Finite Group Databasebeta

← #26032417

Subgroups — #26032417

Order 11392.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×Dic178C211yesmaximal, characteristic, complemented
2#19151968C211yesmaximal, characteristic, complemented
3#19151939C211yesmaximal, characteristic, complemented
4#19151933C211yesmaximal, characteristic, complemented
5#19151902C211yesmaximal, characteristic, complemented
6#19151842C211yesmaximal, characteristic, complemented
7#19151825C211yesmaximal, characteristic, complemented
8#19151803C211yesmaximal, characteristic, complemented
9#19151736C211yesmaximal, characteristic, complemented
10#19151736C211yesmaximal, characteristic, complemented
11#19151698C211yesmaximal, characteristic, complemented
12#19151670C211yesmaximal, characteristic, complemented
13#19151606C211yesmaximal, characteristic, complemented
14C2×#15667935C211yesmaximal, characteristic, complemented
15#553×C89C211yesmaximal, characteristic
16C22×Dic178C2211yescharacteristic
17C22×Dic178C2211yescharacteristic
18C2×#13083562C2211yescharacteristic
19C2×#13083562C2211yescharacteristic
20C2×#13083562C2211yescharacteristic
21C2×#13083558C2211yescharacteristic
22C22×#10623494C2211yescharacteristic
23#15668012C2211yescharacteristic
24#15668011—21no—
25#15667991C2211yescharacteristic
26#15667991C2211yescharacteristic
27#15667989—21no—
28#15667989—21no—
29C4×Dic178C2211yescharacteristic
30D4×Dic89—21no—
31#15667945C2211yescharacteristic
32#15667945C2211yescharacteristic
33#15667943C2211yescharacteristic
34C4⋊#10623494C2211yescharacteristic
35#15667935C2211yescharacteristic
36#15667935C2211yescharacteristic
37#15667935C2211yescharacteristic
38#15667935C2211yescharacteristic
39#15667935C2211yescharacteristic
40#15667935C2211yescharacteristic
41#15667935C2211yescharacteristic
42#15667935C2211yescharacteristic
43#15667935C2211yescharacteristic
44#15667933C2211yescharacteristic
45C22⋊#10623494C2211yescharacteristic
46C22⋊#10623494C2211yescharacteristic
47#15667923C2211yescharacteristic
48#126×C89—21no—
49#125×C89C2211yescharacteristic
50#123×C89—21no—
51#121×C89—21no—
52#121×C89C2211yescharacteristic
53#121×C89C2211yescharacteristic
54C4×D4×C89—21no—
55C4×D4×C89C2211yescharacteristic
56C4×D4×C89C2211yescharacteristic
57#117×C89C2211yescharacteristic
58C2×C4⋊C4×C89C2211yescharacteristic
59C2×Dic178—22no—
60C2×Dic178—22no—
61C2×Dic178—21no—
62C2×Dic178—21no—
63C2×Dic178—41no—
64C2×Dic178—41no—
65C2×Dic178C2311yescharacteristic
66C2×Dic178C2311yescharacteristic
67C2×Dic178C2311yescharacteristic
68C22×Dic89—21no—
69C22×Dic89—21no—
70#13083562—22no—
71#13083562—41no—
72#13083562—41no—
73#13083558—41no—
74C2×#10623494—41no—
75C2×#10623494C2311yescharacteristic
76C2×#10623494C2311yescharacteristic
77C2×#10623494C2311yescharacteristic
78C2×#10623494C2311yescharacteristic
79C4⋊Dic89—21no—
80C4⋊Dic89—21no—
81C4⋊Dic89C2311yescharacteristic
82#13083553—21no—
83#13083553—21no—
84#13083553—21no—
85C22⋊Dic89—21no—
86C22⋊Dic89—21no—
87C4×Dic89—21no—
88C2×D4×C89—21no—
89C2×D4×C89—21no—
90C2×D4×C89D411yescharacteristic
91C2×D4×C89C2311yescharacteristic
92C22×C4×C89—21no—
93C22×C4×C89—21no—
94C22×C4×C89C2311yescharacteristic
95C22×C4×C89C2311yescharacteristic
96C4⋊C4×C89—21no—
97C4⋊C4×C89—21no—
98C4⋊C4×C89—21no—
99C4⋊C4×C89D411yescharacteristic
100C4⋊C4×C89C2311yescharacteristic
101C4⋊C4×C89C2311yescharacteristic
102C4⋊C4×C89C2311yescharacteristic
103C22⋊C4×C89—21no—
104C22⋊C4×C89—21no—
105C22⋊C4×C89—21no—
106C22⋊C4×C89—21no—
107C22⋊C4×C89D411yescharacteristic
108C22⋊C4×C89D411yescharacteristic
109C42×C89—21no—
110C42×C89C2311yescharacteristic
111C23×C89—21no—
112C23×C89C2×D411yescharacteristic
113C23×C89C2×D411yescharacteristic
114D4×C89—22no—
115D4×C89—21no—
116D4×C89—21no—
117D4×C89—42no—
118D4×C89—81no—
119C2×C4×C89—21no—
120C2×C4×C89—21no—
121C2×C4×C89—21no—
122C2×C4×C89—21no—
123C2×C4×C89—21no—
124C2×C4×C89—21no—
125C2×C4×C89—21no—
126C2×C4×C89—81no—
127C2×C4×C89—81no—
128C2×C4×C89C2×D411yescharacteristic
129C2×C4×C89C2×D411yescharacteristic
130C2×C4×C89C2×D411yescharacteristic
131C2×C4×C89C2×D411yescharacteristic
132C2×C4×C89C2411yescharacteristic
133C2×Dic89—21no—
134C2×Dic89—21no—
135C2×Dic89—21no—
136C2×Dic89—21no—
137C2×Dic89—42no—
138C2×Dic89—42no—
139Dic178—21no—
140Dic178—21no—
141Dic178—21no—
142Dic178—21no—
143Dic178—21no—
144Dic178—42no—
145Dic178—42no—
146Dic178#4112yes—
147#10623494—21no—
148#10623494—21no—
149#10623494—21no—
150#10623494—21no—
151C22×C89—21no—
152C22×C89—21no—
153C22×C89—21no—
154C22×C89—21no—
155C22×C89—21no—
156C22×C89—41no—
157C22×C89—41no—
158C22×C89—41no—
159C22×C89#13712yes—
160C22×C89C22×D411yescharacteristic
161C4×C89—21no—
162C4×C89—21no—
163C4×C89—21no—
164C4×C89—21no—
165C4×C89—41no—
166C4×C89—41no—
167C4×C89—41no—
168C4×C89C2×#4111yescharacteristic
169C4×C89H(5,𝔽2)11yescharacteristic
170Dic89—22no—
171Dic89—41no—
172Dic89—41no—
173Dic89—41no—
174C2×C89—22no—
175C2×C89—41no—
176C2×C89—81no—
177C2×C89#54611yescharacteristic
178C2×C89C2×#13711yescharacteristic
179C2×C89#57711yescharacteristic
180#3320—891nomaximal, Sylow
181C89#332011yescharacteristic, complemented, Sylow
182#553D8911yescharacteristic
183D4×Q8—891no—
184#491—891no—
185#485—891no—
186#484—891no—
187#479—891no—
188#468—891no—
189#465—891no—
190#461—891no—
191#448—891no—
192#448—891no—
193#440—891no—
194#435—891no—
195#419—891no—
196C2×#103—891no—
197C22×Q8—891no—
198C22×Q8—891no—
199C2×SD16—891no—
200C2×SD16—891no—
201C2×SD16—891no—
202C2×M16—891no—
203C22×C8—891no—
204C4⋊Q8—891no—
205C4⋊Q8—1781no—
206#126—21no—
207#125C2×D8911yescharacteristic
208#123—21no—
209C22⋊Q8—891no—
210C22⋊Q8—891no—
211C22⋊Q8—1781no—
212C22⋊Q8—1781no—
213#121—21no—
214#121C2×D8911yescharacteristic
215#121C2×D8911yescharacteristic
216C4×Q8—891no—
217C4×D4—21no—
218C4×D4—1781no—
219C4×D4C2×D8911yescharacteristic
220C4×D4C2×D8911yescharacteristic
221#117C2×D8911yescharacteristic
222C2×C4⋊C4C2×D8911yescharacteristic
223#107—891no—
224#107—891no—
225#106—891no—
226C4⋊C8—891no—
227#103—891no—
228#103—891no—
229#103—891no—
230#103—891no—
231#103—891no—
232#103—891no—
233#103—891no—
234#103—891no—
235#103—891no—
236#102—891no—
237C22⋊C8—891no—
238C22⋊C8—891no—
239#97—891no—
240C2×Q8—891no—
241C2×Q8—891no—
242C2×Q8—891no—
243C2×Q8—3561no—
244C2×Q8—3561no—
245C2×Q8—1782no—
246C2×Q8—1782no—
247C2×Q8—1781no—
248C2×Q8—1781no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4C22⋊D8911yescharacteristic
252C2×D4C22×D8911yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—1781no—
256C22×C4—1781no—
257C22×C4C22×D8911yescharacteristic
258C22×C4C22×D8911yescharacteristic
259SD16—3561no—
260SD16—3561no—
261SD16—1782no—
262M16—3561no—
263C2×C8—891no—
264C2×C8—891no—
265C2×C8—891no—
266C2×C8—891no—
267C2×C8—3561no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—891no—
272C4⋊C4—1781no—
273C4⋊C4—1781no—
274C4⋊C4—1781no—
275C4⋊C4—1781no—
276C4⋊C4—1781no—
277C4⋊C4C22⋊D8911yescharacteristic
278C4⋊C4C22×D8911yescharacteristic
279C4⋊C4C22×D8911yescharacteristic
280C4⋊C4C22×D8911yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—1781no—
286C22⋊C4—1781no—
287C22⋊C4C22⋊D8911yescharacteristic
288C22⋊C4C22⋊D8911yescharacteristic
289C42—21no—
290C42—1781no—
291C42C22×D8911yescharacteristic
292C23—21no—
293C23C2×C22⋊D8911yescharacteristic
294C23C2×C22⋊D8911yescharacteristic
295Q8—892no—
296Q8—3562no—
297Q8—3562no—
298Q8—1781no—
299Q8—1781no—
300Q8—1781no—
301Q8—1781no—
302Q8—1781no—
303D4—22no—
304D4—21no—
305D4—21no—
306D4—42no—
307D4—81no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—3562no—
318C2×C4—3562no—
319C2×C4—1781no—
320C2×C4—1781no—
321C2×C4—1781no—
322C2×C4—1781no—
323C2×C4C2×C22⋊D8911yescharacteristic
324C2×C4C2×C22⋊D8911yescharacteristic
325C2×C4C2×C22⋊D8911yescharacteristic
326C2×C4C2×C22⋊D8911yescharacteristic
327C2×C4C23×D8911yescharacteristic
328C8—1781no—
329C8—1781no—
330C8—1781no—
331C8—1781no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22#1566804212yes—
341C22C22×C22⋊D8911yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—3561no—
350C4—3561no—
351C4—3561no—
352C4—1782no—
353C4#41×D8911yescharacteristic
354C4#1566805811yescharacteristic
355C2—22no—
356C2—41no—
357C2—81no—
358C2#1915223611yescharacteristic
359C2C2×#1566804211yescharacteristic
360C2#1915240911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D89
commutator, cosocleC2×C4×C89C24
FrattiniC2×C4C23×D89
Fitting, prime radical#553×C89C2
socleC22×C89C22×D4
hypercenter#553D89
hypocenterC89#3320
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. D4×Dic178C2semidirect
2. #19151968C2semidirect
3. #19151939C2semidirect
4. #19151933C2semidirect
5. #19151902C2semidirect
6. #19151842C2semidirect
7. #19151825C2semidirect
8. #19151803C2semidirect
9. #19151736C2semidirect
10. #19151736C2semidirect
11. #19151698C2semidirect
12. #19151670C2semidirect
13. #19151606C2semidirect
14. C2×#15667935C2semidirect
181. C89#3320semidirect

Maximal subgroups

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

subgroupindex
D4×Dic1782
#191519682
#191519392
#191519332
#191519022
#191518422
#191518252
#191518032
#191517362
#191516982
#191516702
#191516062
C2×#156679352
#553×C892
#332089