Finite Group Databasebeta

← C2×C3×C4×GL(2,𝔽3) #11922607

Subgroups — C2×C3×C4×GL(2,𝔽3) #11922607

Order 1152.

801 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 801 conjugacy classes of subgroups — too many to draw. Only its 124 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×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C2×C3×#859C211yesmaximal, characteristic, complemented
4C3×C4×GL(2,𝔽3)C214yesmaximal, direct product factor
5C2×C4×GL(2,𝔽3)C311yesmaximal, characteristic, direct product factor
6C2×C3×C4×SD16—31nomaximal
7C22×C3×C4×D3—41nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C2×C3×GL(2,𝔽3)C414yesdirect product factor
10C2×C3×GL(2,𝔽3)C2212yes—
11C3×C4×SL(2,𝔽3)C2212yescomplemented
12C3×#859C2212yescomplemented
13C22×GL(2,𝔽3)C2×C311yescharacteristic
14C22×C3×SD16—31no—
15C2×C3×C4×Q8D311yescharacteristic, complemented
16C2×C3×C4×D4—31no—
17C2×C4×SL(2,𝔽3)—21no—
18C2×C4×SL(2,𝔽3)C2×C311yescharacteristic, complemented
19C2×#859C2×C311yescharacteristic, complemented
20C4×GL(2,𝔽3)C2×C314yesdirect product factor
21C3×C4×SD16—34no—
22C3×C4×SD16—34no—
23C2×C3×#106—31no—
24C2×C3×#103—31no—
25C2×C3×#102—31no—
26C2×C3×C4×C8—31no—
27C23×C3×D3—41no—
28C22×C32×C4—41no—
29C22×C3×Dic3—41no—
30C2×C3×C4×D3—44no—
31C2×C3×C4×D3—48no—
32C2×C3×SL(2,𝔽3)C2×C412yescomplemented
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C3×GL(2,𝔽3)C2×C414yesdirect product factor
35C2×C4×SD16—31noSylow
36C22×C3×Q8C2×D311yescharacteristic
37C22×C3×D4—31no—
38C23×C3×C4—61no—
39C22×C4×D3—41no—
40C22×SL(2,𝔽3)—21no—
41C22×SL(2,𝔽3)C22×C311yescharacteristic
42C2×GL(2,𝔽3)C3×C414yesdirect product factor
43C2×GL(2,𝔽3)C22×C312yes—
44C2×C3×SD16—34no—
45C2×C3×SD16—32no—
46C2×C3×SD16—34no—
47C2×C3×SD16—32no—
48C22×C3×C8—31no—
49C22×C3×C8—31no—
50C3×C4×Q8—32no—
51C3×C4×Q8C2×D312yescomplemented
52C3×C4×D4—34no—
53C3×C4×D4—62no—
54C2×C3×C4⋊C4—31no—
55C2×C3×C4⋊C4—31no—
56C2×C3×C22⋊C4—61no—
57C2×C3×C42—31no—
58C4×SL(2,𝔽3)—22no—
59C4×SL(2,𝔽3)C22×C312yescomplemented
60#859C22×C312yescomplemented
61C3×#106—34no—
62C3×#103—32no—
63C3×#103—32no—
64C3×#102—34no—
65C3×C4×C8—34no—
66C23×C32—41no—
67C22×C3×D3—42no—
68C22×C3×D3—44no—
69C22×C3×D3—44no—
70C22×C3×D3—44no—
71C2×C32×C4—44no—
72C2×C32×C4—42no—
73C2×C3×Dic3—42no—
74C2×C3×Dic3—44no—
75C3×C4×D3—416no—
76C3×SL(2,𝔽3)C22×C411yescharacteristic, complemented
77C22×SD16—31no—
78C2×C4×Q8C3×D311yescharacteristic, complemented
79C2×C4×D4—31no—
80C4×SD16—34no—
81C4×SD16—34no—
82C2×#106—31no—
83C2×#103—31no—
84C2×#102—31no—
85C2×C4×C8—31no—
86C24×C3—61no—
87C23×D3—41no—
88C2×C3×Q8—31no—
89C2×C3×Q8—32no—
90C2×C3×Q8C4×D312yescomplemented
91C2×C3×Q8C22×D311yescharacteristic
92C2×C3×D4—32no—
93C2×C3×D4—34no—
94C2×C3×D4—61no—
95C2×C3×D4—62no—
96C22×C3×C4—31no—
97C22×C3×C4—31no—
98C22×C3×C4—41no—
99C22×C3×C4—61no—
100C22×C3×C4—64no—
101C22×C3×C4—81no—
102C22×C3×C4—124no—
103C22×C3×C4S411yescharacteristic
104C22×Dic3—41no—
105C2×C4×D3—44no—
106C2×C4×D3—48no—
107C2×SL(2,𝔽3)—22no—
108C2×SL(2,𝔽3)—21no—
109C2×SL(2,𝔽3)C2×C3×C412yescomplemented
110C2×SL(2,𝔽3)C23×C311yescharacteristic
111GL(2,𝔽3)C2×C3×C414yesdirect product factor
112C3×SD16—38no—
113C3×SD16—34no—
114C3×SD16—34no—
115C2×C3×C8—32no—
116C2×C3×C8—34no—
117C2×C3×C8—32no—
118C2×C3×C8—62no—
119C3×C4⋊C4—32no—
120C3×C4⋊C4—32no—
121C3×C4⋊C4—61no—
122C3×C4⋊C4—61no—
123C3×C22⋊C4—64no—
124C3×C42—32no—
125C3×C42—32no—
126C22×C32—42no—
127C22×C32—42no—
128C22×C32—42no—
129C22×C32—41no—
130C2×C3×D3—44no—
131C2×C3×D3—44no—
132C2×C3×D3—416no—
133C2×C3×D3—44no—
134C32×C4—44no—
135C3×Dic3—44no—
136C22×Q8C2×C3×D311yescharacteristic
137C22×D4—31no—
138C23×C4—61no—
139C2×SD16—32no—
140C2×SD16—34no—
141C2×SD16—34no—
142C2×SD16—32no—
143C22×C8—31no—
144C22×C8—31no—
145C4×Q8—32no—
146C4×Q8C2×C3×D312yescomplemented
147C4×D4—34no—
148C4×D4—62no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—31no—
151C2×C22⋊C4—61no—
152C2×C42—31no—
153#106—34no—
154#103—32no—
155#103—32no—
156#102—34no—
157C4×C8—34no—
158C23×C3—41no—
159C23×C3—64no—
160C23×C3—62no—
161C23×C3—81no—
162C23×C3—122no—
163C23×C3—122no—
164C23×C3C2×S411yescharacteristic
165C22×D3—44no—
166C22×D3—42no—
167C22×D3—44no—
168C22×D3—44no—
169C3×Q8—32no—
170C3×Q8—31no—
171C3×Q8—61no—
172C3×Q8C2×C4×D311yescharacteristic, complemented
173C3×D4—34no—
174C3×D4—62no—
175C3×D4—64no—
176C2×C3×C4—32no—
177C2×C3×C4—32no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—32no—
181C2×C3×C4—42no—
182C2×C3×C4—44no—
183C2×C3×C4—62no—
184C2×C3×C4—62no—
185C2×C3×C4—82no—
186C2×C3×C4—84no—
187C2×C3×C4—122no—
188C2×C3×C4—128no—
189C2×C3×C4GL(2,𝔽3)14yesdirect product factor
190C2×C3×C4C2×S412yes—
191C2×Dic3—42no—
192C2×Dic3—44no—
193C4×D3—416no—
194SL(2,𝔽3)—21no—
195SL(2,𝔽3)C22×C3×C411yescharacteristic, complemented
196C3×C8—34no—
197C3×C8—62no—
198C2×C32—44no—
199C2×C32—41no—
200C2×C32—41no—
201C2×C32—41no—
202C3×D3—48no—
203C24—61no—
204C2×Q8—31no—
205C2×Q8—32no—
206C2×Q8C3×C4×D312yescomplemented
207C2×Q8C22×C3×D311yescharacteristic
208C2×D4—32no—
209C2×D4—34no—
210C2×D4—61no—
211C2×D4—62no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—61no—
215C22×C4—64no—
216C22×C4—124no—
217C22×C4C3×S411yescharacteristic
218SD16—38no—
219SD16—34no—
220SD16—34no—
221C2×C8—32no—
222C2×C8—34no—
223C2×C8—32no—
224C2×C8—62no—
225C4⋊C4—32no—
226C4⋊C4—32no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C22⋊C4—64no—
230C42—32no—
231C42—32no—
232C22×C3—41no—
233C22×C3—42no—
234C22×C3—42no—
235C22×C3—42no—
236C22×C3—64no—
237C22×C3—81no—
238C22×C3—82no—
239C22×C3—82no—
240C22×C3—82no—
241C22×C3—128no—
242C22×C3—122no—
243C22×C3—122no—
244C22×C3C4×S412yes—
245C22×C3C2×GL(2,𝔽3)12yes—
246C22×C3#98512yes—
247C22×C3C22×S411yescharacteristic
248C2×D3—44no—
249C2×D3—44no—
250C2×D3—416no—
251C2×D3—44no—
252C3×C4—31no—
253C3×C4—31no—
254C3×C4—32no—
255C3×C4—44no—
256C3×C4—62no—
257C3×C4—84no—
258C3×C4—122no—
259C3×C4C2×GL(2,𝔽3)14yesdirect product factor
260Dic3—44no—
261C32—41noSylow
262C23—62no—
263C23—64no—
264C23—122no—
265C23—122no—
266C23C2×C3×S411yescharacteristic
267Q8—32no—
268Q8—31no—
269Q8—61no—
270Q8C2×C3×C4×D311yescharacteristic, complemented
271D4—34no—
272D4—62no—
273D4—64no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—32no—
277C2×C4—32no—
278C2×C4—32no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—122no—
282C2×C4—128no—
283C2×C4C3×GL(2,𝔽3)14yesdirect product factor
284C2×C4C2×C3×S412yes—
285C8—34no—
286C8—62no—
287C2×C3—44no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—41no—
291C2×C3—84no—
292C2×C3—81no—
293C2×C3—81no—
294C2×C3—81no—
295C2×C3—124no—
296C2×C3C4×GL(2,𝔽3)14yesdirect product factor
297C2×C3C2×C4×S411yescharacteristic
298C2×C3C22×GL(2,𝔽3)11yescharacteristic
299C2×C3C2×#98511yescharacteristic
300D3—48no—
301C22—64no—
302C22—128no—
303C22—122no—
304C22—122no—
305C22C3×C4×S412yes—
306C22C2×C3×GL(2,𝔽3)12yes—
307C22C3×#98512yes—
308C22C22×C3×S411yescharacteristic
309C4—31no—
310C4—31no—
311C4—32no—
312C4—62no—
313C4—122no—
314C4C2×C3×GL(2,𝔽3)14yesdirect product factor
315C3—41no—
316C3—81no—
317C3C2×C4×GL(2,𝔽3)11yescharacteristic, direct product factor
318C2—124no—
319C2C3×C4×GL(2,𝔽3)14yesdirect product factor
320C2C2×C3×C4×S411yescharacteristic
321C2C22×C3×GL(2,𝔽3)11yescharacteristic
322C2C2×C3×#98511yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C4S4
commutator, hypocenterSL(2,𝔽3)C22×C3×C4
FrattiniC22C22×C3×S4
FittingC2×C3×C4×Q8D3
socleC23×C3C2×S4
prime radicalC22×C3×GL(2,𝔽3)C2
cosocleC2×SL(2,𝔽3)C23×C3
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×SL(2,𝔽3)C2semidirect
3. C2×C3×#859C2semidirect
4. C3×C4×GL(2,𝔽3)C2direct (N × H)
5. C2×C4×GL(2,𝔽3)C3direct (N × H)
9. C2×C3×GL(2,𝔽3)C4direct (N × H)
11. C3×C4×SL(2,𝔽3)C22semidirect
12. C3×#859C22semidirect
15. C2×C3×C4×Q8D3semidirect
18. C2×C4×SL(2,𝔽3)C2×C3semidirect
19. C2×#859C2×C3semidirect
20. C4×GL(2,𝔽3)C2×C3direct (N × H)
32. C2×C3×SL(2,𝔽3)C2×C4semidirect
34. C3×GL(2,𝔽3)C2×C4direct (N × H)
42. C2×GL(2,𝔽3)C3×C4direct (N × H)
51. C3×C4×Q8C2×D3semidirect
59. C4×SL(2,𝔽3)C22×C3semidirect
60. #859C22×C3semidirect
76. C3×SL(2,𝔽3)C22×C4semidirect
78. C2×C4×Q8C3×D3semidirect
90. C2×C3×Q8C4×D3semidirect
109. C2×SL(2,𝔽3)C2×C3×C4semidirect
111. GL(2,𝔽3)C2×C3×C4direct (N × H)
146. C4×Q8C2×C3×D3semidirect
172. C3×Q8C2×C4×D3semidirect
189. C2×C3×C4GL(2,𝔽3)direct (N × H)
195. SL(2,𝔽3)C22×C3×C4semidirect
206. C2×Q8C3×C4×D3semidirect
259. C3×C4C2×GL(2,𝔽3)direct (N × H)
270. Q8C2×C3×C4×D3semidirect
283. C2×C4C3×GL(2,𝔽3)direct (N × H)
296. C2×C3C4×GL(2,𝔽3)direct (N × H)
314. C4C2×C3×GL(2,𝔽3)direct (N × H)
317. C3C2×C4×GL(2,𝔽3)direct (N × H)
319. C2C3×C4×GL(2,𝔽3)direct (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×C3×C4×SL(2,𝔽3)2
C2×C3×#8592
C3×C4×GL(2,𝔽3)2
C2×C4×GL(2,𝔽3)3
C2×C3×C4×SD163
C22×C3×C4×D34