Finite Group Databasebeta

← C22×#63857 #11900340

Subgroups — C22×#63857 #11900340

Order 1152.

4105 conjugacy classes of subgroups.

The subgroup lattice has 345 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#634C211yesmaximal, characteristic, complemented
2C23×C3×C4×D3C211yesmaximal, characteristic
3C22×C3×C4⋊Dic3C211yesmaximal, characteristic, complemented
4C22×Dic3⋊Dic3C212yesmaximal, complemented
5C22×(C2×D3)⋊Dic3C212yesmaximal
6C2×#63857C2124yesmaximal, direct product factor
7C22×#924—31nomaximal
8C22×#897—31nomaximal
9C24×C3×D3C2211yescharacteristic
10C23×C32×C4C2211yescharacteristic
11C23×C3⋊Dic3C2212yes—
12C22×#634—28no—
13C22×#634C22112yescomplemented
14C23×C3×Dic3C2212yes—
15C23×C3×Dic3C2211yescharacteristic
16C22×C3×C4×D3—28no—
17C22×C3×C4×D3C22112yes—
18C2×C3×C4⋊Dic3C22112yescomplemented
19C2×Dic3⋊Dic3C22124yescomplemented
20C2×(C2×D3)⋊Dic3C22124yes—
21#63857C22164yesdirect product factor
22C24×C3×C4—31no—
23C23×C4×D3D311yescharacteristic
24C23×Dic6—31no—
25C23×Dic6—31no—
26C23×Dic6—61no—
27C22×C3×C4⋊C4—31no—
28C22×#217—32no—
29C22×(C2×D3)⋊C4—32no—
30C22×C4⋊Dic3—32no—
31C22×C4⋊Dic3D311yescharacteristic, complemented
32C2×#924—324no—
33C22×Dic3⋊C4—32no—
34C2×#897—324no—
35C24×C32C2311yescharacteristic
36C23×C3×D3—24no—
37C23×C3×D3—24no—
38C23×C3×D3Q818yescomplemented
39C23×C3×D3C2316yes—
40C22×C32×C4D418yes—
41C22×C32×C4C2316yes—
42C22×C3⋊Dic3—28no—
43C22×C3⋊Dic3C23112yes—
44C2×#634—248no—
45C2×#634C23116yescomplemented
46C22×C3×Dic3—24no—
47C22×C3×Dic3—28no—
48C22×C3×Dic3C23112yes—
49C22×C3×Dic3C2316yes—
50C2×C3×C4×D3—248no—
51C2×C3×C4×D3C23116yes—
52C3×C4⋊Dic3C23116yescomplemented
53Dic3⋊Dic3C23132yescomplemented
54(C2×D3)⋊Dic3C23132yes—
55C22×C22⋊Q8—91noSylow
56C25×C3—31no—
57C24×D3C2×D311yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4—31no—
60C23×C3×C4—32no—
61C23×C3×C4—312no—
62C23×C3×C4—68no—
63C23×C3×C4C2×D311yescharacteristic
64C23×C3×C4C2×D311yescharacteristic
65C23×Dic3—32no—
66C23×Dic3—32no—
67C23×Dic3—62no—
68C23×Dic3C2×D311yescharacteristic
69C23×Dic3C2×D312yes—
70C22×C4×D3—28no—
71C22×C4×D3C2×D3112yes—
72C22×Dic6—312no—
73C22×Dic6—312no—
74C22×Dic6—612no—
75C22×Dic6—616no—
76C22×Dic6—68no—
77C22×Dic6—68no—
78C2×C3×C4⋊C4—312no—
79C2×#217—324no—
80C2×(C2×D3)⋊C4—324no—
81C2×C4⋊Dic3—324no—
82C2×C4⋊Dic3C2×D3112yescomplemented
83#924—364no—
84C2×Dic3⋊C4—324no—
85#897—364no—
86C23×C32C2×D414yes—
87C23×C32C2×Q814yes—
88C23×C32#4114yes—
89C23×C32C2413yes—
90C22×C3×D3—232no—
91C22×C3×D3—212no—
92C22×C3×D3—212no—
93C22×C3×D3C2×Q8124yescomplemented
94C22×C3×D3C2414yes—
95C2×C32×C4C2×D4124yes—
96C2×C32×C4C2414yes—
97C2×C3⋊Dic3—224no—
98C2×C3⋊Dic3C2418yes—
99#634—232no—
100C2×C3×Dic3—212no—
101C2×C3×Dic3—224no—
102C2×C3×Dic3C2418yes—
103C2×C3×Dic3C2414yes—
104C3×C4×D3—232no—
105C23×Q8—91no—
106C24×C4—31no—
107C2×C22⋊Q8—924no—
108C22×C4⋊C4—31no—
109C22×C4⋊C4—92no—
110C22×C22⋊C4—92no—
111C24×C3—21no—
112C24×C3—38no—
113C24×C3—36no—
114C24×C3—64no—
115C24×C3—64no—
116C24×C3C22×D311yescharacteristic
117C24×C3C22×D311yescharacteristic
118C23×D3—24no—
119C23×D3—24no—
120C23×D3Dic618yescomplemented
121C23×D3C22×D316yes—
122C22×C3×C4—28no—
123C22×C3×C4—26no—
124C22×C3×C4—36no—
125C22×C3×C4—312no—
126C22×C3×C4—316no—
127C22×C3×C4—64no—
128C22×C3×C4—68no—
129C22×C3×C4—648no—
130C22×C3×C4D1218yes—
131C22×C3×C4#6718yes—
132C22×C3×C4C22×D316yes—
133C22×C3×C4C22×D316yes—
134C22×Dic3—28no—
135C22×Dic3—24no—
136C22×Dic3—312no—
137C22×Dic3—312no—
138C22×Dic3—612no—
139C22×Dic3—616no—
140C22×Dic3—68no—
141C22×Dic3—68no—
142C22×Dic3C22×D3112yes—
143C22×Dic3C22×D316yes—
144C2×C4×D3—248no—
145C2×C4×D3C22×D3116yes—
146C2×Dic6—316no—
147C2×Dic6—316no—
148C2×Dic6—616no—
149C2×Dic6—696no—
150C2×Dic6—648no—
151C2×Dic6—648no—
152C3×C4⋊C4—316no—
153#217—332no—
154(C2×D3)⋊C4—332no—
155C4⋊Dic3—332no—
156C4⋊Dic3C22×D3116yescomplemented
157Dic3⋊C4—332no—
158C22×C32C22⋊Q8116yescomplemented
159C22×C32C22×D416yes—
160C22×C32C22×Q816yes—
161C22×C32C2×#4116yes—
162C22×C32C2511yescharacteristic
163C2×C3×D3—248no—
164C2×C3×D3—24no—
165C2×C3×D3—24no—
166C2×C3×D3C22×Q818yescomplemented
167C32×C4C22×D418yes—
168C3⋊Dic3—28no—
169C3×Dic3—24no—
170C3×Dic3—28no—
171C25—31no—
172C22×Q8—912no—
173C22×Q8—188no—
174C23×C4—32no—
175C23×C4—31no—
176C23×C4—312no—
177C23×C4—68no—
178C23×C4—92no—
179C23×C4D3211yescharacteristic
180C22⋊Q8—964no—
181C2×C4⋊C4—312no—
182C2×C4⋊C4—924no—
183C2×C22⋊C4—924no—
184C23×C3—23no—
185C23×C3—24no—
186C23×C3—24no—
187C23×C3—24no—
188C23×C3—324no—
189C23×C3—34no—
190C23×C3—632no—
191C23×C3—612no—
192C23×C3—612no—
193C23×C3C2×Dic614yes—
194C23×C3C2×D1214yes—
195C23×C3#23514yes—
196C23×C3#23714yes—
197C23×C3D3×Q814yes—
198C23×C3C2×#6714yes—
199C23×C3C23×D313yes—
200C23×C3C23×D313yes—
201C22×D3—232no—
202C22×D3—212no—
203C22×D3—212no—
204C22×D3C2×Dic6124yescomplemented
205C22×D3C23×D314yes—
206C2×C3×C4—24no—
207C2×C3×C4—224no—
208C2×C3×C4—34no—
209C2×C3×C4—38no—
210C2×C3×C4—612no—
211C2×C3×C4—624no—
212C2×C3×C4—632no—
213C2×C3×C4C2×D12124yes—
214C2×C3×C4C2×#67124yes—
215C2×C3×C4C23×D314yes—
216C2×C3×C4C23×D314yes—
217C2×Dic3—224no—
218C2×Dic3—212no—
219C2×Dic3—38no—
220C2×Dic3—38no—
221C2×Dic3—68no—
222C2×Dic3—648no—
223C2×Dic3—624no—
224C2×Dic3—624no—
225C2×Dic3C23×D318yes—
226C2×Dic3C23×D314yes—
227C4×D3—232no—
228Dic6—664no—
229Dic6—632no—
230Dic6—632no—
231C2×C32C2×C22⋊Q8112yescomplemented
232C2×C32C23×D411yescharacteristic
233C2×C32C23×Q811yescharacteristic
234C2×C32C22×#4111yescharacteristic
235C3×D3—28no—
236C24—36no—
237C24—38no—
238C24—64no—
239C24—64no—
240C24C2×D3211yescharacteristic
241C2×Q8—916no—
242C2×Q8—1848no—
243C22×C4—312no—
244C22×C4—36no—
245C22×C4—316no—
246C22×C4—68no—
247C22×C4—64no—
248C22×C4—648no—
249C22×C4—912no—
250C22×C4—188no—
251C22×C4#62618yes—
252C22×C4C2×D3216yes—
253C4⋊C4—316no—
254C4⋊C4—932no—
255C22⋊C4—932no—
256C22×C3—26no—
257C22×C3—26no—
258C22×C3—26no—
259C22×C3—216no—
260C22×C3—21no—
261C22×C3—38no—
262C22×C3—64no—
263C22×C3—64no—
264C22×C3—648no—
265C22×C3#897116yescomplemented
266C22×C3#924116yescomplemented
267C22×C3C22×Dic616yes—
268C22×C3C22×D1216yes—
269C22×C3C2×#23516yes—
270C22×C3C2×#23716yes—
271C22×C3C2×D3×Q816yes—
272C22×C3C22×#6716yes—
273C22×C3C24×D311yescharacteristic
274C22×C3C24×D311yescharacteristic
275C2×D3—24no—
276C2×D3—24no—
277C2×D3—248no—
278C2×D3C22×Dic618yescomplemented
279C3×C4—28no—
280C3×C4—64no—
281C3×C4—68no—
282C3×C4C22×D1218yes—
283C3×C4C22×#6718yes—
284Dic3—28no—
285Dic3—24no—
286Dic3—616no—
287Dic3—68no—
288Dic3—68no—
289C32C22×C22⋊Q811yescharacteristic, complemented, Sylow
290C23—324no—
291C23—34no—
292C23—612no—
293C23—612no—
294C23—632no—
295C23D3×Dic614yes—
296C23#379514yes—
297C23C2×#62614yes—
298C23C22×D3213yes—
299Q8—1832no—
300C2×C4—38no—
301C2×C4—34no—
302C2×C4—624no—
303C2×C4—612no—
304C2×C4—632no—
305C2×C4—98no—
306C2×C4—1824no—
307C2×C4C2×#626124yes—
308C2×C4C22×D3214yes—
309C2×C3—212no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—68no—
314C2×C3C2×#897112yescomplemented
315C2×C3C2×#924112yescomplemented
316C2×C3C23×Dic611yescharacteristic
317C2×C3C23×D1211yescharacteristic
318C2×C3C22×#23511yescharacteristic
319C2×C3C22×#23711yescharacteristic
320C2×C3C22×D3×Q811yescharacteristic
321C2×C3C23×#6711yescharacteristic
322D3—28no—
323C22—38no—
324C22—648no—
325C22—64no—
326C22—64no—
327C22#63857116yesdirect product factor
328C22C2×D3×Dic616yes—
329C22C2×#379516yes—
330C22C22×#62616yes—
331C22C23×D3211yescharacteristic
332C4—68no—
333C4—64no—
334C4—188no—
335C4C22×#62618yes—
336C3—21no—
337C3C22×#89711yescharacteristic, complemented
338C3C22×#92411yescharacteristic, complemented
339C2—68no—
340C2C2×#63857112yesdirect product factor
341C2C22×D3×Dic611yescharacteristic
342C2C22×#379511yescharacteristic
343C2C23×#62611yescharacteristic
subgroupG/H
centerC24C2×D32
commutator, cosocleC22×C32C25
FrattiniC22C23×D32
FittingC23×C32×C4C22
socleC24×C32C23
prime radicalC24×C3×D3C22
hypercenterC23×C4D32
hypocenterC32C22×C22⋊Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C24 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C23×#634C2semidirect
3. C22×C3×C4⋊Dic3C2semidirect
4. C22×Dic3⋊Dic3C2semidirect
6. C2×#63857C2direct (N × H)
13. C22×#634C22semidirect
18. C2×C3×C4⋊Dic3C22semidirect
19. C2×Dic3⋊Dic3C22semidirect
21. #63857C22direct (N × H)
31. C22×C4⋊Dic3D3semidirect
38. C23×C3×D3Q8semidirect
45. C2×#634C23semidirect
52. C3×C4⋊Dic3C23semidirect
53. Dic3⋊Dic3C23semidirect
82. C2×C4⋊Dic3C2×D3semidirect
93. C22×C3×D3C2×Q8semidirect
120. C23×D3Dic6semidirect
156. C4⋊Dic3C22×D3semidirect
158. C22×C32C22⋊Q8semidirect
166. C2×C3×D3C22×Q8semidirect
204. C22×D3C2×Dic6semidirect
231. C2×C32C2×C22⋊Q8semidirect
265. C22×C3#897semidirect
266. C22×C3#924semidirect
278. C2×D3C22×Dic6semidirect
289. C32C22×C22⋊Q8semidirect
314. C2×C3C2×#897semidirect
315. C2×C3C2×#924semidirect
327. C22#63857direct (N × H)
337. C3C22×#897semidirect
338. C3C22×#924semidirect
340. C2C2×#63857direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#6342
C23×C3×C4×D32
C22×C3×C4⋊Dic32
C22×Dic3⋊Dic32
C22×(C2×D3)⋊Dic32
C2×#638572
C22×#9243
C22×#8973