Finite Group Databasebeta

← C2×C3×#5931 #11923748

Subgroups — C2×C3×#5931 #11923748

Order 1152.

640 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 42 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×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3×#5931C212yesmaximal, direct product factor
3C2×#5931C311yesmaximal, characteristic, direct product factor
4C2×C3×#448—31nomaximal
5C22×C3×S4—41nomaximal
6C3×C22⋊SL(2,𝔽3)C2211yescharacteristic, complemented
7C2×C3×GL(2,𝔽3)—42nomaximal
8C23×C3×Q8D311yescharacteristic, complemented
9C2×C22⋊SL(2,𝔽3)—21no—
10C2×C22⋊SL(2,𝔽3)C2×C311yescharacteristic, complemented
11#5931C2×C312yesdirect product factor
12C22×C3×SD16—32no—
13C2×C3×#121—31no—
14C3×#448—32no—
15C3×#448—34no—
16C3×#448—32no—
17C2×C3×#103—32no—
18C2×C3×C22⋊C8—31no—
19C22×C3×A4—41no—
20C2×C3×S4—42no—
21C2×C3×S4—44no—
22C2×C3×SL(2,𝔽3)—42no—
23C2×C3×SL(2,𝔽3)—81no—
24C3×GL(2,𝔽3)—44no—
25C2×#448—31noSylow
26C22×S4—41no—
27C22×C3×Q8—31no—
28C22×C3×Q8—32no—
29C22×C3×Q8—32no—
30C22×C3×Q8—61no—
31C22×C3×Q8—61no—
32C22×C3×Q8C2×D311yescharacteristic, complemented
33C22×C3×D4—31no—
34C22×C3×D4—61no—
35C23×C3×C4—31no—
36C22⋊SL(2,𝔽3)—21no—
37C22⋊SL(2,𝔽3)C22×C311yescharacteristic, complemented
38C2×GL(2,𝔽3)—42no—
39C2×C3×SD16—34no—
40C2×C3×SD16—34no—
41C2×C3×SD16—62no—
42C2×C3×SD16—62no—
43C2×C3×SD16—62no—
44C2×C3×SD16—62no—
45C22×C3×C8—32no—
46C3×#121—32no—
47C3×#121—32no—
48C3×#121—61no—
49C3×#121—61no—
50C2×C3×C4⋊C4—31no—
51C2×C3×C22⋊C4—61no—
52C3×#103—34no—
53C3×#103—34no—
54C3×C22⋊C8—32no—
55C3×C22⋊C8—32no—
56C22×C3×D3—161no—
57C2×C3×A4—42no—
58C2×C3×A4—41no—
59C3×S4—44no—
60C3×SL(2,𝔽3)—42no—
61C3×SL(2,𝔽3)—81no—
62C23×Q8C3×D311yescharacteristic, complemented
63C22×SD16—32no—
64C2×#121—31no—
65#448—32no—
66#448—34no—
67#448—32no—
68C2×#103—32no—
69C2×C22⋊C8—31no—
70C24×C3—61no—
71C24×C3S411yescharacteristic
72C22×A4—41no—
73C22×A4—81no—
74C2×S4—42no—
75C2×S4—44no—
76C2×C3×Q8—21no—
77C2×C3×Q8—32no—
78C2×C3×Q8—32no—
79C2×C3×Q8—32no—
80C2×C3×Q8—32no—
81C2×C3×Q8—32no—
82C2×C3×Q8—61no—
83C2×C3×Q8—61no—
84C2×C3×Q8—62no—
85C2×C3×Q8—62no—
86C2×C3×Q8—62no—
87C2×C3×Q8—61no—
88C2×C3×Q8—61no—
89C2×C3×Q8—61no—
90C2×C3×Q8—62no—
91C2×C3×Q8S412yescomplemented
92C2×C3×D4—32no—
93C2×C3×D4—62no—
94C2×C3×D4—62no—
95C2×C3×D4—61no—
96C2×C3×D4—62no—
97C2×C3×D4—62no—
98C2×C3×D4—122no—
99C2×C3×D4—122no—
100C22×C3×C4—31no—
101C22×C3×C4—31no—
102C22×C3×C4—31no—
103C22×C3×C4—31no—
104C22×C3×C4—31no—
105C22×C3×C4—31no—
106C22×C3×C4—62no—
107C22×C3×C4—62no—
108C22×C3×C4—61no—
109C2×SL(2,𝔽3)—42no—
110C2×SL(2,𝔽3)—81no—
111C2×SL(2,𝔽3)—82no—
112C2×SL(2,𝔽3)—82no—
113GL(2,𝔽3)—44no—
114C3×SD16—64no—
115C3×SD16—64no—
116C3×SD16—64no—
117C3×SD16—64no—
118C2×C3×C8—34no—
119C2×C3×C8—62no—
120C2×C3×C8—62no—
121C3×C4⋊C4—32no—
122C3×C4⋊C4—61no—
123C3×C22⋊C4—62no—
124C3×C22⋊C4—62no—
125C22×C32—161no—
126C2×C3×D3—162no—
127C2×C3×D3—164no—
128C3×A4—41no—
129C22×Q8—31no—
130C22×Q8—32no—
131C22×Q8—32no—
132C22×Q8—61no—
133C22×Q8—61no—
134C22×Q8C2×C3×D311yescharacteristic, complemented
135C22×D4—31no—
136C22×D4—61no—
137C23×C4—31no—
138C2×SD16—34no—
139C2×SD16—34no—
140C2×SD16—62no—
141C2×SD16—62no—
142C2×SD16—62no—
143C2×SD16—62no—
144C22×C8—32no—
145#121—32no—
146#121—32no—
147#121—61no—
148#121—61no—
149C2×C4⋊C4—31no—
150C2×C22⋊C4—61no—
151#103—34no—
152#103—34no—
153C22⋊C8—32no—
154C22⋊C8—32no—
155C23×C3—31no—
156C23×C3—31no—
157C23×C3—32no—
158C23×C3—62no—
159C23×C3—121no—
160C23×C3—122no—
161C23×C3—122no—
162C23×C3—121no—
163C23×C3GL(2,𝔽3)12yescomplemented
164C23×C3C2×S411yescharacteristic
165C22×D3—161no—
166C2×A4—41no—
167C2×A4—42no—
168C2×A4—81no—
169C2×A4—82no—
170S4—44no—
171C3×Q8—21no—
172C3×Q8—32no—
173C3×Q8—32no—
174C3×Q8—32no—
175C3×Q8—61no—
176C3×Q8—62no—
177C3×Q8—61no—
178C3×Q8—62no—
179C3×Q8—61no—
180C3×Q8C2×S412yescomplemented
181C3×D4—64no—
182C3×D4—122no—
183C3×D4—124no—
184C3×D4—122no—
185C3×D4—122no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—32no—
189C2×C3×C4—32no—
190C2×C3×C4—31no—
191C2×C3×C4—31no—
192C2×C3×C4—62no—
193C2×C3×C4—62no—
194C2×C3×C4—62no—
195C2×C3×C4—62no—
196C2×C3×C4—61no—
197C2×C3×C4—61no—
198C2×C3×C4—62no—
199C2×C3×C4—122no—
200SL(2,𝔽3)—42no—
201SL(2,𝔽3)—81no—
202SL(2,𝔽3)—82no—
203SL(2,𝔽3)—82no—
204C3×C8—64no—
205C2×C32—162no—
206C2×C32—161no—
207C3×D3—164no—
208C24—61no—
209C24C3×S411yescharacteristic
210C2×Q8—21no—
211C2×Q8—32no—
212C2×Q8—32no—
213C2×Q8—32no—
214C2×Q8—32no—
215C2×Q8—32no—
216C2×Q8—61no—
217C2×Q8—61no—
218C2×Q8—62no—
219C2×Q8—62no—
220C2×Q8—62no—
221C2×Q8—61no—
222C2×Q8—61no—
223C2×Q8—61no—
224C2×Q8—62no—
225C2×Q8C3×S412yescomplemented
226C2×D4—32no—
227C2×D4—62no—
228C2×D4—62no—
229C2×D4—61no—
230C2×D4—62no—
231C2×D4—62no—
232C2×D4—122no—
233C2×D4—122no—
234C22×C4—31no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—62no—
241C22×C4—62no—
242C22×C4—61no—
243SD16—64no—
244SD16—64no—
245SD16—64no—
246SD16—64no—
247C2×C8—34no—
248C2×C8—62no—
249C2×C8—62no—
250C4⋊C4—32no—
251C4⋊C4—61no—
252C22⋊C4—62no—
253C22⋊C4—62no—
254C22×C3—32no—
255C22×C3—31no—
256C22×C3—32no—
257C22×C3—32no—
258C22×C3—31no—
259C22×C3—31no—
260C22×C3—61no—
261C22×C3—122no—
262C22×C3—122no—
263C22×C3—122no—
264C22×C3—161no—
265C22×C3—242no—
266C22×C3—242no—
267C22×C3—321no—
268C22×C3C2×GL(2,𝔽3)11yescharacteristic, complemented
269C22×C3#102011yescharacteristic
270C2×D3—162no—
271C2×D3—164no—
272A4—41no—
273A4—81no—
274C3×C4—32no—
275C3×C4—32no—
276C3×C4—61no—
277C3×C4—61no—
278C3×C4—122no—
279C32—161noSylow
280C23—31no—
281C23—31no—
282C23—32no—
283C23—62no—
284C23—121no—
285C23—122no—
286C23—122no—
287C23—121no—
288C23C3×GL(2,𝔽3)12yescomplemented
289C23C2×C3×S411yescharacteristic
290Q8—21no—
291Q8—32no—
292Q8—32no—
293Q8—32no—
294Q8—61no—
295Q8—62no—
296Q8—61no—
297Q8—62no—
298Q8—61no—
299Q8C2×C3×S412yescomplemented
300D4—64no—
301D4—122no—
302D4—122no—
303D4—124no—
304D4—122no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—32no—
308C2×C4—32no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—62no—
312C2×C4—62no—
313C2×C4—62no—
314C2×C4—62no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—62no—
318C2×C4—122no—
319C8—64no—
320C2×C3—31no—
321C2×C3—31no—
322C2×C3—32no—
323C2×C3—161no—
324C2×C3—162no—
325C2×C3—242no—
326C2×C3—321no—
327C2×C3—322no—
328C2×C3#593112yesdirect product factor
329C2×C3C2×#102011yescharacteristic
330D3—164no—
331C22—31no—
332C22—32no—
333C22—31no—
334C22—31no—
335C22—32no—
336C22—32no—
337C22—61no—
338C22—122no—
339C22—122no—
340C22—122no—
341C22—242no—
342C22—242no—
343C22C2×C3×GL(2,𝔽3)11yescharacteristic, complemented
344C22C3×#102011yescharacteristic
345C4—32no—
346C4—32no—
347C4—61no—
348C4—61no—
349C4—122no—
350C3—161no—
351C3—321no—
352C3C2×#593111yescharacteristic, direct product factor
353C2—31no—
354C2—31no—
355C2—32no—
356C2—242no—
357C2C3×#593112yesdirect product factor
358C2C2×C3×#102011yescharacteristic
subgroupG/H
center, hypercenterC22×C3#1020
commutator, cosocle, hypocenterC22⋊SL(2,𝔽3)C22×C3
FrattiniC2C2×C3×#1020
FittingC23×C3×Q8D3
socleC24×C3S4
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊SL(2,𝔽3)C2semidirect
2. C3×#5931C2direct (N × H)
3. C2×#5931C3direct (N × H)
6. C3×C22⋊SL(2,𝔽3)C22semidirect
8. C23×C3×Q8D3semidirect
10. C2×C22⋊SL(2,𝔽3)C2×C3semidirect
11. #5931C2×C3direct (N × H)
32. C22×C3×Q8C2×D3semidirect
37. C22⋊SL(2,𝔽3)C22×C3semidirect
62. C23×Q8C3×D3semidirect
91. C2×C3×Q8S4semidirect
134. C22×Q8C2×C3×D3semidirect
163. C23×C3GL(2,𝔽3)semidirect
180. C3×Q8C2×S4semidirect
225. C2×Q8C3×S4semidirect
268. C22×C3C2×GL(2,𝔽3)semidirect
288. C23C3×GL(2,𝔽3)semidirect
299. Q8C2×C3×S4semidirect
328. C2×C3#5931direct (N × H)
343. C22C2×C3×GL(2,𝔽3)semidirect
352. C3C2×#5931direct (N × H)
357. C2C3×#5931direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊SL(2,𝔽3)2
C3×#59312
C2×#59313
C2×C3×#4483
C22×C3×S44
C2×C3×GL(2,𝔽3)4