Finite Group Databasebeta

← C22×#63912 #11900305

Subgroups — C22×#63912 #11900305

Order 1152.

5737 conjugacy classes of subgroups; only the 852 normal subgroups are shown.

normal subgroups only

This group has 5737 conjugacy classes of subgroups — too many to draw. Only its 852 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
1C23×C4×C3⋊D3C211yesmaximal, characteristic, complemented
2C23×#626C212yesmaximal, complemented
3C23×#625C211yesmaximal, characteristic, complemented
4C22×C3×(C2×D3)⋊C4C212yesmaximal, complemented
5C22×#3724C211yesmaximal, characteristic, complemented
6C2×#63912C2124yesmaximal, direct product factor
7C22×#884—32nomaximal
8C24×C3⋊D3C2211yescharacteristic
9C24×C3×D3C2212yes—
10C23×C32×C4C2211yescharacteristic
11C23×C3⋊Dic3C2211yescharacteristic
12C22×C4×C3⋊D3—28no—
13C22×C4×C3⋊D3C22112yescomplemented
14C23×C3×Dic3C2212yes—
15C22×#626—216no—
16C22×#626C22124yescomplemented
17C22×#625—28no—
18C22×#625C22112yescomplemented
19C2×C3×(C2×D3)⋊C4C22124yescomplemented
20C2×#3724C22112yescomplemented
21#63912C22164yesdirect product factor
22C23×#67—32no—
23C23×#67—32no—
24C23×D12—32no—
25C23×C4×D3—32no—
26C23×C4×D3—61no—
27C22×C3×C22⋊C4—32no—
28C22×(C2×D3)⋊C4D312yescomplemented
29C22×Dic3⋊C4—32no—
30C2×#884—348no—
31C24×C32C2311yescharacteristic
32C23×C3⋊D3—24no—
33C23×C3⋊D3—24no—
34C23×C3⋊D3D418yescomplemented
35C23×C3⋊D3C2316yes—
36C23×C3×D3—28no—
37C23×C3×D3—28no—
38C23×C3×D3—28no—
39C23×C3×D3C23112yes—
40C22×C32×C4—24no—
41C22×C32×C4C2316yes—
42C22×C3⋊Dic3D418yescomplemented
43C22×C3⋊Dic3C2316yes—
44C2×C4×C3⋊D3—248no—
45C2×C4×C3⋊D3C23116yescomplemented
46C22×C3×Dic3—28no—
47C22×C3×Dic3C23112yes—
48C2×#626—296no—
49C2×#626C23132yescomplemented
50C2×#625—248no—
51C2×#625C23116yescomplemented
52C3×(C2×D3)⋊C4C23132yescomplemented
53#3724C23116yescomplemented
54C22×#121—91noSylow
55C25×C3—32no—
56C24×D3—32no—
57C24×D3—61no—
58C24×D3C2×D312yes—
59C23×C3×C4—21no—
60C23×C3×C4—32no—
61C23×C3×C4C2×D312yes—
62C22×#67—324no—
63C22×#67—324no—
64C22×#67—616no—
65C22×#67—616no—
66C23×Dic3—32no—
67C23×Dic3—61no—
68C23×Dic3C2×D312yes—
69C22×D12—324no—
70C22×D12—616no—
71C22×C4×D3—324no—
72C22×C4×D3—612no—
73C22×C4×D3—616no—
74C22×C4×D3—616no—
75C2×C3×C22⋊C4—324no—
76C2×(C2×D3)⋊C4C2×D3124yescomplemented
77C2×Dic3⋊C4—324no—
78#884—3128no—
79C23×C32C2×D414yes—
80C23×C32C2×D414yes—
81C23×C32#4114yes—
82C23×C32C2413yes—
83C22×C3⋊D3—232no—
84C22×C3⋊D3—212no—
85C22×C3⋊D3—212no—
86C22×C3⋊D3C2×D4124yescomplemented
87C22×C3⋊D3C2414yes—
88C22×C3×D3—224no—
89C22×C3×D3—224no—
90C22×C3×D3—224no—
91C22×C3×D3—432no—
92C22×C3×D3C2418yes—
93C2×C32×C4—212no—
94C2×C32×C4C2414yes—
95C2×C3⋊Dic3C2×D4124yescomplemented
96C2×C3⋊Dic3C2414yes—
97C4×C3⋊D3—232no—
98C2×C3×Dic3—224no—
99C2×C3×Dic3C2418yes—
100#626—264no—
101#625—232no—
102C23×D4—92no—
103C23×D4—91no—
104C24×C4—91no—
105C2×#121—924no—
106C22×C4⋊C4—91no—
107C22×C22⋊C4—32no—
108C24×C3—21no—
109C24×C3—312no—
110C24×C3—68no—
111C24×C3—68no—
112C24×C3—68no—
113C24×C3C22×D312yes—
114C23×D3—28no—
115C23×D3—28no—
116C23×D3—28no—
117C23×D3—316no—
118C23×D3—312no—
119C23×D3—68no—
120C23×D3—66no—
121C23×D3—68no—
122C23×D3—68no—
123C23×D3—68no—
124C23×D3—68no—
125C23×D3C22×D3112yes—
126C22×C3×C4—28no—
127C22×C3×C4—28no—
128C22×C3×C4—26no—
129C22×C3×C4—312no—
130C22×C3×C4—68no—
131C22×C3×C4C22×D3112yes—
132C2×#67—332no—
133C2×#67—332no—
134C2×#67—696no—
135C2×#67—696no—
136C22×Dic3—28no—
137C22×Dic3—316no—
138C22×Dic3—312no—
139C22×Dic3—68no—
140C22×Dic3—66no—
141C22×Dic3C22×D3112yes—
142C2×D12—332no—
143C2×D12—696no—
144C2×C4×D3—332no—
145C2×C4×D3—616no—
146C2×C4×D3—696no—
147C2×C4×D3—696no—
148C3×C22⋊C4—332no—
149(C2×D3)⋊C4C22×D3132yescomplemented
150Dic3⋊C4—332no—
151C22×C32#121116yescomplemented
152C22×C32C22×D416yes—
153C22×C32C22×D416yes—
154C22×C32C2×#4116yes—
155C22×C32C2511yescharacteristic
156C2×C3⋊D3—248no—
157C2×C3⋊D3—24no—
158C2×C3⋊D3—24no—
159C2×C3⋊D3C22×D418yescomplemented
160C2×C3×D3—28no—
161C2×C3×D3—28no—
162C2×C3×D3—28no—
163C2×C3×D3—448no—
164C32×C4—24no—
165C3⋊Dic3C22×D418yescomplemented
166C3×Dic3—28no—
167C25—32no—
168C25—91no—
169C22×D4—924no—
170C22×D4—912no—
171C22×D4—1816no—
172C22×D4—188no—
173C23×C4—32no—
174C23×C4—91no—
175C23×C4—912no—
176C23×C4—188no—
177C23×C4D3211yescharacteristic
178#121—964no—
179C2×C4⋊C4—912no—
180C2×C22⋊C4—324no—
181C23×C3—23no—
182C23×C3—24no—
183C23×C3—24no—
184C23×C3—24no—
185C23×C3—38no—
186C23×C3—624no—
187C23×C3—624no—
188C23×C3—624no—
189C23×C3—1232no—
190C23×C3#23518yes—
191C23×C3D3×D418yes—
192C23×C3D3×D418yes—
193C23×C3C23×D316yes—
194C22×D3—224no—
195C22×D3—224no—
196C22×D3—224no—
197C22×D3—348no—
198C22×D3—38no—
199C22×D3—432no—
200C22×D3—624no—
201C22×D3—64no—
202C22×D3—624no—
203C22×D3—624no—
204C22×D3—664no—
205C22×D3—664no—
206C22×D3—624no—
207C22×D3—624no—
208C22×D3C23×D318yes—
209C2×C3×C4—224no—
210C2×C3×C4—224no—
211C2×C3×C4—24no—
212C2×C3×C4—38no—
213C2×C3×C4—624no—
214C2×C3×C4C23×D318yes—
215#67—664no—
216#67—664no—
217C2×Dic3—224no—
218C2×Dic3—348no—
219C2×Dic3—38no—
220C2×Dic3—624no—
221C2×Dic3—64no—
222C2×Dic3C23×D318yes—
223D12—664no—
224C4×D3—664no—
225C4×D3—664no—
226C2×C32C2×#121112yescomplemented
227C2×C32C23×D411yescharacteristic
228C2×C32C23×D411yescharacteristic
229C2×C32C22×#4111yescharacteristic
230C3⋊D3—28no—
231C3×D3—48no—
232C24—312no—
233C24—68no—
234C24—68no—
235C24—68no—
236C24—96no—
237C24—98no—
238C24—184no—
239C24—184no—
240C24C2×D3211yescharacteristic
241C2×D4—932no—
242C2×D4—916no—
243C2×D4—1896no—
244C2×D4—1848no—
245C22×C4—24no—
246C22×C4—312no—
247C22×C4—68no—
248C22×C4—96no—
249C22×C4—98no—
250C22×C4—916no—
251C22×C4—1848no—
252C22×C4C2×D3216yes—
253C4⋊C4—916no—
254C22⋊C4—332no—
255C22×C3—26no—
256C22×C3—26no—
257C22×C3—26no—
258C22×C3—21no—
259C22×C3—216no—
260C22×C3—68no—
261C22×C3—68no—
262C22×C3—68no—
263C22×C3—1248no—
264C22×C3#884132yescomplemented
265C22×C3C2×#235112yes—
266C22×C3C2×D3×D4112yes—
267C22×C3C2×D3×D4112yes—
268C22×C3C24×D312yes—
269C2×D3—28no—
270C2×D3—28no—
271C2×D3—28no—
272C2×D3—316no—
273C2×D3—448no—
274C2×D3—68no—
275C2×D3—68no—
276C2×D3—68no—
277C2×D3—696no—
278C2×D3—68no—
279C2×D3—68no—
280C2×D3—696no—
281C3×C4—28no—
282C3×C4—28no—
283C3×C4—68no—
284Dic3—28no—
285Dic3—316no—
286Dic3—68no—
287C32C22×#12111yescharacteristic, complemented, Sylow
288C23—38no—
289C23—624no—
290C23—624no—
291C23—624no—
292C23—924no—
293C23—94no—
294C23—1232no—
295C23—1812no—
296C23—1812no—
297C23—1832no—
298C23#379814yes—
299C23#380214yes—
300C23#381114yes—
301C23C22×D3213yes—
302D4—1864no—
303D4—1832no—
304C2×C4—212no—
305C2×C4—38no—
306C2×C4—624no—
307C2×C4—924no—
308C2×C4—94no—
309C2×C4—1832no—
310C2×C4C22×D3214yes—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—212no—
315C2×C3—128no—
316C2×C3C2×#884124yescomplemented
317C2×C3C22×#23512yes—
318C2×C3C22×D3×D412yes—
319C2×C3C22×D3×D412yes—
320D3—48no—
321D3—616no—
322D3—616no—
323C22—68no—
324C22—68no—
325C22—68no—
326C22—98no—
327C22—1248no—
328C22—184no—
329C22—184no—
330C22—1848no—
331C22#63912116yesdirect product factor
332C22C2×#379816yes—
333C22C2×#380216yes—
334C22C2×#381116yes—
335C22C23×D3211yescharacteristic
336C4—24no—
337C4—68no—
338C4—98no—
339C3—21no—
340C3C22×#88412yescomplemented
341C2—128no—
342C2—188no—
343C2C2×#63912112yesdirect product factor
344C2C22×#379811yescharacteristic
345C2C22×#380211yescharacteristic
346C2C22×#381111yescharacteristic
subgroupG/H
centerC24C2×D32
commutator, cosocleC22×C32C25
FrattiniC22C23×D32
FittingC23×C32×C4C22
socleC24×C32C23
hypercenterC23×C4D32
hypocenterC32C22×#121
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C24 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C23×C4×C3⋊D3C2semidirect
2. C23×#626C2semidirect
3. C23×#625C2semidirect
4. C22×C3×(C2×D3)⋊C4C2semidirect
5. C22×#3724C2semidirect
6. C2×#63912C2direct (N × H)
13. C22×C4×C3⋊D3C22semidirect
16. C22×#626C22semidirect
18. C22×#625C22semidirect
19. C2×C3×(C2×D3)⋊C4C22semidirect
20. C2×#3724C22semidirect
21. #63912C22direct (N × H)
28. C22×(C2×D3)⋊C4D3semidirect
34. C23×C3⋊D3D4semidirect
42. C22×C3⋊Dic3D4semidirect
45. C2×C4×C3⋊D3C23semidirect
49. C2×#626C23semidirect
51. C2×#625C23semidirect
52. C3×(C2×D3)⋊C4C23semidirect
53. #3724C23semidirect
76. C2×(C2×D3)⋊C4C2×D3semidirect
86. C22×C3⋊D3C2×D4semidirect
95. C2×C3⋊Dic3C2×D4semidirect
149. (C2×D3)⋊C4C22×D3semidirect
151. C22×C32#121semidirect
159. C2×C3⋊D3C22×D4semidirect
165. C3⋊Dic3C22×D4semidirect
226. C2×C32C2×#121semidirect
264. C22×C3#884semidirect
287. C32C22×#121semidirect
316. C2×C3C2×#884semidirect
331. C22#63912direct (N × H)
340. C3C22×#884semidirect
343. C2C2×#63912direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4×C3⋊D32
C23×#6262
C23×#6252
C22×C3×(C2×D3)⋊C42
C22×#37242
C2×#639122
C22×#8843