Finite Group Databasebeta

← C22×#63927 #11918664

Subgroups — C22×#63927 #11918664

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×#3830C211yesmaximal, characteristic, complemented
2C22×#3802C211yesmaximal, characteristic, complemented
3C22×C3×#213C212yesmaximal, complemented
4C2×#63927C2124yesmaximal, direct product factor
5C22×#3716C212yesmaximal, complemented
6C22×#3712C211yesmaximal, characteristic, complemented
7C22×#911—32nomaximal
8C23×D32—21no—
9C22×C32×D4C2211yescharacteristic, complemented
10C22×#638—21no—
11C23×C3⋊Dic3—21no—
12C2×#3830—26no—
13C2×#3830C22112yescomplemented
14C22×C4×C3⋊D3C2211yescharacteristic
15C22×#634C2211yescharacteristic, complemented
16C22×C3×D12C2212yes—
17C22×#625—21no—
18C2×#3802—26no—
19C2×#3802C22112yescomplemented
20C2×C3×#213C22124yescomplemented
21C22×C3×C3⋊C8C2212yescomplemented
22#63927C22164yesdirect product factor
23C2×#3716C22124yescomplemented
24C2×#3712C22112yescomplemented
25C22×#237—32no—
26C22×#237—61no—
27C22×D3×D4—32no—
28C22×C3×D8—32no—
29C22×#214—32no—
30C22×#213D312yescomplemented
31C2×#911—348no—
32C22×#204—32no—
33C22×ΓL(1,𝔽25)—32no—
34C24×C32—21no—
35C23×C3⋊D3D411yescharacteristic
36C23×C3×D3—22no—
37C22×D32—212no—
38C22×D32—48no—
39C2×C32×D4—23no—
40C2×C32×D4C2316yescomplemented
41C22×C32×C4C2311yescharacteristic
42C2×#638—212no—
43C22×C3⋊Dic3—21no—
44C22×C3⋊Dic3—212no—
45C22×C3⋊Dic3D411yescharacteristic
46#3830—224no—
47#3830C23116yescomplemented
48C2×C4×C3⋊D3D416yes—
49C2×C4×C3⋊D3C2316yes—
50C2×#634—23no—
51C2×#634C2316yescomplemented
52C2×C3×D12—26no—
53C2×C3×D12C23112yes—
54C2×#625—212no—
55#3802—224no—
56#3802C23116yescomplemented
57C3×#213C23132yescomplemented
58C2×C3×C3⋊C8C23112yescomplemented
59#3716C23132yescomplemented
60#3712C23116yescomplemented
61C22×AGL(1,ℤ/8)—91noSylow
62C24×D3—62no—
63C22×C3×D4—21no—
64C22×C3×D4—32no—
65C22×C3×D4C2×D312yescomplemented
66C22×#67—62no—
67C22×#67—62no—
68C22×#67—62no—
69C23×Dic3—62no—
70C23×Dic3—62no—
71C2×#237—324no—
72C2×#237—612no—
73C2×#237—612no—
74C2×#237—612no—
75C2×D3×D4—324no—
76C2×D3×D4—612no—
77C22×D12C2×D312yes—
78C22×C4×D3—32no—
79C22×C4×D3—61no—
80C22×Dic6—32no—
81C22×Dic6—61no—
82C2×C3×D8—324no—
83C22×C3×C8—32no—
84C2×#214—324no—
85C2×#213C2×D3124yescomplemented
86C22×C3⋊C8C2×D312yescomplemented
87#911—3128no—
88C2×#204—324no—
89C2×ΓL(1,𝔽25)—324no—
90C23×C32—26no—
91C23×C32—44no—
92C23×C32C2×D411yescharacteristic
93C22×C3⋊D3—24no—
94C22×C3⋊D3C2×D416yes—
95C22×C3×D3—212no—
96C22×C3×D3—48no—
97C2×D32—216no—
98C2×D32—448no—
99C32×D4—26no—
100C32×D4C2414yescomplemented
101C2×C32×C4C2×D413yes—
102C2×C32×C4C2413yes—
103#638—216no—
104C2×C3⋊Dic3—26no—
105C2×C3⋊Dic3—216no—
106C2×C3⋊Dic3C2×D416yes—
107C4×C3⋊D3C2×D4112yes—
108C4×C3⋊D3C2414yes—
109#634—26no—
110#634C2414yescomplemented
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—92no—
119C22×D8—32no—
120C22×M16—91no—
121C24×C3—22no—
122C24×C3—22no—
123C24×C3—62no—
124C23×D3—22no—
125C23×D3—32no—
126C23×D3—61no—
127C23×D3—624no—
128C23×D3—1216no—
129C2×C3×D4—26no—
130C2×C3×D4—26no—
131C2×C3×D4—26no—
132C2×C3×D4—312no—
133C2×C3×D4—66no—
134C2×C3×D4C22×D3112yescomplemented
135C22×C3×C4—21no—
136C22×C3×C4C22×D312yes—
137C2×#67—624no—
138C2×#67—624no—
139C2×#67—624no—
140C22×Dic3—32no—
141C22×Dic3—61no—
142C22×Dic3—624no—
143C22×Dic3—62no—
144C22×Dic3—624no—
145C22×Dic3—62no—
146#237—332no—
147#237—616no—
148#237—648no—
149#237—648no—
150D3×D4—332no—
151D3×D4—648no—
152C2×D12—26no—
153C2×D12C22×D3112yes—
154C2×C4×D3—312no—
155C2×C4×D3—312no—
156C2×C4×D3—66no—
157C2×C4×D3—66no—
158C2×Dic6—312no—
159C2×Dic6—66no—
160C2×Dic6—66no—
161C2×Dic6—66no—
162C3×D8—332no—
163C2×C3×C8—312no—
164#214—332no—
165#213C22×D3132yescomplemented
166C2×C3⋊C8C22×D3112yescomplemented
167#204—332no—
168ΓL(1,𝔽25)—332no—
169C22×C32—24no—
170C22×C32—412no—
171C22×C32AGL(1,ℤ/8)14yescomplemented
172C22×C32C22×D413yes—
173C2×C3⋊D3—212no—
174C2×C3⋊D3C22×D414yes—
175C2×C3×D3—28no—
176C2×C3×D3—424no—
177D32—432no—
178C32×C4C22×D413yes—
179C32×C4C2511yescharacteristic
180C3⋊Dic3—24no—
181C3⋊Dic3C22×D414yes—
182C25—181no—
183C2×#41—912no—
184C2×#41—186no—
185C22×Q8—91no—
186C22×D4—32no—
187C22×D4—912no—
188C22×D4—186no—
189C22×D4—181no—
190C22×D4—181no—
191C22×D4D3211yescharacteristic, complemented
192C23×C4—91no—
193C23×C4—181no—
194AGL(1,ℤ/8)—964no—
195C2×SD16—924no—
196C2×D8—324no—
197C2×M16—912no—
198C22×C8—32no—
199C23×C3—212no—
200C23×C3—212no—
201C23×C3—21no—
202C23×C3—48no—
203C23×C3—48no—
204C23×C3—612no—
205C23×C3—128no—
206C23×C3D3×D412yes—
207C22×D3—212no—
208C22×D3—312no—
209C22×D3—48no—
210C22×D3—66no—
211C22×D3—632no—
212C22×D3—68no—
213C22×D3—124no—
214C22×D3—1296no—
215C3×D4—212no—
216C3×D4—212no—
217C3×D4—24no—
218C3×D4—38no—
219C3×D4—612no—
220C3×D4C23×D318yescomplemented
221C2×C3×C4—23no—
222C2×C3×C4—23no—
223C2×C3×C4D3×D416yes—
224C2×C3×C4C23×D316yes—
225#67—632no—
226#67—632no—
227#67—632no—
228C2×Dic3—312no—
229C2×Dic3—66no—
230C2×Dic3—632no—
231C2×Dic3—612no—
232C2×Dic3—632no—
233C2×Dic3—612no—
234D12—212no—
235D12C23×D318yes—
236C4×D3—324no—
237C4×D3—38no—
238C4×D3—612no—
239C4×D3—64no—
240Dic6—38no—
241Dic6—64no—
242Dic6—612no—
243Dic6—612no—
244C3×C8—38no—
245C3⋊C8C23×D318yescomplemented
246C2×C32—44no—
247C2×C32C2×AGL(1,ℤ/8)16yescomplemented
248C2×C32C23×D411yescharacteristic
249C3⋊D3—24no—
250C3×D3—48no—
251C24—21no—
252C24—62no—
253C24—91no—
254C24—1812no—
255C24—368no—
256#41—916no—
257#41—1824no—
258C2×Q8—96no—
259C2×Q8—183no—
260C2×D4—23no—
261C2×D4—312no—
262C2×D4—66no—
263C2×D4—916no—
264C2×D4—1824no—
265C2×D4—1812no—
266C2×D4—1812no—
267C2×D4C2×D3216yescomplemented
268C22×C4—91no—
269C22×C4—96no—
270C22×C4—96no—
271C22×C4—1812no—
272C22×C4—181no—
273C22×C4C2×D3211yescharacteristic
274SD16—932no—
275D8—332no—
276M16—916no—
277C2×C8—312no—
278C22×C3—28no—
279C22×C3—28no—
280C22×C3—23no—
281C22×C3—24no—
282C22×C3—424no—
283C22×C3—424no—
284C22×C3—68no—
285C22×C3—1224no—
286C22×C3#91118yescomplemented
287C22×C3C2×D3×D416yes—
288C2×D3—28no—
289C2×D3—38no—
290C2×D3—424no—
291C2×D3—64no—
292C2×D3—624no—
293C2×D3—1212no—
294C2×D3—1264no—
295C3×C4—23no—
296C3×C4—21no—
297C3×C4C2×D3×D416yes—
298C3×C4C24×D312yes—
299Dic3—38no—
300Dic3—64no—
301Dic3—68no—
302Dic3—68no—
303C32C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
304C23—26no—
305C23—44no—
306C23—612no—
307C23—96no—
308C23—128no—
309C23—1816no—
310C23—184no—
311C23—3648no—
312C23#381111yescharacteristic
313Q8—94no—
314Q8—186no—
315D4—26no—
316D4—38no—
317D4—612no—
318D4—1816no—
319D4—1816no—
320D4C22×D3214yescomplemented
321C2×C4—96no—
322C2×C4—912no—
323C2×C4—94no—
324C2×C4—1816no—
325C2×C4—186no—
326C2×C4#381113yes—
327C2×C4C22×D3213yes—
328C8—38no—
329C2×C3—26no—
330C2×C3—21no—
331C2×C3—48no—
332C2×C3—48no—
333C2×C3—128no—
334C2×C3C2×#911112yescomplemented
335C2×C3C22×D3×D412yes—
336D3—48no—
337D3—68no—
338D3—124no—
339C22—24no—
340C22—412no—
341C22—68no—
342C22—94no—
343C22—1224no—
344C22—1812no—
345C22—3632no—
346C22#6392714yesdirect product factor
347C22C2×#381113yes—
348C4—94no—
349C4—184no—
350C4C2×#381113yes—
351C4C23×D3211yescharacteristic
352C3—21no—
353C3C22×#91112yescomplemented
354C2—44no—
355C2—128no—
356C2—184no—
357C2C2×#6392716yesdirect product factor
358C2C22×#381111yescharacteristic
subgroupG/H
centerC23#3811
commutator, cosocleC32×C4C25
FrattiniC4C23×D32
FittingC22×C32×D4C22
socleC23×C32C2×D4
hypercenterC22×D4D32
hypocenterC32C22×AGL(1,ℤ/8)
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×#3830C2semidirect
2. C22×#3802C2semidirect
3. C22×C3×#213C2semidirect
4. C2×#63927C2direct (N × H)
5. C22×#3716C2semidirect
6. C22×#3712C2semidirect
9. C22×C32×D4C22semidirect
13. C2×#3830C22semidirect
15. C22×#634C22semidirect
19. C2×#3802C22semidirect
20. C2×C3×#213C22semidirect
21. C22×C3×C3⋊C8C22semidirect
22. #63927C22direct (N × H)
23. C2×#3716C22semidirect
24. C2×#3712C22semidirect
30. C22×#213D3semidirect
40. C2×C32×D4C23semidirect
47. #3830C23semidirect
51. C2×#634C23semidirect
56. #3802C23semidirect
57. C3×#213C23semidirect
58. C2×C3×C3⋊C8C23semidirect
59. #3716C23semidirect
60. #3712C23semidirect
65. C22×C3×D4C2×D3semidirect
85. C2×#213C2×D3semidirect
86. C22×C3⋊C8C2×D3semidirect
100. C32×D4C24semidirect
110. #634C24semidirect
114. C3×C3⋊C8C24semidirect
134. C2×C3×D4C22×D3semidirect
165. #213C22×D3semidirect
166. C2×C3⋊C8C22×D3semidirect
171. C22×C32AGL(1,ℤ/8)semidirect
191. C22×D4D32semidirect
220. C3×D4C23×D3semidirect
245. C3⋊C8C23×D3semidirect
247. C2×C32C2×AGL(1,ℤ/8)semidirect
267. C2×D4C2×D32semidirect
286. C22×C3#911semidirect
303. C32C22×AGL(1,ℤ/8)semidirect
320. D4C22×D32semidirect
334. C2×C3C2×#911semidirect
346. C22#63927direct (N × H)
353. C3C22×#911semidirect
357. C2C2×#63927direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#38302
C22×#38022
C22×C3×#2132
C2×#639272
C22×#37162
C22×#37122
C22×#9113