Finite Group Databasebeta

← #36554493

Subgroups — #36554493

Order 22784.

1306 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151655C211yesmaximal, characteristic
2#26022650C211yesmaximal, characteristic
3#26020342C212yesmaximal
4#26020267C214yesmaximal, complemented
5#26020187C212yesmaximal
6#26020082C214yesmaximal, complemented
7C22×C4×Q8×C89C211yesmaximal, characteristic
8#19151655—24no—
9C2×C4⋊#10623494C2212yes—
10C2×#15667936C2218yes—
11C2×C22⋊#10623494C2212yes—
12C2×#15667923C2212yes—
13#19151466—21no—
14#19151464C2212yes—
15#19151461—21no—
16#19151458C2212yes—
17C4×C4⋊Dic89C2212yes—
18C4×C22⋊Dic89—21no—
19#19151444—21no—
20#19151360C2218yes—
21C23×Q8×C89C2211yescharacteristic
22C2×C4×Q8×C89—24no—
23C2×C4×Q8×C89—21no—
24C2×C4×Q8×C89—21no—
25C2×C4×Q8×C89C418yescomplemented
26C2×C4×Q8×C89C2212yes—
27C2×C4×Q8×C89C2212yes—
28C22×C4⋊C4×C89—21no—
29C22×C4⋊C4×C89C2211yescharacteristic
30C22×C42×C89—21no—
31C22×C42×C89C2211yescharacteristic
32C22×#10623494C2314yes—
33C2×C4⋊Dic89C2314yes—
34C2×C22⋊Dic89—22no—
35C2×C4×Dic89—22no—
36C4⋊#10623494—24no—
37#15667936—216no—
38C22⋊#10623494—24no—
39#15667923—24no—
40#15667919—22no—
41#15667919—22no—
42#15667919—22no—
43#15667919—22no—
44C22×Q8×C89—28no—
45C22×Q8×C89—22no—
46C22×Q8×C89—21no—
47C22×Q8×C89C2×C414yes—
48C22×Q8×C89C2312yes—
49C23×C4×C89—21no—
50C23×C4×C89—21no—
51C23×C4×C89D411yescharacteristic
52C23×C4×C89D411yescharacteristic
53C23×C4×C89C2311yescharacteristic
54C4×Q8×C89—28no—
55C4×Q8×C89—28no—
56C4×Q8×C89—24no—
57C4×Q8×C89—24no—
58C4×Q8×C89C2×C418yes—
59C4×Q8×C89C2×C418yes—
60C2×C4⋊C4×C89—28no—
61C2×C4⋊C4×C89—22no—
62C2×C4⋊C4×C89—22no—
63C2×C4⋊C4×C89—21no—
64C2×C4⋊C4×C89—22no—
65C2×C4⋊C4×C89C2×C414yes—
66C2×C4⋊C4×C89C2312yes—
67C2×C42×C89—28no—
68C2×C42×C89—22no—
69C2×C42×C89—22no—
70C2×C42×C89C2×C414yes—
71C2×C42×C89D411yescharacteristic
72C2×C42×C89D411yescharacteristic
73C2×C42×C89D414yes—
74C2×C42×C89C2311yescharacteristic
75C2×C42×C89C2311yescharacteristic
76C22×Dic89—24no—
77C2×#10623494—24no—
78C2×#10623494—28no—
79C4⋊Dic89—28no—
80C22⋊Dic89—44no—
81C4×Dic89—44no—
82C24×C89C2×D411yescharacteristic
83C2×Q8×C89—216no—
84C2×Q8×C89—28no—
85C2×Q8×C89—216no—
86C2×Q8×C89—24no—
87C2×Q8×C89—22no—
88C2×Q8×C89—24no—
89C2×Q8×C89C22×C414yes—
90C2×Q8×C89#4118yes—
91C22×C4×C89—28no—
92C22×C4×C89—24no—
93C22×C4×C89—24no—
94C22×C4×C89—22no—
95C22×C4×C89—24no—
96C22×C4×C89—24no—
97C22×C4×C89—22no—
98C22×C4×C89—42no—
99C22×C4×C89—44no—
100C22×C4×C89C22⋊C414yes—
101C22×C4×C89C22⋊C414yes—
102C22×C4×C89C22×C412yes—
103C22×C4×C89C2×D411yescharacteristic
104C22×C4×C89C2×D411yescharacteristic
105C22×C4×C89C2×D411yescharacteristic
106C22×C4×C89C2×D411yescharacteristic
107C22×C4×C89C2×D411yescharacteristic
108C22×C4×C89C2×D412yes—
109C22×C4×C89C2411yescharacteristic
110C4⋊C4×C89—28no—
111C4⋊C4×C89—28no—
112C4⋊C4×C89—24no—
113C4⋊C4×C89—22no—
114C4⋊C4×C89C22×C414yes—
115C42×C89—28no—
116C42×C89—28no—
117C42×C89C22⋊C414yes—
118C42×C89C22⋊C414yes—
119C42×C89C22×C412yes—
120C42×C89C22×C412yes—
121C42×C89C2×D412yes—
122C42×C89C2×D412yes—
123C23×C89—22no—
124C23×C89—22no—
125C23×C89C2×C22⋊C412yes—
126C23×C89#13714yes—
127C23×C89C22×D411yescharacteristic
128Q8×C89—216no—
129Q8×C89—24no—
130Q8×C89—28no—
131Q8×C89#11718yes—
132C2×C4×C89—216no—
133C2×C4×C89—24no—
134C2×C4×C89—28no—
135C2×C4×C89—22no—
136C2×C4×C89—22no—
137C2×C4×C89—22no—
138C2×C4×C89—22no—
139C2×C4×C89—22no—
140C2×C4×C89—28no—
141C2×C4×C89—24no—
142C2×C4×C89—24no—
143C2×C4×C89—24no—
144C2×C4×C89—44no—
145C2×C4×C89—48no—
146C2×C4×C89—44no—
147C2×C4×C89C2×C22⋊C412yes—
148C2×C4×C89C2×C22⋊C412yes—
149C2×C4×C89C2×C22⋊C412yes—
150C2×C4×C89C2×C22⋊C412yes—
151C2×C4×C89C2×C22⋊C412yes—
152C2×C4×C89C23×C411yescharacteristic
153C2×C4×C89C22×D411yescharacteristic
154C2×C4×C89C2×#4114yes—
155C2×Dic89—44no—
156C2×Dic89—44no—
157C2×Dic89—44no—
158#10623494—44no—
159C22×C89—28no—
160C22×C89—22no—
161C22×C89—22no—
162C22×C89#41914yes—
163C22×C89C22×C22⋊C411yescharacteristic
164C22×C89#52512yes—
165C22×C89C2×#13712yes—
166C22×C89#57512yes—
167C4×C89—22no—
168C4×C89—22no—
169C4×C89—22no—
170C4×C89—22no—
171C4×C89—28no—
172C4×C89—22no—
173C4×C89—44no—
174C4×C89C2×#11712yes—
175C4×C89#52512yes—
176Dic89—84no—
177#20741—891nomaximal, Sylow
178C2×C89—24no—
179C2×C89#228011yescharacteristic
180C2×C89C2×#41911yescharacteristic
181C2×C89#289611yescharacteristic
182C2×C89#311714yes—
183C22×C4×Q8D8911yescharacteristic
184C2×#432—891no—
185#2283—891no—
186#1824—892no—
187#1805—894no—
188#1787—892no—
189#1763—894no—
190C89#2074111yescharacteristic, complemented, Sylow
191C23×Q8C2×D8911yescharacteristic
192C2×C4×Q8—24no—
193C2×C4×Q8—21no—
194C2×C4×Q8—21no—
195C2×C4×Q8Dic8918yescomplemented
196C2×C4×Q8C2×D8912yes—
197C2×C4×Q8C2×D8912yes—
198C22×C4⋊C4—21no—
199C22×C4⋊C4C2×D8911yescharacteristic
200C22×C42—21no—
201C22×C42C2×D8911yescharacteristic
202#432—1784no—
203C2×C4⋊C8—892no—
204C2×#103—898no—
205C2×C22⋊C8—892no—
206C2×#97—892no—
207#383—1781no—
208#382—892no—
209#381—1781no—
210#380—892no—
211C4×C4⋊C4—892no—
212C4×C22⋊C4—1781no—
213#376—1781no—
214#340—898no—
215C22×Q8—28no—
216C22×Q8—21no—
217C22×Q8—22no—
218C22×Q8C2×Dic8914yes—
219C22×Q8C22×D8912yes—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4C22⋊D8911yescharacteristic
223C23×C4C22⋊D8911yescharacteristic
224C23×C4C22×D8911yescharacteristic
225C22×C8—894no—
226C4×Q8—28no—
227C4×Q8—28no—
228C4×Q8—24no—
229C4×Q8—24no—
230C4×Q8C2×Dic8918yes—
231C4×Q8C2×Dic8918yes—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—894no—
238C2×C4⋊C4C2×Dic8914yes—
239C2×C4⋊C4C22×D8912yes—
240C2×C22⋊C4—1782no—
241C2×C42—28no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—1782no—
245C2×C42C2×Dic8914yes—
246C2×C42C22⋊D8911yescharacteristic
247C2×C42C22⋊D8911yescharacteristic
248C2×C42C22⋊D8914yes—
249C2×C42C22×D8911yescharacteristic
250C2×C42C22×D8911yescharacteristic
251C4⋊C8—1784no—
252#103—17816no—
253C22⋊C8—1784no—
254#97—1784no—
255#95—1782no—
256#95—1782no—
257#95—1782no—
258#95—1782no—
259C24C2×C22⋊D8911yescharacteristic
260C2×Q8—216no—
261C2×Q8—28no—
262C2×Q8—216no—
263C2×Q8—22no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8C22×Dic8914yes—
267C2×Q8#1308357418yes—
268C22×C4—28no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—1784no—
278C22×C4C22⋊Dic8914yes—
279C22×C4C22⋊Dic8914yes—
280C22×C4C22×Dic8912yes—
281C22×C4C2×C22⋊D8912yes—
282C22×C4C2×C22⋊D8911yescharacteristic
283C22×C4C2×C22⋊D8911yescharacteristic
284C22×C4C2×C22⋊D8911yescharacteristic
285C22×C4C2×C22⋊D8911yescharacteristic
286C22×C4C2×C22⋊D8911yescharacteristic
287C22×C4C23×D8911yescharacteristic
288C2×C8—1784no—
289C2×C8—1788no—
290C4⋊C4—22no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C4⋊C4—24no—
294C4⋊C4—1788no—
295C4⋊C4C22×Dic8914yes—
296C22⋊C4—3564no—
297C42—28no—
298C42—28no—
299C42—3564no—
300C42C22⋊Dic8914yes—
301C42C22⋊Dic8914yes—
302C42C22×Dic8912yes—
303C42C22×Dic8912yes—
304C42C2×C22⋊D8912yes—
305C42C2×C22⋊D8912yes—
306C23—22no—
307C23—22no—
308C23C2×C22⋊Dic8912yes—
309C23#1566804314yes—
310C23C22×C22⋊D8911yescharacteristic
311Q8—216no—
312Q8—24no—
313Q8—28no—
314Q8#1566797118yes—
315C2×C4—216no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—28no—
322C2×C4—22no—
323C2×C4—28no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—44no—
328C2×C4—48no—
329C2×C4—44no—
330C2×C4—3564no—
331C2×C4—3564no—
332C2×C4—3564no—
333C2×C4C2×C22⋊Dic8912yes—
334C2×C4C2×C22⋊Dic8912yes—
335C2×C4C2×C22⋊Dic8912yes—
336C2×C4C2×C22⋊Dic8912yes—
337C2×C4C2×C22⋊Dic8912yes—
338C2×C4C23×Dic8911yescharacteristic
339C2×C4C22×C22⋊D8911yescharacteristic
340C2×C4C2×#1308357414yes—
341C8—3564no—
342C22—28no—
343C22—22no—
344C22—22no—
345C22#1915160714yes—
346C22C22×C22⋊Dic8911yescharacteristic
347C22#1915210912yes—
348C22C2×#1566804312yes—
349C22#1915239612yes—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—28no—
355C4—22no—
356C4—44no—
357C4—7124no—
358C4C2×#1566797112yes—
359C4#1915210912yes—
360C2—24no—
361C2#2602263511yescharacteristic
362C2C2×#1915160711yescharacteristic
363C2#2602877111yescharacteristic
364C2#2603042314yes—
subgroupG/H
centerC23C22×C22⋊D89
commutatorC2×C4×C89C23×C4
FrattiniC22×C4C23×D89
FittingC22×C4×Q8×C89C2
socleC23×C89C22×D4
prime radicalC24×C89C2×D4
cosocleC22×C4×C89C24
hypercenterC22×C4×Q8D89
hypocenterC89#20741
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
4. #26020267C2semidirect
6. #26020082C2semidirect
25. C2×C4×Q8×C89C4semidirect
190. C89#20741semidirect
195. C2×C4×Q8Dic89semidirect

Maximal subgroups

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

subgroupindex
C2×#191516552
#260226502
#260203422
#260202672
#260201872
#260200822
C22×C4×Q8×C892
#2074189