽3)semidirect
321. C2#11724211direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×SL(2,𝔽3)2
C2×C3×C3⋊GL(2,𝔽3)2
#117242112
C2×D3×GL(2,𝔽3)3
C2×#639403
C22×#11134