Finite Group Databasebeta

← #36445047

Subgroups — #36445047

Order 22784.

610 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×Q8×Dic89C211yesmaximal, characteristic
2#26020107C211yesmaximal, characteristic
3C4×#15667922C211yesmaximal, characteristic
4C2×Q8⋊#10623494C211yesmaximal, characteristic
5#26018092C211yesmaximal, characteristic
6#26018091C211yesmaximal, characteristic
7C2×Q8⋊C8×C89C211yesmaximal, characteristic
8C2×Q8×Dic89C412yes—
9C2×C4⋊#10623494C2211yescharacteristic
10C2×#15667922—22no—
11C2×#15667922—21no—
12C2×C4×#10623494C2211yescharacteristic
13C4⋊C4×Dic89—22no—
14C4⋊C4×Dic89C412yes—
15C4×C4⋊Dic89—21no—
16C4×C4⋊Dic89C2211yescharacteristic
17C42×Dic89—21no—
18C42×Dic89C2211yescharacteristic
19#19151352—21no—
20Q8⋊#10623494C414yescomplemented
21C2×C4×Q8×C89C2211yescharacteristic
22C2×C4⋊C8×C89C2211yescharacteristic
23C2×C4×C8×C89C2211yescharacteristic
24Q8⋊C8×C89C414yescomplemented
25C22×#10623494—21no—
26C22×#10623494—21no—
27Q8×Dic89—24no—
28Q8×Dic89—24no—
29C2×C4⋊Dic89—22no—
30C2×C4⋊Dic89C2×C412yes—
31C2×C4×Dic89—22no—
32C2×C4×Dic89—22no—
33C2×C4×Dic89—21no—
34C2×C4×Dic89C2×C412yes—
35C2×C4×Dic89D411yescharacteristic
36C2×C4×Dic89D411yescharacteristic
37C2×C4×Dic89D412yes—
38C4⋊#10623494—21no—
39C4⋊#10623494C2×C412yes—
40#15667922—28no—
41#15667922—42no—
42C4×#10623494—21no—
43C4×#10623494C2×C412yes—
44#15667919—22no—
45#15667919—21no—
46C22×Q8×C89C2×C411yescharacteristic
47C22×C8×C89—21no—
48C22×C8×C89—21no—
49C4×Q8×C89—21no—
50C4×Q8×C89—21no—
51C4×Q8×C89C2×C412yes—
52C4×Q8×C89C2×C412yes—
53C2×C4⋊C4×C89—21no—
54C2×C4⋊C4×C89C2×C411yescharacteristic
55C2×C42×C89—21no—
56C2×C42×C89C2311yescharacteristic
57C4⋊C8×C89—21no—
58C4⋊C8×C89C2×C412yes—
59C4×C8×C89—21no—
60C4×C8×C89C2×C412yes—
61C22×Dic89—22no—
62C22×Dic89C22⋊C412yes—
63C22×Dic89C22⋊C412yes—
64C2×#10623494—24no—
65C2×#10623494—22no—
66C2×#10623494—22no—
67C2×#10623494—41no—
68C2×#10623494—41no—
69C4⋊Dic89—28no—
70C4⋊Dic89—22no—
71C4⋊Dic89M1614yes—
72C4×Dic89—28no—
73C4×Dic89—22no—
74C4×Dic89—22no—
75C4×Dic89—21no—
76C4×Dic89—21no—
77C4×Dic89—21no—
78C4×Dic89—21no—
79C4×Dic89—24no—
80C4×Dic89—41no—
81C4×Dic89—41no—
82C4×Dic89—44no—
83C2×Q8×C89—21no—
84C2×Q8×C89—22no—
85C2×Q8×C89C4212yes—
86C2×Q8×C89M1614yes—
87C22×C4×C89—21no—
88C22×C4×C89—21no—
89C22×C4×C89C22×C411yescharacteristic
90C22×C4×C89C2×D411yescharacteristic
91C22×C4×C89C2×D411yescharacteristic
92C2×C8×C89—24no—
93C2×C8×C89—22no—
94C2×C8×C89—22no—
95C2×C8×C89—41no—
96C2×C8×C89—41no—
97C4⋊C4×C89—21no—
98C4⋊C4×C89—22no—
99C4⋊C4×C89—22no—
100C4⋊C4×C89C4212yes—
101C42×C89—22no—
102C42×C89—22no—
103C42×C89C22×C411yescharacteristic
104C42×C89C22×C411yescharacteristic
105C42×C89#4111yescharacteristic
106C42×C89#4111yescharacteristic
107C23×C89C2×C22⋊C411yescharacteristic
108Q8×C89—22no—
109Q8×C89—24no—
110Q8×C89#9714yescomplemented
111C2×C4×C89—22no—
112C2×C4×C89—24no—
113C2×C4×C89—22no—
114C2×C4×C89—22no—
115C2×C4×C89—42no—
116C2×C4×C89C2×C4211yescharacteristic
117C2×C4×C89#11711yescharacteristic
118C2×C4×C89C4×D411yescharacteristic
119C2×C4×C89C4×D411yescharacteristic
120C2×C4×C89C4×D411yescharacteristic
121C2×C4×C89C4×D411yescharacteristic
122C2×C4×C89C2×M1612yes—
123C2×C4×C89C2×M1612yes—
124C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C89#13711yescharacteristic
126C2×C4×C89#13711yescharacteristic
127C2×C4×C89#13711yescharacteristic
128C2×Dic89—22no—
129C2×Dic89—22no—
130C2×Dic89—22no—
131C2×Dic89—24no—
132C2×Dic89—22no—
133C2×Dic89—22no—
134C2×Dic89—42no—
135C2×Dic89—42no—
136C2×Dic89#10414yes—
137C8×C89—24no—
138C8×C89—42no—
139#10623494—24no—
140#10623494—42no—
141C22×C89C4×C22⋊C411yescharacteristic
142C22×C89#40712yes—
143C22×C89#41711yescharacteristic
144C22×C89#41911yescharacteristic
145C22×C89C2×#10412yes—
146C4×C89—24no—
147C4×C89—21no—
148C4×C89—21no—
149C4×C89—42no—
150C4×C89C2×#9711yescharacteristic
151C4×C89C4×M1611yescharacteristic
152C4×C89#43212yes—
153C4×C89#44011yescharacteristic
154C4×C89#44011yescharacteristic
155C4×C89#44111yescharacteristic
156C4×C89#44111yescharacteristic
157Dic89—24no—
158Dic89—42no—
159Dic89—44no—
160#8173—891nomaximal, Sylow
161C2×C89#148912yes—
162C2×C89#149112yes—
163C2×C89#174911yescharacteristic
164C2×C89C4×#10411yescharacteristic
165C2×C89#176311yescharacteristic
166C42×Q8—891no—
167#1767—891no—
168C4×#97—891no—
169C2×Q8⋊C8—891no—
170C2×Q8⋊C8D8911yescharacteristic
171#1279—891no—
172#1279—891no—
173C89#817311yescharacteristic, complemented, Sylow
174C2×C4×Q8—892no—
175C2×C4×Q8C2×D8911yescharacteristic
176C2×C4⋊C8—891no—
177C2×C4⋊C8C2×D8911yescharacteristic
178C2×#97—1781no—
179C2×#97—1782no—
180C2×C4×C8—891no—
181C2×C4×C8C2×D8911yescharacteristic
182C4×C4⋊C4—892no—
183C4×C4⋊C4—891no—
184C4×C4⋊C4—1781no—
185C4×C4⋊C4—1782no—
186C43—891no—
187C43—1781no—
188#336—1781no—
189Q8⋊C8—894no—
190Q8⋊C8Dic8914yescomplemented
191C22×Q8C4×D8911yescharacteristic
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—1781no—
195C22×C8—1781no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—1784no—
199C4×Q8—1784no—
200C4×Q8C4×D8912yes—
201C4×Q8C2×Dic8912yes—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—892no—
204C2×C4⋊C4—1782no—
205C2×C4⋊C4C4×D8911yescharacteristic
206C2×C42—21no—
207C2×C42—892no—
208C2×C42—892no—
209C2×C42—891no—
210C2×C42—891no—
211C2×C42—1782no—
212C2×C42—1781no—
213C2×C42—1782no—
214C2×C42C22×D8911yescharacteristic
215C4⋊C8—21no—
216C4⋊C8—892no—
217C4⋊C8—1781no—
218C4⋊C8C2×Dic8912yes—
219#97—3562no—
220#97—1788no—
221C4×C8—21no—
222C4×C8—892no—
223C4×C8—1781no—
224C4×C8C2×Dic8912yes—
225#95—1782no—
226#95—1781no—
227C2×Q8—21no—
228C2×Q8—22no—
229C2×Q8C4×Dic8912yes—
230C2×Q8#1308355714yes—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—892no—
234C22×C4—892no—
235C22×C4—1782no—
236C22×C4C2×C4×D8911yescharacteristic
237C22×C4D4×D8911yescharacteristic
238C22×C4D4×D8911yescharacteristic
239C2×C8—24no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—3561no—
245C2×C8—3561no—
246C2×C8—1782no—
247C2×C8—1784no—
248C2×C8—1782no—
249C4⋊C4—21no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—894no—
253C4⋊C4—1788no—
254C4⋊C4—1782no—
255C4⋊C4C4×Dic8912yes—
256C42—22no—
257C42—22no—
258C42—3564no—
259C42—3561no—
260C42—3561no—
261C42—1784no—
262C42—1788no—
263C42—1782no—
264C42—1782no—
265C42—1781no—
266C42—1781no—
267C42—1781no—
268C42—1781no—
269C42C2×C4×D8911yescharacteristic
270C42C22×Dic8911yescharacteristic
271C42#1308357311yescharacteristic
272C42#1308357311yescharacteristic
273C23C22⋊C4×D8911yescharacteristic
274Q8—24no—
275Q8—22no—
276Q8#1566792214yescomplemented
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—42no—
282C2×C4—894no—
283C2×C4—3562no—
284C2×C4—3562no—
285C2×C4—1784no—
286C2×C4—1782no—
287C2×C4—1782no—
288C2×C4—1782no—
289C2×C4—1782no—
290C2×C4—1782no—
291C2×C4C2×C4×Dic8911yescharacteristic
292C2×C4#1566796811yescharacteristic
293C2×C4#1566797211yescharacteristic
294C2×C4#1566797211yescharacteristic
295C2×C4D4×Dic8911yescharacteristic
296C2×C4D4×Dic8911yescharacteristic
297C2×C4M16×D8912yes—
298C2×C4C2×#1308355712yes—
299C2×C4#1566803911yescharacteristic
300C2×C4#1566804111yescharacteristic
301C2×C4#1566804411yescharacteristic
302C2×C4#1566804411yescharacteristic
303C8—24no—
304C8—42no—
305C8—3562no—
306C8—1784no—
307C22C22⋊C4×Dic8911yescharacteristic
308C22#1915156012yes—
309C22#1915159811yescharacteristic
310C22#1915160511yescharacteristic
311C22#104×D8912yes—
312C4—24no—
313C4—21no—
314C4—21no—
315C4—42no—
316C4—3564no—
317C4—3562no—
318C4—1784no—
319C4C2×#1566792211yescharacteristic
320C4M16×Dic8911yescharacteristic
321C4#1915165212yes—
322C4#1915169911yescharacteristic
323C4#1915170111yescharacteristic
324C4#1915170211yescharacteristic
325C4#1915170511yescharacteristic
326C2#2601869012yes—
327C2#2601869712yes—
328C2#2602001611yescharacteristic
329C2#104×Dic8911yescharacteristic
330C2#2602008411yescharacteristic
subgroupG/H
centerC22×C4D4×D89
commutatorC2×C4×C89C2×C42
FrattiniC2×C42C22×D89
FittingC2×Q8⋊C8×C89C2
socle, prime radicalC23×C89C2×C22⋊C4
cosocleC2×C42×C89C23
hypercenterC2×Q8⋊C8D89
hypocenterC89#8173
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
20. Q8⋊#10623494C4semidirect
24. Q8⋊C8×C89C4semidirect
110. Q8×C89#97semidirect
173. C89#8173semidirect
190. Q8⋊C8Dic89semidirect
276. Q8#15667922semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×Dic892
#260201072
C4×#156679222
C2×Q8⋊#106234942
#260180922
#260180912
C2×Q8⋊C8×C892
#817389