Finite Group Databasebeta

← C22×#63941 #11918675

Subgroups — C22×#63941 #11918675

Order 1152.

3885 conjugacy classes of subgroups; only the 652 normal subgroups are shown.

normal subgroups only

This group has 3885 conjugacy classes of subgroups — too many to draw. Only its 652 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×#3832C211yesmaximal, characteristic, complemented
2C22×#3802C211yesmaximal, characteristic, complemented
3C22×C3×#215C212yesmaximal, complemented
4C2×#63941C2124yesmaximal, direct product factor
5C22×#3714C212yesmaximal, complemented
6C22×#3712C211yesmaximal, characteristic, complemented
7C22×#914—32nomaximal
8C23×D32—21no—
9C22×C32×Q8C2211yescharacteristic, complemented
10C2×#3832—26no—
11C2×#3832C22112yescomplemented
12C22×C4⋊(C3⋊D3)—21no—
13C22×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
14C22×C4×C3⋊D3—21no—
15C22×C4×C3⋊D3C2211yescharacteristic
16C22×C3×D12C2212yes—
17C22×#625—21no—
18C2×#3802—26no—
19C2×#3802C22112yescomplemented
20C2×C3×#215C22124yescomplemented
21C22×C3×C3⋊C8C2212yescomplemented
22#63941C22164yesdirect product factor
23C2×#3714C22124yescomplemented
24C2×#3712C22112yescomplemented
25C22×#239—32no—
26C22×#239—61no—
27C22×D3×D4—32no—
28C22×C3×SD16—32no—
29C22×#215D312yescomplemented
30C22×#213—32no—
31C2×#914—348no—
32C22×D24—32no—
33C22×ΓL(1,𝔽25)—32no—
34C23×C3⋊D3—21no—
35C23×C3⋊D3D411yescharacteristic
36C23×C3×D3—22no—
37C22×D32—212no—
38C22×D32—48no—
39C2×C32×Q8—23no—
40C2×C32×Q8C2316yescomplemented
41C22×C32×C4—21no—
42C22×C32×C4C2311yescharacteristic
43C22×C3⋊Dic3D411yescharacteristic
44#3832—224no—
45#3832C23116yescomplemented
46C2×C4⋊(C3⋊D3)—212no—
47C2×C4⋊(C3⋊D3)—23no—
48C2×C4⋊(C3⋊D3)C2316yescomplemented
49C2×C4×C3⋊D3—212no—
50C2×C4×C3⋊D3D416yes—
51C2×C4×C3⋊D3C2316yes—
52C2×C3×D12—26no—
53C2×C3×D12C23112yes—
54C2×#625—212no—
55#3802—224no—
56#3802C23116yescomplemented
57C3×#215C23132yescomplemented
58C2×C3×C3⋊C8C23112yescomplemented
59#3714C23132yescomplemented
60#3712C23116yescomplemented
61C22×AGL(1,ℤ/8)—91noSylow
62C24×D3—62no—
63C22×C3×Q8—21no—
64C22×C3×Q8C2×D312yescomplemented
65C22×C3×D4—32no—
66C22×#67—62no—
67C2×#239—324no—
68C2×#239—612no—
69C2×#239—612no—
70C2×#239—612no—
71C2×D3×D4—324no—
72C2×D3×D4—612no—
73C22×D12—32no—
74C22×D12—61no—
75C22×D12—62no—
76C22×D12—62no—
77C22×D12C2×D312yes—
78C22×C4×D3—32no—
79C22×C4×D3—61no—
80C22×C4×D3—62no—
81C22×C4×D3—62no—
82C2×C3×SD16—324no—
83C22×C3×C8—32no—
84C2×#215C2×D3124yescomplemented
85C2×#213—324no—
86C22×C3⋊C8C2×D312yescomplemented
87#914—3128no—
88C2×D24—324no—
89C2×ΓL(1,𝔽25)—324no—
90C23×C32C2×D411yescharacteristic
91C22×C3⋊D3—26no—
92C22×C3⋊D3—24no—
93C22×C3⋊D3—44no—
94C22×C3⋊D3C2×D416yes—
95C22×C3×D3—212no—
96C22×C3×D3—48no—
97C2×D32—216no—
98C2×D32—448no—
99C32×Q8—26no—
100C32×Q8C2414yescomplemented
101C2×C32×C4—26no—
102C2×C32×C4C2×D413yes—
103C2×C32×C4C2413yes—
104C2×C3⋊Dic3C2×D416yes—
105C4⋊(C3⋊D3)—216no—
106C4⋊(C3⋊D3)—26no—
107C4⋊(C3⋊D3)C2414yescomplemented
108C4×C3⋊D3—216no—
109C4×C3⋊D3C2×D4112yes—
110C4×C3⋊D3C2414yes—
111C3×D12—212no—
112C3×D12C2418yes—
113#625—216no—
114C3×C3⋊C8C2418yescomplemented
115C22×#41—91no—
116C23×D4—91no—
117C2×AGL(1,ℤ/8)—924no—
118C22×SD16—32no—
119C22×D8—92no—
120C22×M16—91no—
121C24×C3—62no—
122C23×D3—22no—
123C23×D3—32no—
124C23×D3—61no—
125C23×D3—624no—
126C23×D3—62no—
127C23×D3—62no—
128C23×D3—1216no—
129C2×C3×Q8—26no—
130C2×C3×Q8—26no—
131C2×C3×Q8—26no—
132C2×C3×Q8C22×D3112yescomplemented
133C2×C3×D4—312no—
134C2×C3×D4—66no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4C22×D312yes—
139C2×#67—624no—
140C22×Dic3—32no—
141C22×Dic3—61no—
142#239—332no—
143#239—616no—
144#239—648no—
145#239—648no—
146D3×D4—332no—
147D3×D4—648no—
148C2×D12—26no—
149C2×D12—312no—
150C2×D12—66no—
151C2×D12—624no—
152C2×D12—624no—
153C2×D12—66no—
154C2×D12—66no—
155C2×D12C22×D3112yes—
156C2×C4×D3—312no—
157C2×C4×D3—312no—
158C2×C4×D3—66no—
159C2×C4×D3—66no—
160C2×C4×D3—624no—
161C2×C4×D3—624no—
162C3×SD16—332no—
163C2×C3×C8—312no—
164#215C22×D3132yescomplemented
165#213—332no—
166C2×C3⋊C8C22×D3112yescomplemented
167D24—332no—
168ΓL(1,𝔽25)—332no—
169C22×C32AGL(1,ℤ/8)14yescomplemented
170C22×C32C22×D413yes—
171C2×C3⋊D3—24no—
172C2×C3⋊D3—212no—
173C2×C3⋊D3—412no—
174C2×C3⋊D3C22×D414yes—
175C2×C3×D3—28no—
176C2×C3×D3—424no—
177D32—432no—
178C32×C4—24no—
179C32×C4C22×D413yes—
180C32×C4C2511yescharacteristic
181C3⋊Dic3C22×D414yes—
182C25—181no—
183C2×#41—912no—
184C2×#41—186no—
185C22×Q8D3211yescharacteristic, complemented
186C22×D4—32no—
187C22×D4—912no—
188C22×D4—91no—
189C22×D4—181no—
190C22×D4—186no—
191C22×D4—181no—
192C23×C4—91no—
193C23×C4—181no—
194AGL(1,ℤ/8)—964no—
195C2×SD16—324no—
196C2×D8—924no—
197C2×M16—912no—
198C22×C8—32no—
199C23×C3—21no—
200C23×C3—612no—
201C23×C3—128no—
202C23×C3D3×D412yes—
203C22×D3—212no—
204C22×D3—312no—
205C22×D3—48no—
206C22×D3—66no—
207C22×D3—632no—
208C22×D3—68no—
209C22×D3—612no—
210C22×D3—612no—
211C22×D3—124no—
212C22×D3—1296no—
213C22×D3—128no—
214C22×D3—128no—
215C3×Q8—212no—
216C3×Q8—212no—
217C3×Q8—24no—
218C3×Q8C23×D318yescomplemented
219C3×D4—38no—
220C3×D4—612no—
221C2×C3×C4—212no—
222C2×C3×C4—212no—
223C2×C3×C4—23no—
224C2×C3×C4—23no—
225C2×C3×C4D3×D416yes—
226C2×C3×C4C23×D316yes—
227#67—632no—
228C2×Dic3—312no—
229C2×Dic3—66no—
230D12—212no—
231D12—38no—
232D12—64no—
233D12—632no—
234D12—632no—
235D12—612no—
236D12—612no—
237D12C23×D318yes—
238C4×D3—324no—
239C4×D3—38no—
240C4×D3—612no—
241C4×D3—64no—
242C4×D3—632no—
243C4×D3—632no—
244C3×C8—38no—
245C3⋊C8C23×D318yescomplemented
246C2×C32C2×AGL(1,ℤ/8)16yescomplemented
247C2×C32C23×D411yescharacteristic
248C3⋊D3—24no—
249C3⋊D3—44no—
250C3×D3—48no—
251C24—62no—
252C24—91no—
253C24—1812no—
254C24—181no—
255C24—368no—
256#41—916no—
257#41—1824no—
258C2×Q8—23no—
259C2×Q8C2×D3216yescomplemented
260C2×D4—312no—
261C2×D4—66no—
262C2×D4—916no—
263C2×D4—96no—
264C2×D4—1812no—
265C2×D4—1824no—
266C2×D4—1812no—
267C2×D4—183no—
268C22×C4—21no—
269C22×C4—91no—
270C22×C4—96no—
271C22×C4—96no—
272C22×C4—1812no—
273C22×C4C2×D3211yescharacteristic
274SD16—332no—
275D8—932no—
276M16—916no—
277C2×C8—312no—
278C22×C3—23no—
279C22×C3—24no—
280C22×C3—68no—
281C22×C3—1224no—
282C22×C3#91418yescomplemented
283C22×C3C2×D3×D416yes—
284C2×D3—28no—
285C2×D3—38no—
286C2×D3—424no—
287C2×D3—64no—
288C2×D3—624no—
289C2×D3—68no—
290C2×D3—68no—
291C2×D3—1212no—
292C2×D3—1264no—
293C2×D3—1224no—
294C2×D3—1224no—
295C3×C4—28no—
296C3×C4—28no—
297C3×C4—23no—
298C3×C4—21no—
299C3×C4C2×D3×D416yes—
300C3×C4C24×D312yes—
301Dic3—38no—
302Dic3—64no—
303C32C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
304C23—612no—
305C23—96no—
306C23—128no—
307C23—1816no—
308C23—184no—
309C23—186no—
310C23—3648no—
311C23—364no—
312C23#381111yescharacteristic
313Q8—26no—
314Q8C22×D3214yescomplemented
315D4—38no—
316D4—612no—
317D4—94no—
318D4—1816no—
319D4—1816no—
320D4—186no—
321C2×C4—26no—
322C2×C4—96no—
323C2×C4—912no—
324C2×C4—94no—
325C2×C4—1816no—
326C2×C4#381113yes—
327C2×C4C22×D3213yes—
328C8—38no—
329C2×C3—21no—
330C2×C3—26no—
331C2×C3—128no—
332C2×C3C2×#914112yescomplemented
333C2×C3C22×D3×D412yes—
334D3—48no—
335D3—68no—
336D3—124no—
337D3—128no—
338D3—128no—
339C22—68no—
340C22—94no—
341C22—1224no—
342C22—1812no—
343C22—184no—
344C22—3632no—
345C22—3612no—
346C22#6394114yesdirect product factor
347C22C2×#381113yes—
348C4—24no—
349C4—94no—
350C4C2×#381113yes—
351C4C23×D3211yescharacteristic
352C3—21no—
353C3C22×#91412yescomplemented
354C2—128no—
355C2—184no—
356C2—364no—
357C2C2×#6394116yesdirect product factor
358C2C22×#381111yescharacteristic
subgroupG/H
centerC23#3811
commutator, cosocleC32×C4C25
FrattiniC4C23×D32
FittingC22×C32×Q8C22
socleC23×C32C2×D4
hypercenterC22×Q8D32
hypocenterC32C22×AGL(1,ℤ/8)
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#3832C2semidirect
2. C22×#3802C2semidirect
3. C22×C3×#215C2semidirect
4. C2×#63941C2direct (N × H)
5. C22×#3714C2semidirect
6. C22×#3712C2semidirect
9. C22×C32×Q8C22semidirect
11. C2×#3832C22semidirect
13. C22×C4⋊(C3⋊D3)C22semidirect
19. C2×#3802C22semidirect
20. C2×C3×#215C22semidirect
21. C22×C3×C3⋊C8C22semidirect
22. #63941C22direct (N × H)
23. C2×#3714C22semidirect
24. C2×#3712C22semidirect
29. C22×#215D3semidirect
40. C2×C32×Q8C23semidirect
45. #3832C23semidirect
48. C2×C4⋊(C3⋊D3)C23semidirect
56. #3802C23semidirect
57. C3×#215C23semidirect
58. C2×C3×C3⋊C8C23semidirect
59. #3714C23semidirect
60. #3712C23semidirect
64. C22×C3×Q8C2×D3semidirect
84. C2×#215C2×D3semidirect
86. C22×C3⋊C8C2×D3semidirect
100. C32×Q8C24semidirect
107. C4⋊(C3⋊D3)C24semidirect
114. C3×C3⋊C8C24semidirect
132. C2×C3×Q8C22×D3semidirect
164. #215C22×D3semidirect
166. C2×C3⋊C8C22×D3semidirect
169. C22×C32AGL(1,ℤ/8)semidirect
185. C22×Q8D32semidirect
218. C3×Q8C23×D3semidirect
245. C3⋊C8C23×D3semidirect
246. C2×C32C2×AGL(1,ℤ/8)semidirect
259. C2×Q8C2×D32semidirect
282. C22×C3#914semidirect
303. C32C22×AGL(1,ℤ/8)semidirect
314. Q8C22×D32semidirect
332. C2×C3C2×#914semidirect
346. C22#63941direct (N × H)
353. C3C22×#914semidirect
357. C2C2×#63941direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#38322
C22×#38022
C22×C3×#2152
C2×#639412
C22×#37142
C22×#37122
C22×#9143