Finite Group Databasebeta

← #36523692

Subgroups — #36523692

Order 22784.

588 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26028918C212yesmaximal
2#26019041C212yesmaximal, complemented
3#26019037C212yesmaximal, complemented
4#26019032C212yesmaximal, complemented
5#26019019C212yesmaximal, complemented
6#26019015C211yesmaximal, characteristic, complemented
7#26019004C212yesmaximal, complemented
8#26019000C211yesmaximal, characteristic, complemented
9C89×#2965C211yesmaximal, characteristic
10#19151659—22no—
11#19151659—22no—
12#19151655C2212yes—
13#19151655C2212yes—
14#19151651C2212yes—
15#19151651C2212yes—
16#19151622C2212yes—
17C2×C4⋊#10623494C2212yes—
18#19151565—22no—
19#19151554—22no—
20#19151351C2212yes—
21#19151351C2212yes—
22#19151351C2212yes—
23#19151351C2212yes—
24#19151349C2211yescharacteristic
25#19151349C2211yescharacteristic
26#19151349C2212yes—
27#19151349C2211yescharacteristic
28#19151349C2211yescharacteristic
29#19151349C2212yes—
30C4×#41×C89C2211yescharacteristic
31#446×C89C411yescharacteristic
32#446×C89C411yescharacteristic
33#436×C89C411yescharacteristic
34#436×C89C411yescharacteristic
35#435×C89C412yes—
36#435×C89C412yes—
37#431×C89C2211yescharacteristic
38#431×C89C2212yes—
39C4×M16×C89C2212yes—
40C2×C4×C8×C89C2211yescharacteristic
41C2×#13083558—22no—
42C22×#10623494—22no—
43C4⋊#10623494—22no—
44C4⋊#10623494—22no—
45C4⋊#10623494C2312yes—
46C4⋊#10623494C2312yes—
47C4⋊#10623494C2312yes—
48C4⋊#10623494C2312yes—
49C22⋊#10623494—22no—
50C22⋊#10623494—22no—
51C22⋊#10623494—22no—
52C22⋊#10623494—22no—
53#15667923—22no—
54#15667923—22no—
55C4×#10623494—22no—
56C4×#10623494—22no—
57C2×#41×C89D411yescharacteristic
58C2×M16×C89C2×C412yes—
59C2×M16×C89C2×C412yes—
60C22×C8×C89C2×C411yescharacteristic
61C22×C8×C89C2×C411yescharacteristic
62C4×Q8×C89C2×C411yescharacteristic
63C4×Q8×C89Q811yescharacteristic
64C4×D4×C89C2×C412yes—
65C4×D4×C89C2×C411yescharacteristic
66C4×D4×C89Q812yes—
67C4×D4×C89Q811yescharacteristic
68#117×C89D411yescharacteristic
69#117×C89D412yes—
70C2×C42×C89C2312yes—
71C2×C42×C89C2311yescharacteristic
72C4⋊C8×C89C2×C412yes—
73C4⋊C8×C89C2×C412yes—
74C4⋊C8×C89C2×C411yescharacteristic
75C4⋊C8×C89C2×C411yescharacteristic
76C22⋊C8×C89C2×C412yes—
77C22⋊C8×C89C2×C412yes—
78C22⋊C8×C89C2×C412yes—
79#97×C89C2×C412yes—
80#97×C89C2×C412yes—
81C4×C8×C89C2×C411yescharacteristic
82C4×C8×C89C2×C411yescharacteristic
83C4×C8×C89C2311yescharacteristic
84C4×C8×C89C2311yescharacteristic
85C4×C8×C89C2312yes—
86#13083558—44no—
87C2×#10623494—22no—
88C2×#10623494—22no—
89C2×#10623494—22no—
90C2×#10623494—22no—
91C2×#10623494—44no—
92#41×C89—42no—
93C2×Q8×C89C4⋊C411yescharacteristic
94C2×D4×C89C4⋊C412yes—
95C2×D4×C89C4⋊C411yescharacteristic
96C22×C4×C89C22×C411yescharacteristic
97C22×C4×C89C22×C412yes—
98C22×C4×C89C2×D412yes—
99C22×C4×C89C2×D411yescharacteristic
100C22×C4×C89C2×Q812yes—
101C22×C4×C89C2×Q811yescharacteristic
102M16×C89—24no—
103M16×C89—42no—
104C2×C8×C89—22no—
105C2×C8×C89—41no—
106C2×C8×C89C22×C411yescharacteristic
107C2×C8×C89C22×C411yescharacteristic
108C2×C8×C89C22×C411yescharacteristic
109C2×C8×C89C22×C412yes—
110C2×C8×C89C22×C412yes—
111C2×C8×C89C22×C411yescharacteristic
112C4⋊C4×C89C4⋊C411yescharacteristic
113C4⋊C4×C89C4⋊C412yes—
114C4⋊C4×C89C4⋊C411yescharacteristic
115C4⋊C4×C89C4⋊C412yes—
116C22⋊C4×C89C4⋊C412yes—
117C22⋊C4×C89C4⋊C412yes—
118C22⋊C4×C89C4⋊C412yes—
119C42×C89C22×C411yescharacteristic
120C42×C89C22×C412yes—
121C42×C89C2×D411yescharacteristic
122C42×C89C2×D412yes—
123C42×C89C2×Q812yes—
124C42×C89C2×Q811yescharacteristic
125C42×C89C2411yescharacteristic
126C23×C89C2×C4⋊C411yescharacteristic
127C23×C89C2×C4⋊C412yes—
128Q8×C89—41no—
129D4×C89—41no—
130D4×C89—42no—
131C2×C4×C89—21no—
132C2×C4×C89—22no—
133C2×C4×C89—21no—
134C2×C4×C89—22no—
135C2×C4×C89—22no—
136C2×C4×C89—22no—
137C2×C4×C89—22no—
138C2×C4×C89—42no—
139C2×C4×C89—41no—
140C2×C4×C89C2×C4⋊C411yescharacteristic
141C2×C4×C89C2×C4⋊C412yes—
142C2×C4×C89C2×C4⋊C411yescharacteristic
143C2×C4×C89C2×C4⋊C412yes—
144C2×C4×C89C2×C4⋊C411yescharacteristic
145C2×C4×C89C2×C4⋊C412yes—
146C2×C4×C89C23×C411yescharacteristic
147C2×C4×C89C22×D411yescharacteristic
148C2×C4×C89C22×Q811yescharacteristic
149C8×C89—21no—
150C8×C89—22no—
151C8×C89—21no—
152C8×C89—21no—
153C8×C89—22no—
154C8×C89#13112yes—
155#10623494—42no—
156#10623494—42no—
157#10623494—42no—
158#10623494—42no—
159C22×C89—21no—
160C22×C89—22no—
161C22×C89—21no—
162C22×C89—22no—
163C22×C89—21no—
164C22×C89—22no—
165C22×C89C22×C4⋊C411yescharacteristic
166C4×C89—21no—
167C4×C89—22no—
168C4×C89—21no—
169C4×C89—22no—
170C4×C89—21no—
171C4×C89—21no—
172C4×C89—22no—
173C4×C89C2×#13111yescharacteristic
174C4×C89#56811yescharacteristic
175C4×C89#57512yes—
176#17831—891nomaximal, Sylow
177C2×C89—41no—
178C2×C89—42no—
179C2×C89#290611yescharacteristic
180C2×C89#291111yescharacteristic
181C2×C89#291611yescharacteristic
182#2965D8911yescharacteristic
183#2925—892no—
184#1574—892no—
185#1573—892no—
186#1572—892no—
187#1569—892no—
188#1568—891no—
189#1565—892no—
190#1564—891no—
191C89#1783111yescharacteristic, complemented, Sylow
192C4×#41C2×D8911yescharacteristic
193#446Dic8911yescharacteristic
194#446Dic8911yescharacteristic
195#436Dic8911yescharacteristic
196#436Dic8911yescharacteristic
197#435Dic8912yes—
198#435Dic8912yes—
199#433—1782no—
200#433—1782no—
201#432—892no—
202#432—892no—
203#431—892no—
204#431—892no—
205#431C2×D8911yescharacteristic
206#431C2×D8912yes—
207#423—892no—
208C2×C4⋊C8—892no—
209#408—1782no—
210#405—1782no—
211C4×M16C2×D8912yes—
212C2×C4×C8C2×D8911yescharacteristic
213#335—892no—
214#335—892no—
215#335—892no—
216#335—892no—
217#334—891no—
218#334—891no—
219#334—892no—
220#334—891no—
221#334—891no—
222#334—892no—
223C2×#41D35611yescharacteristic
224C2×M16—1782no—
225C2×M16C2×Dic8912yes—
226C2×M16C2×Dic8912yes—
227C22×C8—1782no—
228C22×C8C2×Dic8911yescharacteristic
229C22×C8C2×Dic8911yescharacteristic
230C4×Q8Dic17811yescharacteristic
231C4×Q8C2×Dic8911yescharacteristic
232C4×D4Dic17811yescharacteristic
233C4×D4Dic17812yes—
234C4×D4C2×Dic8912yes—
235C4×D4C2×Dic8911yescharacteristic
236#117D35612yes—
237#117D35611yescharacteristic
238C2×C42C22×D8911yescharacteristic
239C2×C42C22×D8912yes—
240C4⋊C8—892no—
241C4⋊C8—892no—
242C4⋊C8—892no—
243C4⋊C8—892no—
244C4⋊C8—1782no—
245C4⋊C8—1782no—
246C4⋊C8C2×Dic8912yes—
247C4⋊C8C2×Dic8912yes—
248C4⋊C8C2×Dic8911yescharacteristic
249C4⋊C8C2×Dic8911yescharacteristic
250C22⋊C8—1782no—
251C22⋊C8—1782no—
252C22⋊C8—1782no—
253C22⋊C8—1782no—
254C22⋊C8C2×Dic8912yes—
255C22⋊C8C2×Dic8912yes—
256C22⋊C8C2×Dic8912yes—
257#97—1782no—
258#97—1782no—
259#97C2×Dic8912yes—
260#97C2×Dic8912yes—
261C4×C8—1782no—
262C4×C8—1782no—
263C4×C8C2×Dic8911yescharacteristic
264C4×C8C2×Dic8911yescharacteristic
265C4×C8C22×D8911yescharacteristic
266C4×C8C22×D8911yescharacteristic
267C4×C8C22×D8912yes—
268#41—42no—
269C2×Q8C4⋊Dic8911yescharacteristic
270C2×D4C4⋊Dic8912yes—
271C2×D4C4⋊Dic8911yescharacteristic
272C22×C4C22×Dic8911yescharacteristic
273C22×C4C22×Dic8912yes—
274C22×C4C2×D35611yescharacteristic
275C22×C4C2×D35612yes—
276C22×C4C2×Dic17811yescharacteristic
277C22×C4C2×Dic17812yes—
278M16—24no—
279M16—42no—
280M16—3564no—
281C2×C8—22no—
282C2×C8—41no—
283C2×C8—3564no—
284C2×C8—1782no—
285C2×C8—1782no—
286C2×C8—1782no—
287C2×C8—1782no—
288C2×C8C22×Dic8911yescharacteristic
289C2×C8C22×Dic8911yescharacteristic
290C2×C8C22×Dic8911yescharacteristic
291C2×C8C22×Dic8912yes—
292C2×C8C22×Dic8912yes—
293C2×C8C22×Dic8911yescharacteristic
294C4⋊C4C4⋊Dic8912yes—
295C4⋊C4C4⋊Dic8911yescharacteristic
296C4⋊C4C4⋊Dic8912yes—
297C4⋊C4C4⋊Dic8911yescharacteristic
298C22⋊C4C4⋊Dic8912yes—
299C22⋊C4C4⋊Dic8912yes—
300C22⋊C4C4⋊Dic8912yes—
301C42C22×Dic8911yescharacteristic
302C42C22×Dic8912yes—
303C42C2×D35611yescharacteristic
304C42C2×D35612yes—
305C42C2×Dic17811yescharacteristic
306C42C2×Dic17812yes—
307C42C23×D8911yescharacteristic
308C23C2×C4⋊Dic8911yescharacteristic
309C23C2×C4⋊Dic8912yes—
310Q8—41no—
311D4—42no—
312D4—41no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4C2×C4⋊Dic8911yescharacteristic
323C2×C4C2×C4⋊Dic8912yes—
324C2×C4C2×C4⋊Dic8911yescharacteristic
325C2×C4C2×C4⋊Dic8912yes—
326C2×C4C2×C4⋊Dic8911yescharacteristic
327C2×C4C2×C4⋊Dic8912yes—
328C2×C4C23×Dic8911yescharacteristic
329C2×C4C22×D35611yescharacteristic
330C2×C4C22×Dic17811yescharacteristic
331C8—21no—
332C8—22no—
333C8—21no—
334C8—21no—
335C8—22no—
336C8—3562no—
337C8—3562no—
338C8—3562no—
339C8—3562no—
340C8#1566801912yes—
341C22—21no—
342C22—22no—
343C22—22no—
344C22—21no—
345C22—21no—
346C22—22no—
347C22C22×C4⋊Dic8911yescharacteristic
348C4—21no—
349C4—22no—
350C4—22no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—22no—
355C4C2×#1566801911yescharacteristic
356C4#1915236511yescharacteristic
357C4#1915239812yes—
358C2—41no—
359C2—42no—
360C2#2602882011yescharacteristic
361C2#2602884211yescharacteristic
362C2#2602886311yescharacteristic
subgroupG/H
centerC2×C4C22×D356
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#2965C2
socleC22×C89C22×C4⋊C4
prime radicalC2×#41×C89D4
cosocleC42×C89C24
hypercenter#2965D89
hypocenterC89#17831
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2965

Complements

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

NH ≅ G/Ntype
2. #26019041C2semidirect
3. #26019037C2semidirect
4. #26019032C2semidirect
5. #26019019C2semidirect
6. #26019015C2semidirect
7. #26019004C2semidirect
8. #26019000C2semidirect
191. C89#17831semidirect

Maximal subgroups

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

subgroupindex
#260289182
#260190412
#260190372
#260190322
#260190192
#260190152
#260190042
#260190002
C89×#29652
#1783189