Finite Group Databasebeta

← C22×#63980 #11900300

Subgroups — C22×#63980 #11900300

Order 1152.

5737 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C23×C3×#67C212yesmaximal, complemented
3C23×#625C211yesmaximal, characteristic, complemented
4C2×#63980C2124yesmaximal, direct product factor
5C22×#3724C211yesmaximal, characteristic, complemented
6C22×(C2×D3)⋊Dic3C212yesmaximal, complemented
7C22×#939—32nomaximal
8C25×C32C2211yescharacteristic
9C24×C3×D3C2212yes—
10C23×C3⋊Dic3—28no—
11C23×C3⋊Dic3C2211yescharacteristic
12C23×C3⋊Dic3C2211yescharacteristic
13C23×C3⋊Dic3C22112yescomplemented
14C22×C3×#67—216no—
15C22×C3×#67C22124yescomplemented
16C23×C3×Dic3C2212yes—
17C22×#625—28no—
18C22×#625C22112yescomplemented
19#63980C22164yesdirect product factor
20C2×#3724C22112yescomplemented
21C2×(C2×D3)⋊Dic3C22124yescomplemented
22C23×C3×D4—32no—
23C23×#67—32no—
24C23×#67D312yescomplemented
25C24×Dic3—32no—
26C24×Dic3—61no—
27C22×#217—32no—
28C2×#939—348no—
29C22×(C2×D3)⋊C4—32no—
30C22×Dic3⋊C4—32no—
31C24×C32—24no—
32C24×C32—24no—
33C24×C32D418yescomplemented
34C24×C32C2316yes—
35C24×C32C2311yescharacteristic
36C23×C3×D3—28no—
37C23×C3×D3—28no—
38C23×C3×D3—28no—
39C23×C3×D3C23112yes—
40C22×C3⋊Dic3—24no—
41C22×C3⋊Dic3—248no—
42C22×C3⋊Dic3D418yescomplemented
43C22×C3⋊Dic3C2316yes—
44C22×C3⋊Dic3C2316yes—
45C22×C3⋊Dic3C23116yescomplemented
46C2×C3×#67—296no—
47C2×C3×#67C23132yescomplemented
48C22×C3×Dic3—28no—
49C22×C3×Dic3C23112yes—
50C2×#625—248no—
51C2×#625C23116yescomplemented
52#3724C23116yescomplemented
53(C2×D3)⋊Dic3C23132yescomplemented
54C22×#121—91noSylow
55C25×C3—21no—
56C25×C3—32no—
57C25×C3C2×D312yes—
58C24×D3C2×D312yes—
59C22×C3×D4—324no—
60C22×C3×D4—616no—
61C23×C3×C4—32no—
62C22×#67—216no—
63C22×#67—324no—
64C22×#67—616no—
65C22×#67C2×D3124yescomplemented
66C23×Dic3—32no—
67C23×Dic3—32no—
68C23×Dic3—324no—
69C23×Dic3—61no—
70C23×Dic3—61no—
71C23×Dic3—612no—
72C23×Dic3—616no—
73C23×Dic3—616no—
74C23×Dic3C2×D312yes—
75C2×#217—324no—
76#939—3128no—
77C2×(C2×D3)⋊C4—324no—
78C2×Dic3⋊C4—324no—
79C23×C32—232no—
80C23×C32—212no—
81C23×C32—212no—
82C23×C32C2×D414yes—
83C23×C32C2×D414yes—
84C23×C32C2×D4124yescomplemented
85C23×C32#4114yes—
86C23×C32C2414yes—
87C23×C32C2413yes—
88C22×C3×D3—224no—
89C22×C3×D3—224no—
90C22×C3×D3—224no—
91C22×C3×D3—432no—
92C22×C3×D3C2418yes—
93C2×C3⋊Dic3—212no—
94C2×C3⋊Dic3—232no—
95C2×C3⋊Dic3C2×D4124yescomplemented
96C2×C3⋊Dic3C2414yes—
97C2×C3⋊Dic3C2414yes—
98C3×#67—264no—
99C2×C3×Dic3—224no—
100C2×C3×Dic3C2418yes—
101#625—232no—
102C23×D4—32no—
103C23×D4—91no—
104C24×C4—91no—
105C2×#121—924no—
106C22×C4⋊C4—91no—
107C22×C22⋊C4—92no—
108C24×C3—28no—
109C24×C3—28no—
110C24×C3—28no—
111C24×C3—28no—
112C24×C3—28no—
113C24×C3—26no—
114C24×C3—21no—
115C24×C3—312no—
116C24×C3—68no—
117C24×C3—68no—
118C24×C3—68no—
119C24×C3#67116yescomplemented
120C24×C3C22×D3112yes—
121C24×C3C22×D312yes—
122C23×D3—28no—
123C23×D3—28no—
124C23×D3—28no—
125C23×D3C22×D3112yes—
126C2×C3×D4—332no—
127C2×C3×D4—696no—
128C22×C3×C4—312no—
129C22×C3×C4—68no—
130C2×#67—296no—
131C2×#67—332no—
132C2×#67—696no—
133C2×#67C22×D3132yescomplemented
134C22×Dic3—28no—
135C22×Dic3—316no—
136C22×Dic3—312no—
137C22×Dic3—312no—
138C22×Dic3—332no—
139C22×Dic3—68no—
140C22×Dic3—66no—
141C22×Dic3—66no—
142C22×Dic3—616no—
143C22×Dic3—696no—
144C22×Dic3—696no—
145C22×Dic3—68no—
146C22×Dic3—68no—
147C22×Dic3C22×D3112yes—
148#217—332no—
149(C2×D3)⋊C4—332no—
150Dic3⋊C4—332no—
151C22×C32—248no—
152C22×C32—24no—
153C22×C32—24no—
154C22×C32#121116yescomplemented
155C22×C32C22×D416yes—
156C22×C32C22×D416yes—
157C22×C32C22×D418yescomplemented
158C22×C32C2×#4116yes—
159C22×C32C2511yescharacteristic
160C2×C3×D3—28no—
161C2×C3×D3—28no—
162C2×C3×D3—28no—
163C2×C3×D3—448no—
164C3⋊Dic3—24no—
165C3⋊Dic3C22×D418yescomplemented
166C3×Dic3—28no—
167C25—32no—
168C25D3211yescharacteristic
169C22×D4—324no—
170C22×D4—616no—
171C22×D4—912no—
172C22×D4—188no—
173C23×C4—32no—
174C23×C4—91no—
175C23×C4—912no—
176C23×C4—91no—
177C23×C4—188no—
178#121—964no—
179C2×C4⋊C4—912no—
180C2×C22⋊C4—924no—
181C23×C3—224no—
182C23×C3—224no—
183C23×C3—264no—
184C23×C3—264no—
185C23×C3—224no—
186C23×C3—224no—
187C23×C3—24no—
188C23×C3—23no—
189C23×C3—24no—
190C23×C3—24no—
191C23×C3—224no—
192C23×C3—24no—
193C23×C3—38no—
194C23×C3—624no—
195C23×C3—624no—
196C23×C3—624no—
197C23×C3—1232no—
198C23×C3D3×D418yes—
199C23×C3#23718yes—
200C23×C3C2×#6718yes—
201C23×C3C2×#67148yescomplemented
202C23×C3C23×D318yes—
203C23×C3C23×D316yes—
204C22×D3—224no—
205C22×D3—224no—
206C22×D3—224no—
207C22×D3—432no—
208C22×D3C23×D318yes—
209C3×D4—664no—
210C2×C3×C4—38no—
211C2×C3×C4—624no—
212#67—264no—
213#67—664no—
214C2×Dic3—224no—
215C2×Dic3—348no—
216C2×Dic3—38no—
217C2×Dic3—38no—
218C2×Dic3—624no—
219C2×Dic3—64no—
220C2×Dic3—64no—
221C2×Dic3—664no—
222C2×Dic3—664no—
223C2×Dic3—624no—
224C2×Dic3—624no—
225C2×Dic3C23×D318yes—
226C2×C32—28no—
227C2×C32C2×#121112yescomplemented
228C2×C32C23×D411yescharacteristic
229C2×C32C23×D411yescharacteristic
230C2×C32C22×#4111yescharacteristic
231C3×D3—48no—
232C24—24no—
233C24—24no—
234C24—312no—
235C24—68no—
236C24—68no—
237C24—68no—
238C24#62518yescomplemented
239C24C2×D3216yes—
240C24C2×D3211yescharacteristic
241C2×D4—332no—
242C2×D4—696no—
243C2×D4—916no—
244C2×D4—1848no—
245C22×C4—312no—
246C22×C4—68no—
247C22×C4—96no—
248C22×C4—98no—
249C22×C4—96no—
250C22×C4—916no—
251C22×C4—1848no—
252C22×C4—184no—
253C4⋊C4—916no—
254C22⋊C4—932no—
255C22×C3—28no—
256C22×C3—28no—
257C22×C3—296no—
258C22×C3—28no—
259C22×C3—28no—
260C22×C3—296no—
261C22×C3—26no—
262C22×C3—26no—
263C22×C3—28no—
264C22×C3—26no—
265C22×C3—21no—
266C22×C3—216no—
267C22×C3—68no—
268C22×C3—68no—
269C22×C3—68no—
270C22×C3—1248no—
271C22×C3#939132yescomplemented
272C22×C3C2×D3×D4112yes—
273C22×C3C2×#237112yes—
274C22×C3C22×#67112yes—
275C22×C3C22×#67116yescomplemented
276C22×C3C24×D312yes—
277C2×D3—28no—
278C2×D3—28no—
279C2×D3—28no—
280C2×D3—448no—
281C3×C4—68no—
282Dic3—28no—
283Dic3—316no—
284Dic3—68no—
285Dic3—68no—
286Dic3—68no—
287C32C22×#12111yescharacteristic, complemented, Sylow
288C23—212no—
289C23—212no—
290C23—232no—
291C23—38no—
292C23—624no—
293C23—624no—
294C23—624no—
295C23—1232no—
296C23#380514yes—
297C23C2×#62514yes—
298C23C2×#625124yescomplemented
299C23#381114yes—
300C23C22×D3214yes—
301C23C22×D3213yes—
302D4—664no—
303D4—1832no—
304C2×C4—38no—
305C2×C4—624no—
306C2×C4—924no—
307C2×C4—94no—
308C2×C4—94no—
309C2×C4—1832no—
310C2×C4—1812no—
311C2×C3—216no—
312C2×C3—216no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—212no—
317C2×C3—128no—
318C2×C3C2×#939124yescomplemented
319C2×C3C22×D3×D412yes—
320C2×C3C22×#23712yes—
321C2×C3C23×#6712yes—
322D3—48no—
323C22—24no—
324C22—24no—
325C22—248no—
326C22—68no—
327C22—68no—
328C22—68no—
329C22—1248no—
330C22#63980116yesdirect product factor
331C22C2×#380516yes—
332C22C22×#62516yes—
333C22C22×#62518yescomplemented
334C22C2×#381116yes—
335C22C23×D3211yescharacteristic
336C4—68no—
337C4—98no—
338C4—184no—
339C3—21no—
340C3C22×#93912yescomplemented
341C2—28no—
342C2—128no—
343C2C2×#63980112yesdirect product factor
344C2C22×#380511yescharacteristic
345C2C23×#62511yescharacteristic
346C2C22×#381111yescharacteristic
subgroupG/H
centerC24C2×D32
commutator, cosocleC22×C32C25
FrattiniC22C23×D32
FittingC25×C32C22
socleC24×C32C23
hypercenterC25D32
hypocenterC32C22×#121
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C24 ◁ C25

Complements

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

NH ≅ G/Ntype
1. C24×C3⋊Dic3C2semidirect
2. C23×C3×#67C2semidirect
3. C23×#625C2semidirect
4. C2×#63980C2direct (N × H)
5. C22×#3724C2semidirect
6. C22×(C2×D3)⋊Dic3C2semidirect
13. C23×C3⋊Dic3C22semidirect
15. C22×C3×#67C22semidirect
18. C22×#625C22semidirect
19. #63980C22direct (N × H)
20. C2×#3724C22semidirect
21. C2×(C2×D3)⋊Dic3C22semidirect
24. C23×#67D3semidirect
33. C24×C32D4semidirect
42. C22×C3⋊Dic3D4semidirect
45. C22×C3⋊Dic3C23semidirect
47. C2×C3×#67C23semidirect
51. C2×#625C23semidirect
52. #3724C23semidirect
53. (C2×D3)⋊Dic3C23semidirect
65. C22×#67C2×D3semidirect
84. C23×C32C2×D4semidirect
95. C2×C3⋊Dic3C2×D4semidirect
119. C24×C3#67semidirect
133. C2×#67C22×D3semidirect
154. C22×C32#121semidirect
157. C22×C32C22×D4semidirect
165. C3⋊Dic3C22×D4semidirect
201. C23×C3C2×#67semidirect
227. C2×C32C2×#121semidirect
238. C24#625semidirect
271. C22×C3#939semidirect
275. C22×C3C22×#67semidirect
287. C32C22×#121semidirect
298. C23C2×#625semidirect
318. C2×C3C2×#939semidirect
330. C22#63980direct (N × H)
333. C22C22×#625semidirect
340. C3C22×#939semidirect
343. C2C2×#63980direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3⋊Dic32
C23×C3×#672
C23×#6252
C2×#639802
C22×#37242
C22×(C2×D3)⋊Dic32
C22×#9393