Finite Group Databasebeta

← #37545238

Subgroups — #37545238

Order 23168.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153849C211yesmaximal, characteristic, complemented
2C2×SD16×D181C211yesmaximal, characteristic, complemented
3#26041942C211yesmaximal, characteristic, complemented
4#26041926C211yesmaximal, characteristic
5#26041670C211yesmaximal, characteristic, complemented
6#26041521C211yesmaximal, characteristic, complemented
7#26041500C211yesmaximal, characteristic, complemented
8#26041483C211yesmaximal, characteristic, complemented
9#26041439C211yesmaximal, characteristic, complemented
10#26041436C211yesmaximal, characteristic, complemented
11#26041431C211yesmaximal, characteristic, complemented
12#26041417C211yesmaximal, characteristic, complemented
13#26041378C211yesmaximal, characteristic, complemented
14#26041348C211yesmaximal, characteristic, complemented
15#468×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C22×Dic362C2211yescharacteristic, complemented
18C2×Q8×D181C2211yescharacteristic
19C2×D4×D181C2211yescharacteristic
20#19153849—24no—
21C2×Dic724C2211yescharacteristic, complemented
22C2×Dic724C2211yescharacteristic, complemented
23C2×#15668439C2211yescharacteristic, complemented
24C2×#15668438C2211yescharacteristic
25C2×#15668438C2211yescharacteristic
26C2×#15668437C2211yescharacteristic
27C2×#15668437C2211yescharacteristic
28C2×#15668436C2211yescharacteristic
29SD16×D181—24no—
30C2×C8×D181C2211yescharacteristic
31#19153815—21no—
32#19153814C2211yescharacteristic, complemented
33#19153803C2211yescharacteristic
34#19153798—21no—
35#19153796C2211yescharacteristic
36#19153796C2211yescharacteristic
37#19153794—21no—
38#19153793—21no—
39#19153787—21no—
40#19153784—21no—
41#19153781C2211yescharacteristic, complemented
42#19153778C2211yescharacteristic
43C4×C22⋊D181C2211yescharacteristic
44#19153776—21no—
45C22⋊C4×D181—21no—
46#19153745C2211yescharacteristic, complemented
47#19153739C2211yescharacteristic, complemented
48#19153739C2211yescharacteristic, complemented
49#19153739C2211yescharacteristic, complemented
50#19153736C2211yescharacteristic
51C22⋊#13089678C2211yescharacteristic, complemented
52#19153728C2211yescharacteristic
53#19153726C2211yescharacteristic, complemented
54C2×SD16×C181C2211yescharacteristic
55C2×M16×C181C2211yescharacteristic, complemented
56C22⋊Q8×C181C2211yescharacteristic, complemented
57#121×C181C2211yescharacteristic
58#107×C181C2211yescharacteristic, complemented
59#103×C181C2211yescharacteristic, complemented
60#102×C181C2211yescharacteristic
61C23×D181—21no—
62#15668449—22no—
63#15668449—41no—
64C2×Dic362—21no—
65C2×Dic362—22no—
66C2×Dic362—21no—
67C2×Dic362—41no—
68C2×Dic362C2311yescharacteristic
69C2×Dic362C2311yescharacteristic
70C2×Dic362C2311yescharacteristic
71Q8×D181—22no—
72Q8×D181—41no—
73C2×D724C2311yescharacteristic
74C2×C22⋊D181—21no—
75C2×C22⋊D181—21no—
76C2×C22⋊D181D411yescharacteristic
77D4×D181—22no—
78D4×D181—41no—
79C22×Dic181—21no—
80C2×C4×D181—21no—
81C2×C4×D181—21no—
82C2×C4×D181—21no—
83C2×C4×D181C2311yescharacteristic
84Dic724—22no—
85Dic724—22no—
86#15668439—41no—
87#15668438—22no—
88#15668438D414yescomplemented
89#15668437—22no—
90#15668437—41no—
91#15668436—22no—
92C2×#13089678C2311yescharacteristic
93C2×#13089678C2311yescharacteristic
94C8×D181—22no—
95C4⋊Dic181—21no—
96C4⋊Dic181—21no—
97#15668428—21no—
98#15668428—21no—
99#15668428—21no—
100#15668428D411yescharacteristic
101C22⋊Dic181—21no—
102C22⋊Dic181—21no—
103C22⋊Dic181D411yescharacteristic
104#15668426—21no—
105#15668426—21no—
106#15668426—21no—
107#15668426D411yescharacteristic
108C4×Dic181—21no—
109C4×Dic181C2311yescharacteristic
110C2×Q8×C181C2311yescharacteristic
111C2×D4×C181—21no—
112C2×D4×C181C2311yescharacteristic
113C22×C4×C181C2311yescharacteristic
114SD16×C181—22no—
115M16×C181—22no—
116C2×C8×C181C2311yescharacteristic
117C2×C8×C181C2311yescharacteristic
118C4⋊C4×C181—21no—
119C4⋊C4×C181C2311yescharacteristic
120C4⋊C4×C181C2311yescharacteristic
121C22⋊C4×C181—21no—
122C22⋊C4×C181—21no—
123C22×D181—21no—
124C22×D181—42no—
125C22×D181—42no—
126C22×D181—81no—
127C22×D181C2×D411yescharacteristic
128Dic362—21no—
129Dic362—21no—
130Dic362—21no—
131Dic362—21no—
132Dic362—21no—
133Dic362—42no—
134Dic362—42no—
135Dic362C2×D412yes—
136D724—21no—
137D724C2×D412yes—
138C22⋊D181—22no—
139C22⋊D181—42no—
140C22⋊D181—42no—
141C2×Dic181—21no—
142C2×Dic181—21no—
143C2×Dic181—21no—
144C2×Dic181—42no—
145C2×Dic181C2×D411yescharacteristic
146C2×Dic181C2×D411yescharacteristic
147C2×Dic181C2×D411yescharacteristic
148C4×D181—21no—
149C4×D181—21no—
150C4×D181—42no—
151C4×D181—42no—
152C4×D181—81no—
153#13089678—21no—
154#13089678—21no—
155C23×C181—21no—
156C23×C181C2×D411yescharacteristic
157Q8×C181—21no—
158Q8×C181—21no—
159D4×C181—21no—
160D4×C181—21no—
161D4×C181—81no—
162C2×C4×C181—21no—
163C2×C4×C181—21no—
164C2×C4×C181—21no—
165C2×C4×C181—21no—
166C2×C4×C181—21no—
167C2×C4×C181C2×D411yescharacteristic
168C2×C4×C181C2411yescharacteristic
169C8×C181—21no—
170C8×C181C2×D412yes—
171C22×C181—21no—
172C22×C181—21no—
173C22×C181—21no—
174C22×C181—41no—
175C22×C181—41no—
176C22×C181—41no—
177C22×C181C22×D411yescharacteristic
178C2×D181—21no—
179C2×D181—21no—
180C2×D181—22no—
181C2×D181—41no—
182C2×D181—41no—
183C2×D181—84no—
184C2×D181#13712yescomplemented
185C4×C181—21no—
186C4×C181—41no—
187C4×C181—41no—
188C4×C181—41no—
189C4×C181C22×D411yescharacteristic
190C4×C181H(5,𝔽2)11yescharacteristic
191Dic181—21no—
192Dic181—22no—
193Dic181—21no—
194Dic181—21no—
195Dic181—41no—
196Dic181—41no—
197C2×C181—41no—
198C2×C181—81no—
199C2×C181D4211yescharacteristic
200C2×C181C2×#13711yescharacteristic
201C2×C181#57711yescharacteristic
202D181—22no—
203D181—42no—
204C181#326611yescharacteristic, complemented, Sylow
205#3266—1811nomaximal, Sylow
206C2×#137—1811no—
207C22×SD16—1811no—
208D4×Q8—1811no—
209#546—1811no—
210#497—1811no—
211#468D18111yescharacteristic, complemented
212#467—1811no—
213#464—1811no—
214#461—1811no—
215#452—1811no—
216#451—1811no—
217#450—1811no—
218#448—1811no—
219#440—1811no—
220#435—1811no—
221C2×#41—1811no—
222C22×Q8—1811no—
223C22×Q8—1811no—
224C22×D4—1811no—
225#137—3624no—
226C2×Q16—1811no—
227C2×Q16—1811no—
228C2×Q16—1811no—
229C2×SD16—3624no—
230C2×SD16—1811no—
231C2×SD16—1811no—
232C2×SD16—1811no—
233C2×SD16—1811no—
234C2×SD16—1811no—
235C2×SD16C2×D18111yescharacteristic
236C2×M16C2×D18111yescharacteristic, complemented
237C22×C8—1811no—
238C4⋊Q8—3621no—
239C4⋊Q8—1811no—
240#124—1811no—
241#123—3621no—
242C22⋊Q8—3621no—
243C22⋊Q8—3621no—
244C22⋊Q8—1811no—
245C22⋊Q8—1811no—
246C22⋊Q8C2×D18111yescharacteristic, complemented
247#121—3621no—
248#121C2×D18111yescharacteristic
249#120—3621no—
250C4×Q8—1811no—
251C4×D4—3621no—
252C4×D4—1811no—
253C4×D4—1811no—
254C2×C22⋊C4—3621no—
255#107C2×D18111yescharacteristic, complemented
256C4⋊C8—1811no—
257#103—1811no—
258#103—1811no—
259#103—1811no—
260#103C2×D18111yescharacteristic, complemented
261#102—1811no—
262#102C2×D18111yescharacteristic
263C22⋊C8—1811no—
264C22⋊C8—1811no—
265#97—1811no—
266C24—3621no—
267#41—3622no—
268#41—7241no—
269C2×Q8—3621no—
270C2×Q8—3622no—
271C2×Q8—3621no—
272C2×Q8—3622no—
273C2×Q8—1811no—
274C2×Q8—1811no—
275C2×Q8—1811no—
276C2×Q8—7241no—
277C2×Q8—7241no—
278C2×Q8C22×D18111yescharacteristic
279C2×D4—21no—
280C2×D4—3622no—
281C2×D4—3621no—
282C2×D4—3621no—
283C2×D4—1811no—
284C2×D4—1811no—
285C2×D4—7241no—
286C2×D4C22×D18111yescharacteristic
287C22×C4—3621no—
288C22×C4—3621no—
289C22×C4—3621no—
290C22×C4—3621no—
291C22×C4—1811no—
292C22×C4C22×D18111yescharacteristic
293Q16—3622no—
294Q16—3622no—
295Q16—7241no—
296SD16—22no—
297SD16—3622no—
298SD16—3622no—
299SD16—3622no—
300SD16—1814no—
301SD16—7241no—
302M16—22no—
303C2×C8—3622no—
304C2×C8—1811no—
305C2×C8—1811no—
306C2×C8C22×D18111yescharacteristic
307C2×C8C22×D18111yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—3621no—
310C4⋊C4—3621no—
311C4⋊C4—3621no—
312C4⋊C4—3621no—
313C4⋊C4—3621no—
314C4⋊C4—1811no—
315C4⋊C4C22×D18111yescharacteristic
316C4⋊C4C22×D18111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—3621no—
320C22⋊C4—3621no—
321C22⋊C4—3621no—
322C22⋊C4—3621no—
323C22⋊C4—3621no—
324C22⋊C4—1811no—
325C22⋊C4—1811no—
326C42—3621no—
327C42—1811no—
328C23—21no—
329C23—3621no—
330C23—14481no—
331C23—1811no—
332C23—7242no—
333C23—7242no—
334C23D4×D18111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—3621no—
338Q8—3621no—
339Q8—3621no—
340Q8—3621no—
341Q8—3621no—
342Q8—1812no—
343Q8—7242no—
344Q8—7242no—
345D4—21no—
346D4—21no—
347D4—81no—
348D4—3622no—
349D4—3621no—
350D4—1812no—
351D4—7242no—
352D4—7242no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—3621no—
359C2×C4—3621no—
360C2×C4—3621no—
361C2×C4—3621no—
362C2×C4—3621no—
363C2×C4—14481no—
364C2×C4—1811no—
365C2×C4—1811no—
366C2×C4—1811no—
367C2×C4—7242no—
368C2×C4—7242no—
369C2×C4—7242no—
370C2×C4D4×D18111yescharacteristic
371C2×C4C23×D18111yescharacteristic
372C8—21no—
373C8—3621no—
374C8—3621no—
375C8D4×D18112yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—3621no—
383C22—3621no—
384C22—3622no—
385C22—14484no—
386C22—1812no—
387C22—7241no—
388C22—7241no—
389C22C2×D4×D18111yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—3621no—
395C4—3622no—
396C4—3621no—
397C4—3621no—
398C4—7241no—
399C4—7241no—
400C4C2×D4×D18111yescharacteristic
401C4#1915386211yescharacteristic
402C2—41no—
403C2—81no—
404C2—3622no—
405C2—7242no—
406C2#2604191311yescharacteristic
407C2#137×D18111yescharacteristic
408C2#2604208911yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#468×C181C2
socleC22×C181C22×D4
prime radical#26041926C2
hypercenter#468D181
hypocenterC181#3266
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#19153849C2semidirect
2. C2×SD16×D181C2semidirect
3. #26041942C2semidirect
5. #26041670C2semidirect
6. #26041521C2semidirect
7. #26041500C2semidirect
8. #26041483C2semidirect
9. #26041439C2semidirect
10. #26041436C2semidirect
11. #26041431C2semidirect
12. #26041417C2semidirect
13. #26041378C2semidirect
14. #26041348C2semidirect
15. #468×C181C2semidirect
17. C22×Dic362C22semidirect
21. C2×Dic724C22semidirect
22. C2×Dic724C22semidirect
23. C2×#15668439C22semidirect
32. #19153814C22semidirect
41. #19153781C22semidirect
46. #19153745C22semidirect
47. #19153739C22semidirect
48. #19153739C22semidirect
49. #19153739C22semidirect
51. C22⋊#13089678C22semidirect
53. #19153726C22semidirect
55. C2×M16×C181C22semidirect
56. C22⋊Q8×C181C22semidirect
58. #107×C181C22semidirect
59. #103×C181C22semidirect
88. #15668438D4semidirect
184. C2×D181#137semidirect
204. C181#3266semidirect
211. #468D181semidirect
236. C2×M16C2×D181semidirect
246. C22⋊Q8C2×D181semidirect
255. #107C2×D181semidirect
260. #103C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538492
C2×SD16×D1812
#260419422
#260419262
#260416702
#260415212
#260415002
#260414832
#260414392
#260414362
#260414312
#260414172
#260413782
#260413482
#468×C1812
#3266181