Finite Group Databasebeta

← C22×#63856 #11900342

Subgroups — C22×#63856 #11900342

Order 1152.

3889 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4×D3C211yesmaximal, characteristic
2C23×C3×Dic6C211yesmaximal, characteristic, complemented
3C22×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4C22×#3724C212yesmaximal, complemented
5C22×(C2×D3)⋊Dic3C212yesmaximal
6C2×#63856C2124yesmaximal, direct product factor
7C22×#946—31nomaximal
8C22×#924—31nomaximal
9C24×C3×D3C2211yescharacteristic
10C23×C32×C4C2211yescharacteristic
11C23×C3⋊Dic3C2212yes—
12C23×C3×Dic3C2212yes—
13C23×C3×Dic3C2211yescharacteristic
14C22×C3×C4×D3—28no—
15C22×C3×C4×D3C22112yes—
16C22×C3×Dic6—28no—
17C22×C3×Dic6C22112yescomplemented
18C2×C4⋊(C3⋊Dic3)C22112yescomplemented
19C2×#3724C22124yescomplemented
20C2×(C2×D3)⋊Dic3C22124yes—
21#63856C22164yesdirect product factor
22C23×C3×Q8—31no—
23C24×C3×C4—31no—
24C23×C4×D3D311yescharacteristic
25C23×Dic6D311yescharacteristic, complemented
26C22×#217—32no—
27C2×#946—324no—
28C22×(C2×D3)⋊C4—32no—
29C22×C4⋊Dic3—31no—
30C22×C4⋊Dic3—31no—
31C22×C4⋊Dic3—61no—
32C2×#924—324no—
33C22×Dic3⋊C4—32no—
34C22×Dic3⋊C4—32no—
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—
44C22×C3×Dic3—28no—
45C22×C3×Dic3—24no—
46C22×C3×Dic3C23112yes—
47C22×C3×Dic3C2316yes—
48C2×C3×C4×D3—248no—
49C2×C3×C4×D3C23116yes—
50C2×C3×Dic6—248no—
51C2×C3×Dic6C23116yescomplemented
52C4⋊(C3⋊Dic3)C23116yescomplemented
53#3724C23132yescomplemented
54(C2×D3)⋊Dic3C23132yes—
55C22×C22⋊Q8—91noSylow
56C25×C3—31no—
57C24×D3C2×D311yescharacteristic
58C22×C3×Q8—312no—
59C22×C3×Q8—68no—
60C23×C3×C4—21no—
61C23×C3×C4—31no—
62C23×C3×C4—32no—
63C23×C3×C4—312no—
64C23×C3×C4—68no—
65C23×C3×C4C2×D311yescharacteristic
66C23×C3×C4C2×D311yescharacteristic
67C23×Dic3—32no—
68C23×Dic3—32no—
69C23×Dic3—62no—
70C23×Dic3C2×D312yes—
71C23×Dic3C2×D311yescharacteristic
72C22×C4×D3—28no—
73C22×C4×D3C2×D3112yes—
74C22×Dic6—28no—
75C22×Dic6C2×D3112yescomplemented
76C2×#217—324no—
77#946—364no—
78C2×(C2×D3)⋊C4—324no—
79C2×C4⋊Dic3—312no—
80C2×C4⋊Dic3—312no—
81C2×C4⋊Dic3—612no—
82#924—364no—
83C2×Dic3⋊C4—324no—
84C2×Dic3⋊C4—324no—
85C23×C32C2×D414yes—
86C23×C32C2×Q814yes—
87C23×C32#4114yes—
88C23×C32C2413yes—
89C22×C3×D3—232no—
90C22×C3×D3—212no—
91C22×C3×D3—212no—
92C22×C3×D3C2×Q8124yescomplemented
93C22×C3×D3C2414yes—
94C2×C32×C4C2×D4124yes—
95C2×C32×C4C2414yes—
96C2×C3⋊Dic3—224no—
97C2×C3⋊Dic3C2418yes—
98C2×C3×Dic3—224no—
99C2×C3×Dic3—212no—
100C2×C3×Dic3C2418yes—
101C2×C3×Dic3C2414yes—
102C3×C4×D3—232no—
103C3×Dic6—232no—
104C23×Q8—31no—
105C24×C4—31no—
106C2×C22⋊Q8—924no—
107C22×C4⋊C4—92no—
108C22×C4⋊C4—91no—
109C22×C22⋊C4—92no—
110C24×C3—21no—
111C24×C3—38no—
112C24×C3—36no—
113C24×C3—64no—
114C24×C3—64no—
115C24×C3C22×D311yescharacteristic
116C24×C3C22×D311yescharacteristic
117C23×D3—24no—
118C23×D3—24no—
119C23×D3Dic618yescomplemented
120C23×D3C22×D316yes—
121C2×C3×Q8—316no—
122C2×C3×Q8—648no—
123C22×C3×C4—28no—
124C22×C3×C4—26no—
125C22×C3×C4—36no—
126C22×C3×C4—312no—
127C22×C3×C4—316no—
128C22×C3×C4—64no—
129C22×C3×C4—68no—
130C22×C3×C4—648no—
131C22×C3×C4#6718yes—
132C22×C3×C4#6718yes—
133C22×C3×C4C22×D316yes—
134C22×C3×C4C22×D316yes—
135C22×Dic3—28no—
136C22×Dic3—24no—
137C22×Dic3—312no—
138C22×Dic3—312no—
139C22×Dic3—612no—
140C22×Dic3—68no—
141C22×Dic3—68no—
142C22×Dic3—128no—
143C22×Dic3C22×D316yes—
144C22×Dic3C22×D3112yes—
145C2×C4×D3—248no—
146C2×C4×D3C22×D3116yes—
147C2×Dic6—248no—
148C2×Dic6C22×D3116yescomplemented
149#217—332no—
150(C2×D3)⋊C4—332no—
151C4⋊Dic3—316no—
152C4⋊Dic3—316no—
153C4⋊Dic3—616no—
154Dic3⋊C4—332no—
155Dic3⋊C4—332no—
156C22×C32C22⋊Q8116yescomplemented
157C22×C32C22×D416yes—
158C22×C32C22×Q816yes—
159C22×C32C2×#4116yes—
160C22×C32C2511yescharacteristic
161C2×C3×D3—248no—
162C2×C3×D3—24no—
163C2×C3×D3—24no—
164C2×C3×D3C22×Q818yescomplemented
165C32×C4C22×D418yes—
166C3⋊Dic3—28no—
167C3×Dic3—28no—
168C3×Dic3—24no—
169C25—31no—
170C22×Q8—312no—
171C22×Q8—68no—
172C23×C4—32no—
173C23×C4—31no—
174C23×C4—312no—
175C23×C4—68no—
176C23×C4—92no—
177C23×C4D3211yescharacteristic
178C22⋊Q8—964no—
179C2×C4⋊C4—924no—
180C2×C4⋊C4—912no—
181C2×C22⋊C4—924no—
182C23×C3—23no—
183C23×C3—24no—
184C23×C3—24no—
185C23×C3—24no—
186C23×C3—324no—
187C23×C3—34no—
188C23×C3—632no—
189C23×C3—612no—
190C23×C3—612no—
191C23×C3C2×Dic614yes—
192C23×C3#23514yes—
193C23×C3D3×Q814yes—
194C23×C3#23914yes—
195C23×C3C2×#6714yes—
196C23×C3C2×#6714yes—
197C23×C3C23×D313yes—
198C23×C3C23×D313yes—
199C22×D3—232no—
200C22×D3—212no—
201C22×D3—212no—
202C22×D3C2×Dic6124yescomplemented
203C22×D3C23×D314yes—
204C3×Q8—632no—
205C2×C3×C4—24no—
206C2×C3×C4—224no—
207C2×C3×C4—34no—
208C2×C3×C4—38no—
209C2×C3×C4—612no—
210C2×C3×C4—624no—
211C2×C3×C4—632no—
212C2×C3×C4C2×#67124yes—
213C2×C3×C4C2×#67124yes—
214C2×C3×C4C23×D314yes—
215C2×C3×C4C23×D314yes—
216C2×Dic3—224no—
217C2×Dic3—212no—
218C2×Dic3—38no—
219C2×Dic3—38no—
220C2×Dic3—68no—
221C2×Dic3—624no—
222C2×Dic3—624no—
223C2×Dic3—1224no—
224C2×Dic3C23×D314yes—
225C2×Dic3C23×D318yes—
226C4×D3—232no—
227Dic6—232no—
228C2×C32C2×C22⋊Q8112yescomplemented
229C2×C32C23×D411yescharacteristic
230C2×C32C23×Q811yescharacteristic
231C2×C32C22×#4111yescharacteristic
232C3×D3—28no—
233C24—36no—
234C24—38no—
235C24—64no—
236C24—64no—
237C24C2×D3211yescharacteristic
238C2×Q8—316no—
239C2×Q8—648no—
240C22×C4—312no—
241C22×C4—36no—
242C22×C4—316no—
243C22×C4—64no—
244C22×C4—68no—
245C22×C4—648no—
246C22×C4—912no—
247C22×C4—188no—
248C22×C4#62518yes—
249C22×C4C2×D3216yes—
250C4⋊C4—932no—
251C4⋊C4—916no—
252C22⋊C4—932no—
253C22×C3—26no—
254C22×C3—26no—
255C22×C3—26no—
256C22×C3—216no—
257C22×C3—21no—
258C22×C3—38no—
259C22×C3—64no—
260C22×C3—64no—
261C22×C3—648no—
262C22×C3#924116yescomplemented
263C22×C3#946116yescomplemented
264C22×C3C22×Dic616yes—
265C22×C3C2×#23516yes—
266C22×C3C2×D3×Q816yes—
267C22×C3C2×#23916yes—
268C22×C3C22×#6716yes—
269C22×C3C22×#6716yes—
270C22×C3C24×D311yescharacteristic
271C22×C3C24×D311yescharacteristic
272C2×D3—24no—
273C2×D3—24no—
274C2×D3—248no—
275C2×D3C22×Dic618yescomplemented
276C3×C4—28no—
277C3×C4—64no—
278C3×C4—68no—
279C3×C4C22×#6718yes—
280C3×C4C22×#6718yes—
281Dic3—28no—
282Dic3—24no—
283Dic3—68no—
284Dic3—68no—
285Dic3—128no—
286C32C22×C22⋊Q811yescharacteristic, complemented, Sylow
287C23—324no—
288C23—34no—
289C23—612no—
290C23—612no—
291C23—632no—
292C23D3×Dic614yes—
293C23#379914yes—
294C23C2×#62514yes—
295C23C22×D3213yes—
296Q8—632no—
297C2×C4—38no—
298C2×C4—34no—
299C2×C4—612no—
300C2×C4—624no—
301C2×C4—632no—
302C2×C4—98no—
303C2×C4—1824no—
304C2×C4C2×#625124yes—
305C2×C4C22×D3214yes—
306C2×C3—212no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—68no—
311C2×C3C2×#924112yescomplemented
312C2×C3C2×#946112yescomplemented
313C2×C3C23×Dic611yescharacteristic
314C2×C3C22×#23511yescharacteristic
315C2×C3C22×D3×Q811yescharacteristic
316C2×C3C22×#23911yescharacteristic
317C2×C3C23×#6711yescharacteristic
318C2×C3C23×#6711yescharacteristic
319D3—28no—
320C22—38no—
321C22—648no—
322C22—64no—
323C22—64no—
324C22#63856116yesdirect product factor
325C22C2×D3×Dic616yes—
326C22C2×#379916yes—
327C22C22×#62516yes—
328C22C23×D3211yescharacteristic
329C4—64no—
330C4—68no—
331C4—188no—
332C4C22×#62518yes—
333C3—21no—
334C3C22×#92411yescharacteristic, complemented
335C3C22×#94611yescharacteristic, complemented
336C2—68no—
337C2C2×#63856112yesdirect product factor
338C2C22×D3×Dic611yescharacteristic
339C2C22×#379911yescharacteristic
340C2C23×#62511yescharacteristic
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
2. C23×C3×Dic6C2semidirect
3. C22×C4⋊(C3⋊Dic3)C2semidirect
4. C22×#3724C2semidirect
6. C2×#63856C2direct (N × H)
17. C22×C3×Dic6C22semidirect
18. C2×C4⋊(C3⋊Dic3)C22semidirect
19. C2×#3724C22semidirect
21. #63856C22direct (N × H)
25. C23×Dic6D3semidirect
38. C23×C3×D3Q8semidirect
51. C2×C3×Dic6C23semidirect
52. C4⋊(C3⋊Dic3)C23semidirect
53. #3724C23semidirect
75. C22×Dic6C2×D3semidirect
92. C22×C3×D3C2×Q8semidirect
119. C23×D3Dic6semidirect
148. C2×Dic6C22×D3semidirect
156. C22×C32C22⋊Q8semidirect
164. C2×C3×D3C22×Q8semidirect
202. C22×D3C2×Dic6semidirect
228. C2×C32C2×C22⋊Q8semidirect
262. C22×C3#924semidirect
263. C22×C3#946semidirect
275. C2×D3C22×Dic6semidirect
286. C32C22×C22⋊Q8semidirect
311. C2×C3C2×#924semidirect
312. C2×C3C2×#946semidirect
324. C22#63856direct (N × H)
334. C3C22×#924semidirect
335. C3C22×#946semidirect
337. C2C2×#63856direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4×D32
C23×C3×Dic62
C22×C4⋊(C3⋊Dic3)2
C22×#37242
C22×(C2×D3)⋊Dic32
C2×#638562
C22×#9463
C22×#9243