Finite Group Databasebeta

← #37547528

Subgroups — #37547528

Order 23168.

964 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×Dic362C211yesmaximal, characteristic, complemented
2#26042009C213yesmaximal, complemented
3#26042007C213yesmaximal, complemented
4#26042006C213yesmaximal, complemented
5#26042006C213yesmaximal, complemented
6#26042005C211yesmaximal, characteristic, complemented
7#26042005C211yesmaximal, characteristic, complemented
8#26041909C213yesmaximal
9#26041906C213yesmaximal
10#26041835C213yesmaximal
11#26041833C213yesmaximal
12#26041755C211yesmaximal, characteristic
13#26041755C211yesmaximal, characteristic
14C4×Q8×D181C211yesmaximal, characteristic
15#557×C181C211yesmaximal, characteristic, complemented
16C2×Q8×D181C2211yescharacteristic
17#19153816C2213yes—
18#19153815C2213yes—
19#19153815C2213yes—
20#19153815C2216yes—
21#19153815C2213yes—
22#19153815C2213yes—
23#19153815C2213yes—
24#19153814C2216yes—
25#19153814C2213yes—
26#19153808C2213yes—
27#19153808C2213yes—
28#19153807C2213yes—
29#19153807C2213yes—
30#19153806C2216yes—
31#19153806C2216yes—
32#19153806C2213yes—
33#19153806C2213yes—
34#19153796C2213yes—
35#19153794C2216yes—
36#19153794C2211yescharacteristic
37#19153794C2213yes—
38#19153792C2213yes—
39Q8×Dic181C2211yescharacteristic
40Q8×Dic181C2211yescharacteristic
41Q8×Dic181C2211yescharacteristic
42C4×Dic362C2213yes—
43C4×Dic362C2213yes—
44C4×Dic362C2213yes—
45C4×Dic362C2213yes—
46C4×Dic362C2213yes—
47C4×Dic362C2213yes—
48#19153781C2213yes—
49#19153781C2213yes—
50#19153781C2216yes—
51#19153781C2213yes—
52#19153781C2216yes—
53#19153774C2213yes—
54#19153774C2213yes—
55#19153773C2213yes—
56#19153773C2213yes—
57C4⋊C4×D181C2213yes—
58C42×D181C2213yes—
59C4⋊Q8×C181C2213yes—
60#125×C181C2213yes—
61#125×C181C2213yes—
62C4×Q8×C181C2211yescharacteristic
63C4×Q8×C181C2211yescharacteristic
64C4×Q8×C181C2213yes—
65C4×Q8×C181C2211yescharacteristic
66C2×Dic362C2313yes—
67C2×Dic362C2311yescharacteristic
68C2×Dic362C2316yes—
69C2×Dic362C2313yes—
70C2×Dic362C2313yes—
71Q8×D181Q818yescomplemented
72C2×C4×D181C2311yescharacteristic
73C2×C4×D181C2313yes—
74C2×C4×D181C2313yes—
75C4⋊Dic181C2313yes—
76C4⋊Dic181C2313yes—
77C4⋊Dic181C2313yes—
78C4⋊Dic181C2313yes—
79C4⋊Dic181C2313yes—
80C4⋊Dic181C2313yes—
81#15668428C2316yes—
82#15668428C2313yes—
83#15668428C2313yes—
84#15668428C2313yes—
85#15668428C2313yes—
86#15668428C2313yes—
87#15668428C2313yes—
88#15668428C2316yes—
89#15668428C2316yes—
90#15668428C2311yescharacteristic
91#15668428C2311yescharacteristic
92#15668428C2313yes—
93#15668428C2313yes—
94#15668426C2311yescharacteristic
95#15668426C2311yescharacteristic
96#15668426C2313yes—
97#15668426C2313yes—
98C4×Dic181C2313yes—
99C4×Dic181C2313yes—
100C4×Dic181C2313yes—
101C4×Dic181C2313yes—
102C4×Dic181C2313yes—
103C4×Dic181C2316yes—
104C4×Dic181C2316yes—
105C2×Q8×C181C2313yes—
106C2×Q8×C181C2311yescharacteristic
107C4⋊C4×C181C2311yescharacteristic
108C4⋊C4×C181C2313yes—
109C4⋊C4×C181C2313yes—
110C4⋊C4×C181C2313yes—
111C4⋊C4×C181C2313yes—
112C4⋊C4×C181C2313yes—
113C4⋊C4×C181C2316yes—
114C42×C181C2313yes—
115C42×C181C2313yes—
116C42×C181C2313yes—
117C22×D181C2411yescharacteristic
118Dic362—26no—
119Dic362—22no—
120Dic362—46no—
121Dic362—43no—
122Dic362C2×Q8112yes—
123C2×Dic181C2411yescharacteristic
124C2×Dic181C2413yes—
125C2×Dic181C2411yescharacteristic
126C2×Dic181C2413yes—
127C2×Dic181C2413yes—
128C2×Dic181C2413yes—
129C2×Dic181C2411yescharacteristic
130C4×D181—22no—
131C4×D181—43no—
132C4×D181C2×Q8112yes—
133Q8×C181—43no—
134Q8×C181C2×Q814yes—
135C2×C4×C181C2411yescharacteristic
136C2×C4×C181C2411yescharacteristic
137C2×C4×C181C2411yescharacteristic
138C2×C4×C181C2413yes—
139C2×C4×C181C2413yes—
140C2×C4×C181C2413yes—
141C2×C4×C181C2413yes—
142C22×C181C2511yescharacteristic
143C2×D181—21no—
144C2×D181—21no—
145C2×D181C22×Q812yes—
146C4×C181—21no—
147C4×C181—21no—
148C4×C181—23no—
149C4×C181—21no—
150C4×C181—23no—
151C4×C181—23no—
152C4×C181C22×Q816yes—
153Dic181—21no—
154Dic181—23no—
155Dic181—23no—
156Dic181—21no—
157Dic181—23no—
158Dic181C22×Q816yes—
159Dic1812−1+412yes—
160C2×C181C23×Q811yescharacteristic
161C2×C181C2×2−1+411yescharacteristic
162C2×C181#58511yescharacteristic
163D181—22no—
164C181#350211yescharacteristic, complemented, Sylow
165#3502—1811nomaximal, Sylow
166Q82—1813no—
167Q82—1811no—
168#557—1813no—
169#557—1813no—
170#557—1811no—
171#557—1811no—
172#557—1813no—
173#557D18111yescharacteristic, complemented
174#544—1813no—
175#544—1813no—
176#533—1813no—
177#533—1813no—
178#519—1811no—
179#519—1811no—
180C2×C4×Q8—1811no—
181C22×Q8—1811no—
182C4⋊Q8—1813no—
183C4⋊Q8—1816no—
184C4⋊Q8—1813no—
185C4⋊Q8—1813no—
186C4⋊Q8—1813no—
187C4⋊Q8—1816no—
188C4⋊Q8—1813no—
189C4⋊Q8—1813no—
190C4⋊Q8—1813no—
191C4⋊Q8C2×D18113yes—
192#125—1813no—
193#125—1813no—
194#125—1816no—
195#125—1816no—
196#125—1813no—
197#125—1813no—
198#125—1813no—
199#125—1813no—
200#125C2×D18113yes—
201#125C2×D18113yes—
202C22⋊Q8—1813no—
203C22⋊Q8—1813no—
204C22⋊Q8—1811no—
205C22⋊Q8—1816no—
206C22⋊Q8—1813no—
207C4×Q8—1811no—
208C4×Q8—1811no—
209C4×Q8—1811no—
210C4×Q8—1813no—
211C4×Q8—1813no—
212C4×Q8—1813no—
213C4×Q8—1816no—
214C4×Q8—1816no—
215C4×Q8—1813no—
216C4×Q8—1813no—
217C4×Q8—1813no—
218C4×Q8—1813no—
219C4×Q8—1813no—
220C4×Q8—1813no—
221C4×Q8C2×D18111yescharacteristic
222C4×Q8C2×D18111yescharacteristic
223C4×Q8C2×D18111yescharacteristic
224C4×Q8C2×D18113yes—
225#117—1813no—
226#117—1813no—
227#117—1813no—
228#117—1813no—
229C2×C4⋊C4—1813no—
230C2×C42—1813no—
231C2×Q8—1818no—
232C2×Q8—1813no—
233C2×Q8—1811no—
234C2×Q8—1813no—
235C2×Q8—1813no—
236C2×Q8—1816no—
237C2×Q8C22×D18113yes—
238C2×Q8C22×D18111yescharacteristic
239C22×C4—1811no—
240C22×C4—1813no—
241C22×C4—1813no—
242C4⋊C4—1816no—
243C4⋊C4—1813no—
244C4⋊C4—1813no—
245C4⋊C4—1813no—
246C4⋊C4—1813no—
247C4⋊C4—1813no—
248C4⋊C4—1813no—
249C4⋊C4—1813no—
250C4⋊C4—1813no—
251C4⋊C4—1811no—
252C4⋊C4—1813no—
253C4⋊C4—1811no—
254C4⋊C4—1816no—
255C4⋊C4—1813no—
256C4⋊C4—1816no—
257C4⋊C4—1813no—
258C4⋊C4—1813no—
259C4⋊C4—1813no—
260C4⋊C4—1813no—
261C4⋊C4C22×D18113yes—
262C4⋊C4C22×D18113yes—
263C4⋊C4C22×D18113yes—
264C4⋊C4C22×D18113yes—
265C4⋊C4C22×D18116yes—
266C4⋊C4C22×D18111yescharacteristic
267C4⋊C4C22×D18113yes—
268C22⋊C4—1813no—
269C22⋊C4—1811no—
270C22⋊C4—1813no—
271C22⋊C4—1811no—
272C42—1813no—
273C42—1813no—
274C42—1813no—
275C42—1813no—
276C42—1813no—
277C42—1816no—
278C42—1816no—
279C42C22×D18113yes—
280C42C22×D18113yes—
281C42C22×D18113yes—
282C23—1811no—
283Q8—43no—
284Q8—3626no—
285Q8—3622no—
286Q8—18112no—
287Q8—7243no—
288Q8—7246no—
289Q8Q8×D18114yes—
290C2×C4—3622no—
291C2×C4—1811no—
292C2×C4—1813no—
293C2×C4—1811no—
294C2×C4—1813no—
295C2×C4—1813no—
296C2×C4—1813no—
297C2×C4—1811no—
298C2×C4—18112no—
299C2×C4—7243no—
300C2×C4C23×D18111yescharacteristic
301C2×C4C23×D18113yes—
302C2×C4C23×D18113yes—
303C2×C4C23×D18113yes—
304C2×C4C23×D18111yescharacteristic
305C2×C4C23×D18111yescharacteristic
306C2×C4C23×D18113yes—
307C22—3621no—
308C22—3621no—
309C22—1812no—
310C22C24×D18111yescharacteristic
311C4—23no—
312C4—21no—
313C4—23no—
314C4—21no—
315C4—23no—
316C4—21no—
317C4—3623no—
318C4—3621no—
319C4—3623no—
320C4—3621no—
321C4—3623no—
322C4—1816no—
323C4—1812no—
324C4C2×Q8×D18116yes—
325C2—3622no—
326C2C22×Q8×D18111yescharacteristic
327C22−1+4×D18111yescharacteristic
328C2#2604211511yescharacteristic
subgroupG/H
center, FrattiniC22C24×D181
commutator, socle, cosocleC22×C181C25
Fitting#557×C181C2
prime radicalC22×D181C24
hypercenter#557D181
hypocenterC181#3502
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. Q8×Dic362C2semidirect
2. #26042009C2semidirect
3. #26042007C2semidirect
4. #26042006C2semidirect
5. #26042006C2semidirect
6. #26042005C2semidirect
7. #26042005C2semidirect
15. #557×C181C2semidirect
71. Q8×D181Q8semidirect
164. C181#3502semidirect
173. #557D181semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic3622
#260420092
#260420072
#260420062
#260420052
#260419092
#260419062
#260418352
#260418332
#260417552
C4×Q8×D1812
#557×C1812
#3502181