Finite Group Databasebeta

← #37545237

Subgroups — #37545237

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×#19153845C211yesmaximal, characteristic, complemented
2C2×SD16×D181C211yesmaximal, characteristic, complemented
3#26041944C211yesmaximal, characteristic, complemented
4#26041924C211yesmaximal, characteristic
5#26041671C211yesmaximal, characteristic, complemented
6#26041528C211yesmaximal, characteristic, complemented
7#26041519C211yesmaximal, characteristic, complemented
8#26041502C211yesmaximal, characteristic, complemented
9#26041438C211yesmaximal, characteristic, complemented
10#26041436C211yesmaximal, characteristic, complemented
11#26041428C211yesmaximal, characteristic, complemented
12#26041419C211yesmaximal, characteristic, complemented
13#26041379C211yesmaximal, characteristic, complemented
14#26041349C211yesmaximal, characteristic, complemented
15#461×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic
18C2×Q8×D181C2211yescharacteristic, complemented
19C2×D4×D181C2211yescharacteristic
20#19153845—24no—
21C2×Dic724C2211yescharacteristic, complemented
22C2×#15668439C2211yescharacteristic, complemented
23C2×#15668439C2211yescharacteristic, complemented
24C2×#15668438C2211yescharacteristic
25C2×#15668437C2211yescharacteristic
26C2×#15668436C2211yescharacteristic
27C2×#15668436C2211yescharacteristic
28SD16×D181—24no—
29C2×#15668432C2211yescharacteristic, complemented
30C2×C8×D181C2211yescharacteristic
31#19153815—21no—
32#19153804C2211yescharacteristic
33#19153800—21no—
34#19153796C2211yescharacteristic
35#19153794—21no—
36#19153794—21no—
37#19153792C2211yescharacteristic
38#19153792C2211yescharacteristic, complemented
39#19153791C2211yescharacteristic
40#19153790—21no—
41#19153786—21no—
42C4×Dic362C2211yescharacteristic, complemented
43C4×D724C2211yescharacteristic
44#19153778—21no—
45C2×#15668426—21no—
46#19153749C2211yescharacteristic, complemented
47#19153741C2211yescharacteristic, complemented
48#19153739C2211yescharacteristic, complemented
49#19153738C2211yescharacteristic
50#19153737C2211yescharacteristic
51#19153728C2211yescharacteristic
52#19153728C2211yescharacteristic, complemented
53#19153727C2211yescharacteristic, complemented
54C2×SD16×C181C2211yescharacteristic
55C2×SD16×C181C2211yescharacteristic
56C4⋊Q8×C181C2211yescharacteristic, complemented
57C4×D4×C181C2211yescharacteristic
58C4⋊C8×C181C2211yescharacteristic, complemented
59#103×C181C2211yescharacteristic, complemented
60#103×C181C2211yescharacteristic, complemented
61C23×D181—21no—
62#15668448—22no—
63#15668448—41no—
64C2×Dic362—21no—
65C2×Dic362—21no—
66C2×Dic362C2311yescharacteristic
67C2×Dic362C2311yescharacteristic
68Q8×D181—22no—
69Q8×D181—22no—
70Q8×D181—41no—
71Q8×D181—41no—
72C2×D724—21no—
73C2×D724D411yescharacteristic
74C2×D724C2311yescharacteristic
75C2×C22⋊D181—21no—
76C2×C22⋊D181—21no—
77D4×D181—22no—
78D4×D181—41no—
79C22×Dic181—21no—
80C2×C4×D181—21no—
81C2×C4×D181—21no—
82C2×C4×D181C2311yescharacteristic
83C2×C4×D181C2311yescharacteristic
84Dic724—22no—
85#15668439—22no—
86#15668439—41no—
87#15668438—22no—
88#15668437D414yescomplemented
89#15668436—22no—
90#15668436—41no—
91#15668432—22no—
92C2×#13089678C2311yescharacteristic
93C2×#13089678C2311yescharacteristic
94C8×D181—22no—
95C4⋊Dic181—21no—
96C4⋊Dic181D411yescharacteristic
97C4⋊Dic181C2311yescharacteristic
98#15668428—21no—
99#15668428—21no—
100#15668428—21no—
101#15668426—21no—
102#15668426—21no—
103#15668426—21no—
104#15668426—21no—
105#15668426—21no—
106#15668426—21no—
107#15668426D411yescharacteristic
108#15668426D411yescharacteristic
109C4×Dic181—21no—
110C2×Q8×C181C2311yescharacteristic
111C2×Q8×C181C2311yescharacteristic
112C2×D4×C181C2311yescharacteristic
113C22×C4×C181—21no—
114SD16×C181—22no—
115SD16×C181—22no—
116C2×C8×C181C2311yescharacteristic
117C2×C8×C181C2311yescharacteristic
118C4⋊C4×C181—21no—
119C4⋊C4×C181—21no—
120C4⋊C4×C181C2311yescharacteristic
121C22⋊C4×C181—21no—
122C42×C181C2311yescharacteristic
123C22×D181—21no—
124C22×D181—42no—
125C22×D181—42no—
126C22×D181—81no—
127C22×D181C2×D411yescharacteristic
128C22×D181C2×D411yescharacteristic
129Dic362—21no—
130Dic362—21no—
131Dic362—21no—
132Dic362—42no—
133Dic362—42no—
134Dic362C2×D412yes—
135D724—22no—
136D724—21no—
137D724—21no—
138D724—81no—
139C22⋊D181—42no—
140C22⋊D181—42no—
141C2×Dic181—21no—
142C2×Dic181—21no—
143C2×Dic181—21no—
144C2×Dic181—42no—
145C2×Dic181C2×D411yescharacteristic
146C2×Dic181C2×D411yescharacteristic
147C4×D181—21no—
148C4×D181—21no—
149C4×D181—21no—
150C4×D181—21no—
151C4×D181—42no—
152C4×D181—42no—
153#13089678—21no—
154#13089678C2×D412yes—
155C23×C181—21no—
156Q8×C181—21no—
157Q8×C181—21no—
158Q8×C181—21no—
159Q8×C181—21no—
160D4×C181—21no—
161D4×C181C2×D412yes—
162C2×C4×C181—21no—
163C2×C4×C181—21no—
164C2×C4×C181—21no—
165C2×C4×C181—81no—
166C2×C4×C181C2×D411yescharacteristic
167C2×C4×C181C2×D411yescharacteristic
168C2×C4×C181C2411yescharacteristic
169C8×C181—21no—
170C8×C181—21no—
171C22×C181—22no—
172C22×C181—41no—
173C22×C181—41no—
174C22×C181C22×D411yescharacteristic
175C2×D181—21no—
176C2×D181—21no—
177C2×D181—21no—
178C2×D181—21no—
179C2×D181—21no—
180C2×D181—41no—
181C2×D181—41no—
182C2×D181—41no—
183C2×D181—84no—
184C2×D181#13712yescomplemented
185C4×C181—21no—
186C4×C181—21no—
187C4×C181—41no—
188C4×C181—41no—
189C4×C181—41no—
190C4×C181C22×D411yescharacteristic
191C4×C181H(5,𝔽2)11yescharacteristic
192Dic181—21no—
193Dic181—21no—
194Dic181—22no—
195Dic181—41no—
196Dic181—41no—
197C2×C181—42no—
198C2×C181D4211yescharacteristic
199C2×C181C2×#13711yescharacteristic
200C2×C181#57711yescharacteristic
201D181—22no—
202D181—41no—
203D181—81no—
204C181#326611yescharacteristic, complemented, Sylow
205#3266—1811nomaximal, Sylow
206C2×#137—1811no—
207C22×SD16—1811no—
208D4×Q8—1811no—
209#546—1811no—
210#497—1811no—
211#468—1811no—
212#467—1811no—
213#464—1811no—
214#461D18111yescharacteristic, complemented
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×SD16C2×D18111yescharacteristic
235C2×SD16C2×D18111yescharacteristic
236C2×M16—1811no—
237C22×C8—1811no—
238C4⋊Q8—3621no—
239C4⋊Q8C2×D18111yescharacteristic, complemented
240#124—1811no—
241#123—3621no—
242C22⋊Q8—3621no—
243C22⋊Q8—3621no—
244C22⋊Q8—1811no—
245C22⋊Q8—1811no—
246C22⋊Q8—1811no—
247#121—3621no—
248#121—1811no—
249#120—3621no—
250C4×Q8—1811no—
251C4×D4—3621no—
252C4×D4—1811no—
253C4×D4C2×D18111yescharacteristic
254C2×C22⋊C4—3621no—
255#107—1811no—
256C4⋊C8C2×D18111yescharacteristic, complemented
257#103—1811no—
258#103—1811no—
259#103C2×D18111yescharacteristic, complemented
260#103C2×D18111yescharacteristic, complemented
261#102—1811no—
262#102—1811no—
263C22⋊C8—1811no—
264C22⋊C8—1811no—
265#97—1811no—
266C24—3621no—
267#41—3622no—
268#41—7241no—
269C2×Q8—3621no—
270C2×Q8—3621no—
271C2×Q8—3622no—
272C2×Q8—3622no—
273C2×Q8—1811no—
274C2×Q8—1811no—
275C2×Q8—7241no—
276C2×Q8—7241no—
277C2×Q8C22×D18111yescharacteristic
278C2×Q8C22×D18111yescharacteristic
279C2×D4—3621no—
280C2×D4—3622no—
281C2×D4—3621no—
282C2×D4—3621no—
283C2×D4—1811no—
284C2×D4—1811no—
285C2×D4—7241no—
286C2×D4C22×D18111yescharacteristic
287C22×C4—21no—
288C22×C4—3621no—
289C22×C4—3621no—
290C22×C4—3621no—
291C22×C4—1811no—
292C22×C4—1811no—
293Q16—3622no—
294Q16—3622no—
295Q16—7241no—
296SD16—22no—
297SD16—22no—
298SD16—3622no—
299SD16—3622no—
300SD16—1814no—
301SD16—7241no—
302M16—3622no—
303C2×C8—3622no—
304C2×C8—1811no—
305C2×C8—1811no—
306C2×C8C22×D18111yescharacteristic
307C2×C8C22×D18111yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—3621no—
311C4⋊C4—3621no—
312C4⋊C4—3621no—
313C4⋊C4—3621no—
314C4⋊C4—1811no—
315C4⋊C4—1811no—
316C4⋊C4C22×D18111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—3621no—
319C22⋊C4—3621no—
320C22⋊C4—3621no—
321C22⋊C4—3621no—
322C22⋊C4—3621no—
323C22⋊C4—3621no—
324C22⋊C4—1811no—
325C22⋊C4—1811no—
326C42—3621no—
327C42C22×D18111yescharacteristic
328C23—21no—
329C23—3621no—
330C23—14481no—
331C23—1811no—
332C23—1811no—
333C23—7242no—
334C23—7242no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—3621no—
340Q8—3621no—
341Q8—3621no—
342Q8—1812no—
343Q8—7242no—
344Q8—7242no—
345D4—21no—
346D4—3622no—
347D4—3621no—
348D4—3621no—
349D4—14481no—
350D4—7242no—
351D4—7242no—
352D4D4×D18112yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—81no—
357C2×C4—3621no—
358C2×C4—3621no—
359C2×C4—3621no—
360C2×C4—3621no—
361C2×C4—3621no—
362C2×C4—3621no—
363C2×C4—3621no—
364C2×C4—1811no—
365C2×C4—1811no—
366C2×C4—7242no—
367C2×C4—7242no—
368C2×C4—7242no—
369C2×C4D4×D18111yescharacteristic
370C2×C4D4×D18111yescharacteristic
371C2×C4C23×D18111yescharacteristic
372C8—21no—
373C8—21no—
374C8—3621no—
375C8—1812no—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—3621no—
380C22—3621no—
381C22—3621no—
382C22—3621no—
383C22—3621no—
384C22—14484no—
385C22—1812no—
386C22—7241no—
387C22—7241no—
388C22—7241no—
389C22C2×D4×D18111yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—3621no—
396C4—3621no—
397C4—3622no—
398C4—7241no—
399C4—7241no—
400C4C2×D4×D18111yescharacteristic
401C4#1915386211yescharacteristic
402C2—42no—
403C2—3622no—
404C2—14481no—
405C2—7241no—
406C2#2604191311yescharacteristic
407C2#137×D18111yescharacteristic
408C2#2604208511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#461×C181C2
socleC22×C181C22×D4
prime radical#26041924C2
hypercenter#461D181
hypocenterC181#3266
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#19153845C2semidirect
2. C2×SD16×D181C2semidirect
3. #26041944C2semidirect
5. #26041671C2semidirect
6. #26041528C2semidirect
7. #26041519C2semidirect
8. #26041502C2semidirect
9. #26041438C2semidirect
10. #26041436C2semidirect
11. #26041428C2semidirect
12. #26041419C2semidirect
13. #26041379C2semidirect
14. #26041349C2semidirect
15. #461×C181C2semidirect
18. C2×Q8×D181C22semidirect
21. C2×Dic724C22semidirect
22. C2×#15668439C22semidirect
23. C2×#15668439C22semidirect
29. C2×#15668432C22semidirect
38. #19153792C22semidirect
42. C4×Dic362C22semidirect
46. #19153749C22semidirect
47. #19153741C22semidirect
48. #19153739C22semidirect
52. #19153728C22semidirect
53. #19153727C22semidirect
56. C4⋊Q8×C181C22semidirect
58. C4⋊C8×C181C22semidirect
59. #103×C181C22semidirect
60. #103×C181C22semidirect
88. #15668437D4semidirect
184. C2×D181#137semidirect
204. C181#3266semidirect
214. #461D181semidirect
239. C4⋊Q8C2×D181semidirect
256. C4⋊C8C2×D181semidirect
259. #103C2×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×#191538452
C2×SD16×D1812
#260419442
#260419242
#260416712
#260415282
#260415192
#260415022
#260414382
#260414362
#260414282
#260414192
#260413792
#260413492
#461×C1812
#3266181