Finite Group Databasebeta

← C22×#64204 #11923617

Subgroups — C22×#64204 #11923617

Order 1152.

1788 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1788 conjugacy classes of subgroups — too many to draw. Only its 191 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×Dic3×A4C211yesmaximal, characteristic
2C22×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic
3C22×C3×SL(2,ℤ/4)C211yesmaximal, characteristic
4C2×#64204C2112yesmaximal, direct product factor
5C22×A4⋊Q8—31nomaximal
6C22×#878—31nomaximal
7C23×C3×A4C2211yescharacteristic
8C22×#627—41nomaximal
9C2×Dic3×A4C2216yes—
10C2×C3⋊SL(2,ℤ/4)C2216yes—
11C2×C3×SL(2,ℤ/4)C2216yes—
12#64204C22116yesdirect product factor
13C24×Dic3D311yescharacteristic
14C23×Dic6—31no—
15C22×C4×A4—31no—
16C22×SL(2,ℤ/4)—31no—
17C22×SL(2,ℤ/4)—61no—
18C22×SL(2,ℤ/4)D311yescharacteristic
19C2×A4⋊Q8—312no—
20C22×C3×C22⋊C4—31no—
21C22×#217—31no—
22C22×C4⋊Dic3—31no—
23C22×Dic3⋊C4—31no—
24C22×Dic3⋊C4—31no—
25C2×#878—312no—
26C2×#878—312no—
27C22×C3×A4Q814yescomplemented
28C22×C3×A4C2313yes—
29C22×C3⋊Dic3—41no—
30C22×C3×Dic3—41no—
31C22×C3×Dic3—41no—
32C2×#627—412no—
33Dic3×A4C2314yes—
34C3⋊SL(2,ℤ/4)C2314yes—
35C3×SL(2,ℤ/4)C2314yes—
36C22×C22⋊Q8—91noSylow
37C25×C3C2×D311yescharacteristic
38C23×A4—21no—
39C23×A4C2×D311yescharacteristic
40C23×C3×C4—31no—
41C23×C3×C4—31no—
42C23×Dic3—31no—
43C23×Dic3—36no—
44C23×Dic3—31no—
45C23×Dic3—31no—
46C23×Dic3—31no—
47C23×Dic3C2×D316yes—
48C22×Dic6—312no—
49C22×Dic6—61no—
50C22×Dic6—66no—
51C22×Dic6—61no—
52C22×Dic6—121no—
53C2×C4×A4—36no—
54C2×SL(2,ℤ/4)—36no—
55C2×SL(2,ℤ/4)—66no—
56C2×SL(2,ℤ/4)C2×D316yes—
57A4⋊Q8—316no—
58C2×C3×C22⋊C4—36no—
59C2×C3×C22⋊C4—36no—
60C2×#217—36no—
61C2×#217—36no—
62C2×C4⋊Dic3—312no—
63C2×Dic3⋊C4—312no—
64C2×Dic3⋊C4—312no—
65#878—348no—
66#878—316no—
67C23×C32—41no—
68C2×C3×A4C2×Q816yescomplemented
69C2×C3×A4C2411yescharacteristic
70C2×C3⋊Dic3—46no—
71C2×C3×Dic3—46no—
72C2×C3×Dic3—46no—
73#627—416no—
74C23×Q8—91no—
75C24×C4—31no—
76C2×C22⋊Q8—912no—
77C2×C22⋊Q8—912no—
78C22×C4⋊C4—91no—
79C22×C4⋊C4—91no—
80C22×C4⋊C4—91no—
81C22×C22⋊C4—31no—
82C22×C22⋊C4—91no—
83C24×C3—31no—
84C24×C3—34no—
85C24×C3—33no—
86C24×C3Dic614yescomplemented
87C24×C3C22×D313yes—
88C22×A4—24no—
89C22×A4—23no—
90C22×A4Dic614yescomplemented
91C22×A4C22×D313yes—
92C22×C3×C4—36no—
93C22×C3×C4—36no—
94C22×C3×C4—63no—
95C22×C3×C4—64no—
96C22×C3×C4—61no—
97C22×C3×C4—121no—
98C22×Dic3—31no—
99C22×Dic3—36no—
100C22×Dic3—36no—
101C22×Dic3—312no—
102C22×Dic3—36no—
103C22×Dic3—36no—
104C22×Dic3—36no—
105C22×Dic3—41no—
106C22×Dic3—64no—
107C22×Dic3—63no—
108C22×Dic3—63no—
109C22×Dic3—64no—
110C22×Dic3—61no—
111C22×Dic3—121no—
112C22×Dic3—241no—
113C22×Dic3S411yescharacteristic
114C22×Dic3C22×D314yes—
115C2×Dic6—316no—
116C2×Dic6—612no—
117C2×Dic6—624no—
118C2×Dic6—612no—
119C2×Dic6—1212no—
120C4×A4—34no—
121SL(2,ℤ/4)—34no—
122SL(2,ℤ/4)—64no—
123SL(2,ℤ/4)C22×D314yes—
124C3×C22⋊C4—312no—
125C3×C22⋊C4—34no—
126#217—312no—
127#217—34no—
128C4⋊Dic3—316no—
129Dic3⋊C4—316no—
130Dic3⋊C4—316no—
131C22×C32—44no—
132C22×C32—43no—
133C3×A4C22×Q811yescharacteristic, complemented
134C3⋊Dic3—44no—
135C3×Dic3—44no—
136C3×Dic3—44no—
137C25D3211yescharacteristic
138C22×Q8—912no—
139C22×Q8—181no—
140C22×Q8—186no—
141C22×Q8—181no—
142C23×C4—31no—
143C23×C4—31no—
144C23×C4—36no—
145C23×C4—91no—
146C23×C4—91no—
147C23×C4—96no—
148C23×C4—91no—
149C23×C4—91no—
150C22⋊Q8—916no—
151C22⋊Q8—948no—
152C2×C4⋊C4—912no—
153C2×C4⋊C4—912no—
154C2×C4⋊C4—912no—
155C2×C22⋊C4—36no—
156C2×C22⋊C4—36no—
157C2×C22⋊C4—96no—
158C2×C22⋊C4—96no—
159C23×C3—34no—
160C23×C3—34no—
161C23×C3—33no—
162C23×C3—33no—
163C23×C3—33no—
164C23×C3—312no—
165C23×C3—36no—
166C23×C3—41no—
167C23×C3—61no—
168C23×C3—66no—
169C23×C3—81no—
170C23×C3C2×Dic616yescomplemented
171C23×C3C2×S411yescharacteristic
172C23×C3C23×D311yescharacteristic
173C2×A4—21no—
174C2×A4—26no—
175C2×A4C2×Dic616yescomplemented
176C2×A4C23×D311yescharacteristic
177C2×C3×C4—34no—
178C2×C3×C4—34no—
179C2×C3×C4—66no—
180C2×C3×C4—612no—
181C2×C3×C4—66no—
182C2×C3×C4—126no—
183C2×Dic3—36no—
184C2×Dic3—34no—
185C2×Dic3—312no—
186C2×Dic3—34no—
187C2×Dic3—34no—
188C2×Dic3—34no—
189C2×Dic3—46no—
190C2×Dic3—66no—
191C2×Dic3—66no—
192C2×Dic3—612no—
193C2×Dic3—66no—
194C2×Dic3—126no—
195C2×Dic3—246no—
196C2×Dic3C2×S416yes—
197Dic6—616no—
198Dic6—616no—
199Dic6—1216no—
200C2×C32—46no—
201C2×C32—41no—
202C24—31no—
203C24—33no—
204C24—34no—
205C24#62714yescomplemented
206C24C2×D3213yes—
207C2×Q8—916no—
208C2×Q8—1812no—
209C2×Q8—1824no—
210C2×Q8—1812no—
211C22×C4—31no—
212C22×C4—36no—
213C22×C4—36no—
214C22×C4—34no—
215C22×C4—63no—
216C22×C4—64no—
217C22×C4—61no—
218C22×C4—96no—
219C22×C4—96no—
220C22×C4—91no—
221C22×C4—96no—
222C22×C4—912no—
223C22×C4—96no—
224C22×C4—96no—
225C22×C4—184no—
226C22×C4—183no—
227C22×C4—183no—
228C22×C4—184no—
229C22×C4—181no—
230C4⋊C4—916no—
231C4⋊C4—916no—
232C4⋊C4—916no—
233C22⋊C4—312no—
234C22⋊C4—34no—
235C22⋊C4—912no—
236C22⋊C4—94no—
237C22×C3—36no—
238C22×C3—31no—
239C22×C3—36no—
240C22×C3—36no—
241C22×C3—312no—
242C22×C3—31no—
243C22×C3—33no—
244C22×C3—43no—
245C22×C3—44no—
246C22×C3—63no—
247C22×C3—64no—
248C22×C3—83no—
249C22×C3—84no—
250C22×C3A4⋊Q814yescomplemented
251C22×C3C22×Dic611yescharacteristic, complemented
252C22×C3C22×S413yes—
253A4—21no—
254A4C22×Dic611yescharacteristic, complemented
255C3×C4—64no—
256C3×C4—64no—
257C3×C4—124no—
258Dic3—34no—
259Dic3—44no—
260Dic3—64no—
261Dic3—64no—
262Dic3—124no—
263Dic3—244no—
264Dic3C22×S414yes—
265C32—41noSylow
266C23—33no—
267C23—33no—
268C23—34no—
269C23—34no—
270C23—33no—
271C23—312no—
272C23—36no—
273C23—61no—
274C23—66no—
275C23C2×#62716yescomplemented
276C23D3×S411yescharacteristic
277C23C22×D3211yescharacteristic
278Q8—1816no—
279Q8—1816no—
280C2×C4—36no—
281C2×C4—34no—
282C2×C4—34no—
283C2×C4—66no—
284C2×C4—612no—
285C2×C4—66no—
286C2×C4—94no—
287C2×C4—912no—
288C2×C4—96no—
289C2×C4—94no—
290C2×C4—94no—
291C2×C4—94no—
292C2×C4—186no—
293C2×C4—186no—
294C2×C4—1812no—
295C2×C4—186no—
296C2×C3—31no—
297C2×C3—31no—
298C2×C3—36no—
299C2×C3—46no—
300C2×C3—41no—
301C2×C3—86no—
302C2×C3—81no—
303C2×C3C2×A4⋊Q816yescomplemented
304C2×C3C23×S411yescharacteristic
305C22—31no—
306C22—36no—
307C22—36no—
308C22—36no—
309C22—312no—
310C22—31no—
311C22—33no—
312C22—63no—
313C22—64no—
314C22#6420414yesdirect product factor
315C22C22×#62711yescharacteristic, complemented
316C22C2×D3×S413yes—
317C4—34no—
318C4—64no—
319C4—64no—
320C4—94no—
321C4—184no—
322C4—184no—
323C3—41no—
324C3—81no—
325C3C22×A4⋊Q811yescharacteristic, complemented
326C2—31no—
327C2—31no—
328C2—36no—
329C2C2×#6420416yesdirect product factor
330C2C22×D3×S411yescharacteristic
subgroupG/H
center, hypercenterC23D3×S4
commutator, cosocleC2×C3×A4C24
FrattiniC2C22×D3×S4
Fitting, socleC25×C3C2×D3
prime radicalC23×C3×A4C22
hypocenterC3×A4C22×Q8
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
4. C2×#64204C2direct (N × H)
12. #64204C22direct (N × H)
27. C22×C3×A4Q8semidirect
68. C2×C3×A4C2×Q8semidirect
86. C24×C3Dic6semidirect
90. C22×A4Dic6semidirect
133. C3×A4C22×Q8semidirect
170. C23×C3C2×Dic6semidirect
175. C2×A4C2×Dic6semidirect
205. C24#627semidirect
250. C22×C3A4⋊Q8semidirect
251. C22×C3C22×Dic6semidirect
254. A4C22×Dic6semidirect
275. C23C2×#627semidirect
303. C2×C3C2×A4⋊Q8semidirect
314. C22#64204direct (N × H)
315. C22C22×#627semidirect
325. C3C22×A4⋊Q8semidirect
329. C2C2×#64204direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Dic3×A42
C22×C3⋊SL(2,ℤ/4)2
C22×C3×SL(2,ℤ/4)2
C2×#642042
C22×A4⋊Q83
C22×#8783
C22×#6274