Finite Group Databasebeta

← C2×C3×C4×#226 #11922599

Subgroups — C2×C3×C4×#226 #11922599

Order 1152.

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

normal subgroups only

This group has 609 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×#226C211yesmaximal, characteristic
2C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic
3C2×C3×#859C211yesmaximal, characteristic
4C3×C4×#226C214yesmaximal, direct product factor
5C2×C4×#226C311yesmaximal, characteristic, direct product factor
6C2×C3×C4×Q16—31nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C2×C3×#226C414yesdirect product factor
9C2×C3×#226C2212yes—
10C2×C3×C4×Dic3—41nomaximal
11C3×C4×SL(2,𝔽3)C2212yes—
12C3×#859C2212yes—
13C22×#226C2×C311yescharacteristic
14C22×C3×Q16—31no—
15C2×C3×C4×Q8—31no—
16C2×C3×C4×Q8D311yescharacteristic
17C2×C4×SL(2,𝔽3)—21no—
18C2×C4×SL(2,𝔽3)C2×C311yescharacteristic
19C2×#859C2×C311yescharacteristic
20C4×#226C2×C314yesdirect product factor
21C3×C4×Q16—34no—
22C3×C4×Q16—34no—
23C2×C3×#107—31no—
24C2×C3×#103—31no—
25C2×C3×#103—31no—
26C2×C3×C4×C8—31no—
27C22×C32×C4—41no—
28C22×C3×Dic3—41no—
29C22×C3×Dic3—41no—
30C2×C3×SL(2,𝔽3)C2×C412yes—
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C3×#226C2×C414yesdirect product factor
33C3×C4×Dic3—44no—
34C2×C4×Q16—31noSylow
35C22×C3×Q8—31no—
36C22×C3×Q8C2×D311yescharacteristic
37C22×SL(2,𝔽3)—21no—
38C22×SL(2,𝔽3)C22×C311yescharacteristic
39C2×#226C3×C414yesdirect product factor
40C2×#226C22×C312yes—
41C2×C3×Q16—34no—
42C2×C3×Q16—32no—
43C2×C3×Q16—34no—
44C2×C3×Q16—32no—
45C22×C3×C8—31no—
46C22×C3×C8—31no—
47C3×C4×Q8—32no—
48C3×C4×Q8—34no—
49C3×C4×Q8—62no—
50C3×C4×Q8C2×D312yes—
51C2×C3×C4⋊C4—31no—
52C2×C3×C4⋊C4—31no—
53C2×C3×C4⋊C4—61no—
54C2×C3×C42—31no—
55C2×C3×C42—61no—
56C2×C4×Dic3—41no—
57C4×SL(2,𝔽3)—22no—
58C4×SL(2,𝔽3)C22×C312yes—
59#859C22×C312yes—
60C3×#107—34no—
61C3×#103—34no—
62C3×#103—32no—
63C3×#103—32no—
64C3×C4×C8—34no—
65C23×C32—41no—
66C2×C32×C4—44no—
67C2×C32×C4—42no—
68C2×C3×Dic3—42no—
69C2×C3×Dic3—42no—
70C2×C3×Dic3—44no—
71C2×C3×Dic3—44no—
72C3×SL(2,𝔽3)C22×C411yescharacteristic
73C22×Q16—31no—
74C2×C4×Q8—31no—
75C2×C4×Q8C3×D311yescharacteristic
76C4×Q16—34no—
77C4×Q16—34no—
78C2×#107—31no—
79C2×#103—31no—
80C2×#103—31no—
81C2×C4×C8—31no—
82C2×C3×Q8—31no—
83C2×C3×Q8—32no—
84C2×C3×Q8—32no—
85C2×C3×Q8—34no—
86C2×C3×Q8—61no—
87C2×C3×Q8—62no—
88C2×C3×Q8C4×D312yes—
89C2×C3×Q8C22×D311yescharacteristic
90C22×C3×C4—31no—
91C22×C3×C4—31no—
92C22×C3×C4—41no—
93C22×C3×C4—61no—
94C22×C3×C4—61no—
95C22×C3×C4—81no—
96C22×C3×C4S411yescharacteristic
97C22×Dic3—41no—
98C22×Dic3—41no—
99C2×SL(2,𝔽3)—22no—
100C2×SL(2,𝔽3)—21no—
101C2×SL(2,𝔽3)C2×C3×C412yes—
102C2×SL(2,𝔽3)C23×C311yescharacteristic
103#226C2×C3×C414yesdirect product factor
104C3×Q16—38no—
105C3×Q16—34no—
106C3×Q16—34no—
107C2×C3×C8—32no—
108C2×C3×C8—34no—
109C2×C3×C8—32no—
110C2×C3×C8—62no—
111C3×C4⋊C4—32no—
112C3×C4⋊C4—32no—
113C3×C4⋊C4—61no—
114C3×C4⋊C4—61no—
115C3×C4⋊C4—64no—
116C3×C42—32no—
117C3×C42—32no—
118C3×C42—64no—
119C4×Dic3—44no—
120C22×C32—42no—
121C22×C32—42no—
122C22×C32—42no—
123C22×C32—41no—
124C32×C4—44no—
125C3×Dic3—44no—
126C3×Dic3—44no—
127C22×Q8—31no—
128C22×Q8C2×C3×D311yescharacteristic
129C2×Q16—32no—
130C2×Q16—34no—
131C2×Q16—34no—
132C2×Q16—32no—
133C22×C8—31no—
134C22×C8—31no—
135C4×Q8—32no—
136C4×Q8—34no—
137C4×Q8—62no—
138C4×Q8C2×C3×D312yes—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—61no—
142C2×C42—31no—
143C2×C42—61no—
144#107—34no—
145#103—34no—
146#103—32no—
147#103—32no—
148C4×C8—34no—
149C23×C3—41no—
150C23×C3—81no—
151C23×C3C2×S411yescharacteristic
152C3×Q8—32no—
153C3×Q8—31no—
154C3×Q8—34no—
155C3×Q8—61no—
156C3×Q8—62no—
157C3×Q8—64no—
158C3×Q8C2×C4×D311yescharacteristic
159C2×C3×C4—32no—
160C2×C3×C4—32no—
161C2×C3×C4—31no—
162C2×C3×C4—31no—
163C2×C3×C4—32no—
164C2×C3×C4—42no—
165C2×C3×C4—44no—
166C2×C3×C4—64no—
167C2×C3×C4—62no—
168C2×C3×C4—62no—
169C2×C3×C4—62no—
170C2×C3×C4—82no—
171C2×C3×C4—84no—
172C2×C3×C4—122no—
173C2×C3×C4#22614yesdirect product factor
174C2×C3×C4C2×S412yes—
175C2×Dic3—42no—
176C2×Dic3—42no—
177C2×Dic3—44no—
178C2×Dic3—44no—
179SL(2,𝔽3)—21no—
180SL(2,𝔽3)C22×C3×C411yescharacteristic
181C3×C8—34no—
182C3×C8—62no—
183C2×C32—44no—
184C2×C32—41no—
185C2×C32—41no—
186C2×C32—41no—
187C2×Q8—31no—
188C2×Q8—32no—
189C2×Q8—32no—
190C2×Q8—34no—
191C2×Q8—61no—
192C2×Q8—62no—
193C2×Q8C3×C4×D312yes—
194C2×Q8C22×C3×D311yescharacteristic
195C22×C4—31no—
196C22×C4—31no—
197C22×C4—61no—
198C22×C4—61no—
199C22×C4C3×S411yescharacteristic
200Q16—38no—
201Q16—34no—
202Q16—34no—
203C2×C8—32no—
204C2×C8—34no—
205C2×C8—32no—
206C2×C8—62no—
207C4⋊C4—32no—
208C4⋊C4—32no—
209C4⋊C4—61no—
210C4⋊C4—61no—
211C4⋊C4—64no—
212C42—32no—
213C42—32no—
214C42—64no—
215C22×C3—41no—
216C22×C3—42no—
217C22×C3—42no—
218C22×C3—42no—
219C22×C3—81no—
220C22×C3—82no—
221C22×C3—82no—
222C22×C3—82no—
223C22×C3C4×S412yes—
224C22×C3C2×#22612yes—
225C22×C3#98512yes—
226C22×C3C22×S411yescharacteristic
227C3×C4—31no—
228C3×C4—31no—
229C3×C4—32no—
230C3×C4—44no—
231C3×C4—64no—
232C3×C4—62no—
233C3×C4—84no—
234C3×C4—122no—
235C3×C4C2×#22614yesdirect product factor
236Dic3—44no—
237Dic3—44no—
238C32—41noSylow
239C23C2×C3×S411yescharacteristic
240Q8—32no—
241Q8—31no—
242Q8—34no—
243Q8—61no—
244Q8—62no—
245Q8—64no—
246Q8C2×C3×C4×D311yescharacteristic
247C2×C4—31no—
248C2×C4—31no—
249C2×C4—32no—
250C2×C4—32no—
251C2×C4—32no—
252C2×C4—62no—
253C2×C4—64no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—122no—
257C2×C4C3×#22614yesdirect product factor
258C2×C4C2×C3×S412yes—
259C8—34no—
260C8—62no—
261C2×C3—44no—
262C2×C3—41no—
263C2×C3—41no—
264C2×C3—41no—
265C2×C3—84no—
266C2×C3—81no—
267C2×C3—81no—
268C2×C3—81no—
269C2×C3C4×#22614yesdirect product factor
270C2×C3C2×C4×S411yescharacteristic
271C2×C3C22×#22611yescharacteristic
272C2×C3C2×#98511yescharacteristic
273C22C3×C4×S412yes—
274C22C2×C3×#22612yes—
275C22C3×#98512yes—
276C22C22×C3×S411yescharacteristic
277C4—31no—
278C4—31no—
279C4—32no—
280C4—64no—
281C4—62no—
282C4—122no—
283C4C2×C3×#22614yesdirect product factor
284C3—41no—
285C3—81no—
286C3C2×C4×#22611yescharacteristic, direct product factor
287C2C3×C4×#22614yesdirect product factor
288C2C2×C3×C4×S411yescharacteristic
289C2C22×C3×#22611yescharacteristic
290C2C2×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×SL(2,𝔽3)C22
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
4. C3×C4×#226C2direct (N × H)
5. C2×C4×#226C3direct (N × H)
8. C2×C3×#226C4direct (N × H)
20. C4×#226C2×C3direct (N × H)
32. C3×#226C2×C4direct (N × H)
39. C2×#226C3×C4direct (N × H)
103. #226C2×C3×C4direct (N × H)
173. C2×C3×C4#226direct (N × H)
235. C3×C4C2×#226direct (N × H)
257. C2×C4C3×#226direct (N × H)
269. C2×C3C4×#226direct (N × H)
283. C4C2×C3×#226direct (N × H)
286. C3C2×C4×#226direct (N × H)
287. C2C3×C4×#226direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#2262
C2×C3×C4×SL(2,𝔽3)2
C2×C3×#8592
C3×C4×#2262
C2×C4×#2263
C2×C3×C4×Q163
C2×C3×C4×Dic34