Finite Group Databasebeta

← C2×C4×C3⋊SL(2,ℤ/4) #11923027

Subgroups — C2×C4×C3⋊SL(2,ℤ/4) #11923027

Order 1152.

1374 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 1374 conjugacy classes of subgroups — too many to draw. Only its 243 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×C4×A4C211yesmaximal, characteristic
2C22×C3⋊SL(2,ℤ/4)C212yesmaximal
3C4×C3⋊SL(2,ℤ/4)C214yesmaximal, direct product factor
4C2×C4×SL(2,ℤ/4)—33nomaximal
5C2×C4×#217—31nomaximal
6C23×C3×A4C2211yescharacteristic
7C2×C3×C4×A4C414yescomplemented
8C2×C3×C4×A4C2212yes—
9C2×C3⋊SL(2,ℤ/4)C418yesdirect product factor
10C2×C3⋊SL(2,ℤ/4)C2214yes—
11C2×C4×C3⋊Dic3—41nomaximal
12C24×C3×C4D311yescharacteristic
13C22×C4×A4D313yes—
14C22×SL(2,ℤ/4)—36no—
15C22×#217—32no—
16C22×C4×Dic3—31no—
17C22×C4×Dic3—31no—
18C4×SL(2,ℤ/4)—312no—
19C4×#217—34no—
20C4×#217—34no—
21C2×C3⋊#95—32no—
22C22×C3×A4C2×C414yes—
23C22×C3×A4C2×C412yes—
24C22×C3×A4C2311yescharacteristic
25C22×C32×C4—41no—
26C22×C3⋊Dic3—42no—
27C3×C4×A4C2×C414yescomplemented
28C3⋊SL(2,ℤ/4)C2×C418yesdirect product factor
29C4×C3⋊Dic3—44no—
30C2×C4×C22⋊C4—91noSylow
31C25×C3C2×D311yescharacteristic
32C23×A4C2×D313yes—
33C23×C3×C4—31no—
34C23×C3×C4—34no—
35C23×C3×C4—31no—
36C23×C3×C4—32no—
37C23×C3×C4Dic314yescomplemented
38C23×C3×C4C2×D312yes—
39C23×Dic3—32no—
40C23×Dic3—32no—
41C2×C4×A4Dic3112yescomplemented
42C2×C4×A4C2×D316yes—
43C2×SL(2,ℤ/4)—324no—
44C2×SL(2,ℤ/4)—312no—
45C2×#217—34no—
46C2×#217—38no—
47C2×#217—38no—
48C2×#217—34no—
49C2×C4×Dic3—34no—
50C2×C4×Dic3—34no—
51C2×C4×Dic3—64no—
52C2×C4×Dic3—61no—
53C2×C4×Dic3—62no—
54C2×C4×Dic3—61no—
55C2×C4×Dic3—123no—
56C3⋊#95—38no—
57C23×C32—41no—
58C2×C3×A4C4214yescomplemented
59C2×C3×A4C22×C412yes—
60C2×C3×A4C22×C411yescharacteristic
61C2×C32×C4—44no—
62C2×C32×C4—42no—
63C2×C3⋊Dic3—48no—
64C2×C3⋊Dic3—44no—
65C24×C4C3⋊D311yescharacteristic
66C22×C22⋊C4—92no—
67C22×C42—91no—
68C22×C42—91no—
69C4×C22⋊C4—94no—
70C4×C22⋊C4—94no—
71C2×#95—92no—
72C24×C3—31no—
73C24×C3—34no—
74C24×C3—31no—
75C24×C3—32no—
76C24×C3C4×D314yes—
77C24×C3C2×Dic312yes—
78C24×C3C22×D311yescharacteristic
79C22×A4C4×D3112yes—
80C22×A4C2×Dic316yes—
81C22×A4C22×D313yes—
82C22×C3×C4—31no—
83C22×C3×C4—34no—
84C22×C3×C4—34no—
85C22×C3×C4—32no—
86C22×C3×C4—32no—
87C22×C3×C4—32no—
88C22×C3×C4—34no—
89C22×C3×C4—38no—
90C22×C3×C4—34no—
91C22×C3×C4—43no—
92C22×C3×C4—64no—
93C22×C3×C4—62no—
94C22×C3×C4—61no—
95C22×C3×C4C2×Dic314yescomplemented
96C22×C3×C4S411yescharacteristic
97C22×Dic3—34no—
98C22×Dic3—34no—
99C22×Dic3—38no—
100C22×Dic3—38no—
101C22×Dic3—64no—
102C22×Dic3—62no—
103C22×Dic3—64no—
104C22×Dic3—64no—
105C22×Dic3—62no—
106C22×Dic3—126no—
107C4×A4C2×Dic3112yescomplemented
108SL(2,ℤ/4)—324no—
109#217—316no—
110#217—38no—
111#217—38no—
112C4×Dic3—68no—
113C4×Dic3—64no—
114C4×Dic3—64no—
115C4×Dic3—1212no—
116C22×C32—44no—
117C22×C32—42no—
118C22×C32—41no—
119C3×A4C2×C4211yescharacteristic, complemented
120C32×C4—44no—
121C3⋊Dic3—48no—
122C25C2×C3⋊D311yescharacteristic
123C23×C4—31no—
124C23×C4—31no—
125C23×C4—32no—
126C23×C4—34no—
127C23×C4—92no—
128C23×C4—92no—
129C23×C4C3⋊Dic314yescomplemented
130C23×C4C2×C3⋊D312yes—
131C2×C22⋊C4—94no—
132C2×C22⋊C4—98no—
133C2×C22⋊C4—98no—
134C2×C22⋊C4—94no—
135C2×C42—94no—
136C2×C42—94no—
137C2×C42—181no—
138C2×C42—184no—
139C2×C42—182no—
140C2×C42—181no—
141#95—98no—
142C23×C3—34no—
143C23×C3—34no—
144C23×C3—32no—
145C23×C3—32no—
146C23×C3—31no—
147C23×C3—31no—
148C23×C3—34no—
149C23×C3—32no—
150C23×C3—32no—
151C23×C3—31no—
152C23×C3—38no—
153C23×C3—34no—
154C23×C3—43no—
155C23×C3—61no—
156C23×C3—62no—
157C23×C3—64no—
158C23×C3C4×Dic314yescomplemented
159C23×C3C2×C4×D312yes—
160C23×C3C22×Dic311yescharacteristic
161C23×C3C2×S411yescharacteristic
162C2×A4C4×Dic3112yescomplemented
163C2×A4C2×C4×D316yes—
164C2×A4C22×Dic313yes—
165C2×C3×C4—34no—
166C2×C3×C4—32no—
167C2×C3×C4—34no—
168C2×C3×C4—34no—
169C2×C3×C4—38no—
170C2×C3×C4—34no—
171C2×C3×C4—46no—
172C2×C3×C4—412no—
173C2×C3×C4—62no—
174C2×C3×C4—64no—
175C2×C3×C4SL(2,ℤ/4)14yescomplemented
176C2×C3×C4C2×S412yes—
177C2×Dic3—38no—
178C2×Dic3—38no—
179C2×Dic3—64no—
180C2×Dic3—64no—
181C2×Dic3—616no—
182C2×Dic3—64no—
183C2×Dic3—68no—
184C2×Dic3—68no—
185C2×Dic3—64no—
186C2×Dic3—1212no—
187C2×Dic3—1224no—
188C2×C32—42no—
189C2×C32—44no—
190C2×C32—41no—
191C24—31no—
192C24—31no—
193C24—34no—
194C24—32no—
195C24C4×C3⋊D314yes—
196C24C2×C3⋊Dic312yes—
197C24C22×C3⋊D311yescharacteristic
198C22×C4—31no—
199C22×C4—32no—
200C22×C4—32no—
201C22×C4—34no—
202C22×C4—34no—
203C22×C4—32no—
204C22×C4—34no—
205C22×C4—38no—
206C22×C4—34no—
207C22×C4—64no—
208C22×C4—62no—
209C22×C4—61no—
210C22×C4—94no—
211C22×C4—94no—
212C22×C4—98no—
213C22×C4—98no—
214C22×C4—184no—
215C22×C4—182no—
216C22×C4—184no—
217C22×C4—184no—
218C22×C4—182no—
219C22×C4C2×C3⋊Dic314yescomplemented
220C22×C4C3⋊S411yescharacteristic
221C22⋊C4—916no—
222C22⋊C4—98no—
223C22⋊C4—98no—
224C42—184no—
225C42—188no—
226C42—184no—
227C22×C3—34no—
228C22×C3—32no—
229C22×C3—31no—
230C22×C3—32no—
231C22×C3—32no—
232C22×C3—34no—
233C22×C3—38no—
234C22×C3—34no—
235C22×C3—34no—
236C22×C3—31no—
237C22×C3—31no—
238C22×C3—32no—
239C22×C3—412no—
240C22×C3—46no—
241C22×C3—43no—
242C22×C3—61no—
243C22×C3—64no—
244C22×C3—62no—
245C22×C3C2×C4×Dic311yescharacteristic, complemented
246C22×C3C4×S414yes—
247C22×C3C2×SL(2,ℤ/4)12yes—
248C22×C3C22×S411yescharacteristic
249A4C2×C4×Dic313yescomplemented
250C3×C4—34no—
251C3×C4—412no—
252C3×C4C2×SL(2,ℤ/4)14yescomplemented
253Dic3—68no—
254Dic3—68no—
255Dic3—1224no—
256C32—41noSylow
257C23—34no—
258C23—34no—
259C23—32no—
260C23—32no—
261C23—31no—
262C23—31no—
263C23—34no—
264C23—32no—
265C23—32no—
266C23—31no—
267C23—38no—
268C23—34no—
269C23—61no—
270C23—64no—
271C23—62no—
272C23C4×C3⋊Dic314yescomplemented
273C23C2×C4×C3⋊D312yes—
274C23C22×C3⋊Dic311yescharacteristic
275C23C2×C3⋊S411yescharacteristic
276C2×C4—34no—
277C2×C4—32no—
278C2×C4—34no—
279C2×C4—34no—
280C2×C4—38no—
281C2×C4—34no—
282C2×C4—62no—
283C2×C4—64no—
284C2×C4—98no—
285C2×C4—98no—
286C2×C4—184no—
287C2×C4—1816no—
288C2×C4—184no—
289C2×C4—184no—
290C2×C4—188no—
291C2×C4—188no—
292C2×C4—184no—
293C2×C4C3⋊SL(2,ℤ/4)14yesdirect product factor
294C2×C4C2×C3⋊S412yes—
295C2×C3—32no—
296C2×C3—31no—
297C2×C3—34no—
298C2×C3—31no—
299C2×C3—412no—
300C2×C3—46no—
301C2×C3—43no—
302C2×C3C4×SL(2,ℤ/4)14yescomplemented
303C2×C3C2×C4×S412yes—
304C2×C3C22×SL(2,ℤ/4)11yescharacteristic
305C22—34no—
306C22—32no—
307C22—31no—
308C22—38no—
309C22—34no—
310C22—34no—
311C22—32no—
312C22—32no—
313C22—34no—
314C22—31no—
315C22—31no—
316C22—32no—
317C22—61no—
318C22—64no—
319C22—62no—
320C22C2×C4×C3⋊Dic311yescharacteristic, complemented
321C22C4×C3⋊S414yes—
322C22C2×C3⋊SL(2,ℤ/4)12yes—
323C22C22×C3⋊S411yescharacteristic
324C4—34no—
325C4—188no—
326C4—188no—
327C4C2×C3⋊SL(2,ℤ/4)14yesdirect product factor
328C3—43no—
329C3C2×C4×SL(2,ℤ/4)11yescharacteristic, complemented
330C2—32no—
331C2—31no—
332C2—34no—
333C2—31no—
334C2C4×C3⋊SL(2,ℤ/4)14yesdirect product factor
335C2C2×C4×C3⋊S412yes—
336C2C22×C3⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, hypercenterC22×C4C3⋊S4
commutator, hypocenterC3×A4C2×C42
FrattiniC22C22×C3⋊S4
FittingC24×C3×C4D3
socleC25×C3C2×D3
prime radicalC23×C3×A4C22
cosocleC22×C3×A4C23
LCSG ▷ C3×A4
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C4×C3⋊SL(2,ℤ/4)C2direct (N × H)
7. C2×C3×C4×A4C4semidirect
9. C2×C3⋊SL(2,ℤ/4)C4direct (N × H)
27. C3×C4×A4C2×C4semidirect
28. C3⋊SL(2,ℤ/4)C2×C4direct (N × H)
37. C23×C3×C4Dic3semidirect
41. C2×C4×A4Dic3semidirect
58. C2×C3×A4C42semidirect
95. C22×C3×C4C2×Dic3semidirect
107. C4×A4C2×Dic3semidirect
119. C3×A4C2×C42semidirect
129. C23×C4C3⋊Dic3semidirect
158. C23×C3C4×Dic3semidirect
162. C2×A4C4×Dic3semidirect
175. C2×C3×C4SL(2,ℤ/4)semidirect
219. C22×C4C2×C3⋊Dic3semidirect
245. C22×C3C2×C4×Dic3semidirect
249. A4C2×C4×Dic3semidirect
252. C3×C4C2×SL(2,ℤ/4)semidirect
272. C23C4×C3⋊Dic3semidirect
293. C2×C4C3⋊SL(2,ℤ/4)direct (N × H)
302. C2×C3C4×SL(2,ℤ/4)semidirect
320. C22C2×C4×C3⋊Dic3semidirect
327. C4C2×C3⋊SL(2,ℤ/4)direct (N × H)
329. C3C2×C4×SL(2,ℤ/4)semidirect
334. C2C4×C3⋊SL(2,ℤ/4)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×C4×A42
C22×C3⋊SL(2,ℤ/4)2
C4×C3⋊SL(2,ℤ/4)2
C2×C4×SL(2,ℤ/4)3
C2×C4×#2173
C2×C4×C3⋊Dic34