Finite Group Databasebeta

← #37544555

Subgroups — #37544555

Order 23168.

414 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041791C211yesmaximal, characteristic
2C22⋊Q8×D181C211yesmaximal, characteristic, complemented
3#26041620C211yesmaximal, characteristic, complemented
4#26041616C211yesmaximal, characteristic, complemented
5#26041599C211yesmaximal, characteristic, complemented
6#26041538C211yesmaximal, characteristic, complemented
7#26041536C211yesmaximal, characteristic, complemented
8#26041528C211yesmaximal, characteristic, complemented
9#26041526C211yesmaximal, characteristic, complemented
10#26041443C211yesmaximal, characteristic, complemented
11#26041439C211yesmaximal, characteristic, complemented
12#26041436C211yesmaximal, characteristic, complemented
13#26041419C211yesmaximal, characteristic, complemented
14#26041241C211yesmaximal, characteristic, complemented
15#485×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic
18C22×C4×D181C2211yescharacteristic
19C2×Dic724C2211yescharacteristic, complemented
20C2×#15668439C2211yescharacteristic, complemented
21C2×#15668438C2211yescharacteristic, complemented
22C2×#15668436C2211yescharacteristic, complemented
23C2×#15668432C2211yescharacteristic
24C2×#15668432C2211yescharacteristic
25#19153800—21no—
26#19153796C2211yescharacteristic
27#19153796C2211yescharacteristic
28#19153796C2211yescharacteristic
29#19153795C2211yescharacteristic
30#19153794—21no—
31#19153793—21no—
32#19153792C2211yescharacteristic
33#19153791C2211yescharacteristic
34#19153791C2211yescharacteristic
35#19153786—21no—
36C4×D724—21no—
37C4×D724—21no—
38C4⋊C4×D181—21no—
39C4⋊C4×D181C2211yescharacteristic
40C22⋊C4×D181—21no—
41#19153749C2211yescharacteristic, complemented
42#19153747C2211yescharacteristic, complemented
43#19153741C2211yescharacteristic, complemented
44#19153739C2211yescharacteristic, complemented
45#19153739C2211yescharacteristic, complemented
46#19153739C2211yescharacteristic, complemented
47#19153738C2211yescharacteristic, complemented
48#19153736C2211yescharacteristic, complemented
49C22⋊#13089678C2211yescharacteristic
50#19153728C2211yescharacteristic
51#19153728C2211yescharacteristic
52C22⋊Q8×C181C2211yescharacteristic
53#117×C181C2211yescharacteristic
54#107×C181C2211yescharacteristic, complemented
55#106×C181C2211yescharacteristic, complemented
56#103×C181C2211yescharacteristic, complemented
57#103×C181C2211yescharacteristic, complemented
58C22⋊C8×C181C2211yescharacteristic
59C23×D181D411yescharacteristic
60#15668449—41no—
61#15668449—41no—
62C2×Dic362—21no—
63C2×Dic362C2311yescharacteristic
64C2×Dic362C2311yescharacteristic
65Q8×D181—41no—
66Q8×D181—41no—
67C2×D724—21no—
68C2×D724—21no—
69C2×D724C2311yescharacteristic
70C2×C22⋊D181—21no—
71C22×Dic181D411yescharacteristic
72C2×C4×D181—21no—
73C2×C4×D181—21no—
74C2×C4×D181—21no—
75C2×C4×D181—41no—
76C2×C4×D181—41no—
77C2×C4×D181D411yescharacteristic
78C2×C4×D181D411yescharacteristic
79C2×C4×D181C2311yescharacteristic
80C2×C4×D181C2311yescharacteristic
81Dic724—41no—
82#15668439—41no—
83#15668438—41no—
84#15668436—41no—
85#15668432—41no—
86#15668432—41no—
87C2×#13089678C2311yescharacteristic
88C2×#13089678C2311yescharacteristic
89C4⋊Dic181—21no—
90C4⋊Dic181—21no—
91C4⋊Dic181—21no—
92C4⋊Dic181C2311yescharacteristic
93#15668428—21no—
94#15668428—21no—
95#15668428—21no—
96C22⋊Dic181—21no—
97#15668426—21no—
98#15668426—21no—
99#15668426—21no—
100#15668426—21no—
101#15668426—21no—
102#15668426—21no—
103C2×Q8×C181C2311yescharacteristic
104C22×C4×C181C2311yescharacteristic
105C2×C8×C181C2311yescharacteristic
106C2×C8×C181C2311yescharacteristic
107C4⋊C4×C181—21no—
108C4⋊C4×C181C2311yescharacteristic
109C4⋊C4×C181C2311yescharacteristic
110C4⋊C4×C181C2311yescharacteristic
111C22⋊C4×C181—21no—
112C22⋊C4×C181—21no—
113C42×C181—21no—
114C22×D181—22no—
115C22×D181—21no—
116C22×D181—41no—
117C22×D181—41no—
118C22×D181C2×D411yescharacteristic
119C22×D181C2×D411yescharacteristic
120Dic362—21no—
121Dic362—21no—
122Dic362—21no—
123Dic362—21no—
124Dic362—42no—
125D724—21no—
126D724—21no—
127D724—42no—
128D724—42no—
129C22⋊D181—42no—
130C2×Dic181—21no—
131C2×Dic181—21no—
132C2×Dic181—21no—
133C2×Dic181—41no—
134C2×Dic181C2×D411yescharacteristic
135C2×Dic181C2×D411yescharacteristic
136C4×D181—21no—
137C4×D181—21no—
138C4×D181—21no—
139C4×D181—21no—
140C4×D181—41no—
141C4×D181—41no—
142C4×D181—42no—
143C4×D181—42no—
144C4×D181—42no—
145#13089678—21no—
146#13089678—21no—
147C23×C181C2×D411yescharacteristic
148Q8×C181—21no—
149Q8×C181—21no—
150C2×C4×C181—21no—
151C2×C4×C181—21no—
152C2×C4×C181—21no—
153C2×C4×C181—21no—
154C2×C4×C181—21no—
155C2×C4×C181—21no—
156C2×C4×C181C2×D411yescharacteristic
157C2×C4×C181C2411yescharacteristic
158C8×C181—21no—
159C8×C181—21no—
160C22×C181—21no—
161C22×C181—21no—
162C22×C181—21no—
163C22×C181C22×D411yescharacteristic
164C2×D181—21no—
165C2×D181—21no—
166C2×D181—21no—
167C2×D181—21no—
168C2×D181—21no—
169C2×D181—44no—
170C2×D181—41no—
171C2×D181—41no—
172C2×D181—41no—
173C2×D181#13712yes—
174C4×C181—21no—
175C4×C181—41no—
176C4×C181—41no—
177C4×C181—41no—
178C4×C181—41no—
179C4×C181H(5,𝔽2)11yescharacteristic
180C4×C181H(5,𝔽2)11yescharacteristic
181Dic181—21no—
182Dic181—21no—
183Dic181—41no—
184Dic181—41no—
185Dic181—41no—
186C2×C181—41no—
187C2×C181#53411yescharacteristic
188C2×C181C2×#13711yescharacteristic
189C2×C181#57511yescharacteristic
190D181—22no—
191D181—41no—
192D181—81no—
193C181#319611yescharacteristic, complemented, Sylow
194#3196—1811nomaximal, Sylow
195#525—1811no—
196C2×C22⋊Q8—1811no—
197#485—1811no—
198#485D18111yescharacteristic, complemented
199#484—1811no—
200#481—1811no—
201#470—1811no—
202#470—1811no—
203#468—1811no—
204#468—1811no—
205#452—1811no—
206#452—1811no—
207#451—1811no—
208#448—1811no—
209#407—1811no—
210C2×#41—1811no—
211C22×Q8—1811no—
212C23×C4—1811no—
213C2×Q16—1811no—
214C2×Q16—1811no—
215C2×SD16—1811no—
216C2×SD16—1811no—
217C2×M16—1811no—
218C2×M16—1811no—
219#123—3621no—
220C22⋊Q8—3621no—
221C22⋊Q8—3621no—
222C22⋊Q8—1811no—
223C22⋊Q8—1811no—
224C22⋊Q8—1811no—
225C22⋊Q8—1811no—
226C22⋊Q8—1811no—
227C22⋊Q8C2×D18111yescharacteristic
228#121—1811no—
229#121—1811no—
230#120—3621no—
231C4×D4—3621no—
232C4×D4—3621no—
233#117C2×D18111yescharacteristic
234C2×C4⋊C4—3621no—
235C2×C4⋊C4—1811no—
236C2×C22⋊C4—3621no—
237#107—1811no—
238#107C2×D18111yescharacteristic, complemented
239#106—1811no—
240#106C2×D18111yescharacteristic, complemented
241#103—1811no—
242#103—1811no—
243#103—1811no—
244#103—1811no—
245#103C2×D18111yescharacteristic, complemented
246#103C2×D18111yescharacteristic, complemented
247#102—1811no—
248#102—1811no—
249C22⋊C8—1811no—
250C22⋊C8—1811no—
251C22⋊C8—1811no—
252C22⋊C8C2×D18111yescharacteristic
253C24—1811no—
254#41—7241no—
255#41—7241no—
256C2×Q8—3621no—
257C2×Q8—1811no—
258C2×Q8—1811no—
259C2×Q8—7241no—
260C2×Q8—7241no—
261C2×Q8C22×D18111yescharacteristic
262C2×D4—3621no—
263C2×D4—3621no—
264C2×D4—3621no—
265C2×D4—1811no—
266C22×C4—3621no—
267C22×C4—3621no—
268C22×C4—3621no—
269C22×C4—1811no—
270C22×C4—1811no—
271C22×C4—1811no—
272C22×C4—1811no—
273C22×C4—1811no—
274C22×C4—7241no—
275C22×C4—7241no—
276C22×C4C22×D18111yescharacteristic
277Q16—7241no—
278Q16—7241no—
279SD16—7241no—
280SD16—7241no—
281M16—7241no—
282M16—7241no—
283C2×C8—1811no—
284C2×C8—1811no—
285C2×C8C22×D18111yescharacteristic
286C2×C8C22×D18111yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—3621no—
289C4⋊C4—3621no—
290C4⋊C4—3621no—
291C4⋊C4—3621no—
292C4⋊C4—3621no—
293C4⋊C4—3621no—
294C4⋊C4—1811no—
295C4⋊C4C22×D18111yescharacteristic
296C4⋊C4C22×D18111yescharacteristic
297C4⋊C4C22×D18111yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3621no—
301C22⋊C4—3621no—
302C22⋊C4—3621no—
303C22⋊C4—3621no—
304C22⋊C4—3621no—
305C22⋊C4—3621no—
306C22⋊C4—3621no—
307C42—21no—
308C23—3622no—
309C23—3621no—
310C23—1811no—
311C23—1811no—
312C23—7241no—
313C23—7241no—
314C23D4×D18111yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—3621no—
318Q8—3621no—
319Q8—3621no—
320Q8—3621no—
321Q8—7242no—
322D4—3621no—
323D4—3621no—
324D4—7242no—
325D4—7242no—
326D4—7242no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—3621no—
334C2×C4—3621no—
335C2×C4—3621no—
336C2×C4—3621no—
337C2×C4—3621no—
338C2×C4—3621no—
339C2×C4—3621no—
340C2×C4—1811no—
341C2×C4—1811no—
342C2×C4—7241no—
343C2×C4—7241no—
344C2×C4—7242no—
345C2×C4—7242no—
346C2×C4—7242no—
347C2×C4—7241no—
348C2×C4D4×D18111yescharacteristic
349C2×C4C23×D18111yescharacteristic
350C8—21no—
351C8—21no—
352C8—3621no—
353C8—3621no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—3621no—
358C22—3621no—
359C22—3621no—
360C22—3621no—
361C22—3621no—
362C22—1812no—
363C22—7244no—
364C22—7241no—
365C22—7241no—
366C22—7241no—
367C22C2×D4×D18111yescharacteristic
368C4—21no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—3621no—
374C4—3621no—
375C4—7241no—
376C4—7241no—
377C4—7241no—
378C4#1915386211yescharacteristic
379C4#1915386211yescharacteristic
380C2—41no—
381C2—3622no—
382C2—14481no—
383C2—7241no—
384C2#2604184011yescharacteristic
385C2#137×D18111yescharacteristic
386C2#2604207611yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#485×C181C2
socleC22×C181C22×D4
prime radical#26041791C2
hypercenter#485D181
hypocenterC181#3196
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
2. C22⋊Q8×D181C2semidirect
3. #26041620C2semidirect
4. #26041616C2semidirect
5. #26041599C2semidirect
6. #26041538C2semidirect
7. #26041536C2semidirect
8. #26041528C2semidirect
9. #26041526C2semidirect
10. #26041443C2semidirect
11. #26041439C2semidirect
12. #26041436C2semidirect
13. #26041419C2semidirect
14. #26041241C2semidirect
15. #485×C181C2semidirect
19. C2×Dic724C22semidirect
20. C2×#15668439C22semidirect
21. C2×#15668438C22semidirect
22. C2×#15668436C22semidirect
41. #19153749C22semidirect
42. #19153747C22semidirect
43. #19153741C22semidirect
44. #19153739C22semidirect
45. #19153739C22semidirect
46. #19153739C22semidirect
47. #19153738C22semidirect
48. #19153736C22semidirect
54. #107×C181C22semidirect
55. #106×C181C22semidirect
56. #103×C181C22semidirect
57. #103×C181C22semidirect
193. C181#3196semidirect
198. #485D181semidirect
238. #107C2×D181semidirect
240. #106C2×D181semidirect
245. #103C2×D181semidirect
246. #103C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260417912
C22⋊Q8×D1812
#260416202
#260416162
#260415992
#260415382
#260415362
#260415282
#260415262
#260414432
#260414392
#260414362
#260414192
#260412412
#485×C1812
#3196181