Finite Group Databasebeta

← #37545236

Subgroups — #37545236

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
2C22×#15668438C211yesmaximal, characteristic, complemented
3D4×Dic362C211yesmaximal, characteristic, complemented
4#26041925C211yesmaximal, characteristic
5#26041673C211yesmaximal, characteristic, complemented
6#26041525C211yesmaximal, characteristic, complemented
7#26041520C211yesmaximal, characteristic, complemented
8#26041502C211yesmaximal, characteristic, complemented
9#26041485C211yesmaximal, characteristic, complemented
10#26041443C211yesmaximal, characteristic, complemented
11#26041437C211yesmaximal, characteristic, complemented
12#26041432C211yesmaximal, characteristic, complemented
13#26041420C211yesmaximal, characteristic, complemented
14#26041381C211yesmaximal, characteristic, complemented
15#435×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C22×Dic362C2211yescharacteristic
18C22×Dic362C2211yescharacteristic, complemented
19C22×D724C2211yescharacteristic
20#19153849—24no—
21C2×Dic724C2211yescharacteristic, complemented
22C2×Dic724C2211yescharacteristic, complemented
23C2×Dic724C2211yescharacteristic, complemented
24C2×#15668438—24no—
25C2×#15668438C2211yescharacteristic
26C2×#15668438C2211yescharacteristic
27C2×#15668438C2211yescharacteristic
28C2×#15668438C2211yescharacteristic
29C2×#15668438C2211yescharacteristic
30C2×#15668438C2211yescharacteristic
31#19153816C2211yescharacteristic, complemented
32#19153815—21no—
33#19153804C2211yescharacteristic
34#19153801—21no—
35#19153795C2211yescharacteristic
36#19153795C2211yescharacteristic
37#19153795C2211yescharacteristic, complemented
38#19153793—21no—
39#19153793—21no—
40#19153790C2211yescharacteristic
41#19153789—21no—
42#19153784—21no—
43C4×Dic362C2211yescharacteristic, complemented
44C4×D724C2211yescharacteristic
45D4×Dic181—21no—
46C22⋊C4×D181—21no—
47#19153750C2211yescharacteristic, complemented
48#19153741C2211yescharacteristic, complemented
49#19153741C2211yescharacteristic, complemented
50#19153741C2211yescharacteristic, complemented
51#19153741C2211yescharacteristic, complemented
52#19153738C2211yescharacteristic
53#19153738C2211yescharacteristic
54C2×M16×C181C2211yescharacteristic, complemented
55C22×C8×C181C2211yescharacteristic
56C4×D4×C181C2211yescharacteristic
57C4⋊C8×C181C2211yescharacteristic, complemented
58C22⋊C8×C181C2211yescharacteristic
59C22⋊C8×C181C2211yescharacteristic, complemented
60#97×C181C2211yescharacteristic, complemented
61C23×D181—21no—
62#15668449—22no—
63#15668449—41no—
64C2×Dic362—22no—
65C2×Dic362—22no—
66C2×Dic362—21no—
67C2×Dic362—21no—
68C2×Dic362—41no—
69C2×Dic362—41no—
70C2×Dic362C2311yescharacteristic
71C2×Dic362C2311yescharacteristic
72C2×Dic362C2311yescharacteristic
73C2×Dic362C2311yescharacteristic
74C2×D724—22no—
75C2×D724—21no—
76C2×D724—41no—
77C2×D724C2311yescharacteristic
78C2×D724C2311yescharacteristic
79C2×C22⋊D181—21no—
80C2×C22⋊D181—21no—
81C22×Dic181—21no—
82C22×Dic181—21no—
83C2×C4×D181—21no—
84C2×C4×D181—21no—
85Dic724—22no—
86Dic724—22no—
87Dic724—41no—
88#15668438—22no—
89#15668438—22no—
90#15668438—22no—
91#15668438—22no—
92#15668438—41no—
93#15668438D414yescomplemented
94C4⋊Dic181—21no—
95C4⋊Dic181—21no—
96C4⋊Dic181C2311yescharacteristic
97C4⋊Dic181C2311yescharacteristic
98#15668428—21no—
99#15668428—21no—
100#15668428—21no—
101#15668428—21no—
102C22⋊Dic181—21no—
103C22⋊Dic181—21no—
104C22⋊Dic181—21no—
105#15668426—21no—
106#15668426—21no—
107#15668426—21no—
108#15668426—21no—
109C4×Dic181—21no—
110C2×D4×C181D411yescharacteristic
111C22×C4×C181C2311yescharacteristic
112C22×C4×C181C2311yescharacteristic
113M16×C181—22no—
114C2×C8×C181—22no—
115C2×C8×C181C2311yescharacteristic
116C2×C8×C181C2311yescharacteristic
117C2×C8×C181C2311yescharacteristic
118C2×C8×C181C2311yescharacteristic
119C4⋊C4×C181D411yescharacteristic
120C22⋊C4×C181D411yescharacteristic
121C22⋊C4×C181D411yescharacteristic
122C42×C181C2311yescharacteristic
123C22×D181—21no—
124C22×D181—21no—
125C22×D181—42no—
126C22×D181—42no—
127C22×D181—81no—
128Dic362—21no—
129Dic362—21no—
130Dic362—21no—
131Dic362—21no—
132Dic362—21no—
133Dic362—21no—
134Dic362—21no—
135Dic362—42no—
136Dic362—42no—
137Dic362C2×D412yes—
138D724—21no—
139D724—21no—
140D724—21no—
141D724—42no—
142D724C2×D412yes—
143C22⋊D181—42no—
144C22⋊D181—81no—
145C2×Dic181—21no—
146C2×Dic181—21no—
147C2×Dic181—21no—
148C2×Dic181—21no—
149C2×Dic181—21no—
150C2×Dic181—21no—
151C2×Dic181—42no—
152C2×Dic181—42no—
153C4×D181—42no—
154C4×D181—81no—
155C23×C181C2×D411yescharacteristic
156C23×C181C2×D411yescharacteristic
157D4×C181—22no—
158C2×C4×C181—21no—
159C2×C4×C181—21no—
160C2×C4×C181—21no—
161C2×C4×C181—21no—
162C2×C4×C181C2×D411yescharacteristic
163C2×C4×C181C2×D411yescharacteristic
164C2×C4×C181C2×D411yescharacteristic
165C2×C4×C181C2×D411yescharacteristic
166C2×C4×C181C2411yescharacteristic
167C8×C181—21no—
168C8×C181—21no—
169C8×C181—21no—
170C8×C181C2×D412yes—
171C22×C181—21no—
172C22×C181—21no—
173C22×C181—21no—
174C22×C181—21no—
175C22×C181—21no—
176C22×C181#13712yescomplemented
177C22×C181C22×D411yescharacteristic
178C2×D181—22no—
179C2×D181—41no—
180C2×D181—41no—
181C2×D181—41no—
182C2×D181—41no—
183C2×D181—41no—
184C2×D181—84no—
185C4×C181—21no—
186C4×C181—21no—
187C4×C181—21no—
188C4×C181—21no—
189C4×C181C22×D411yescharacteristic
190C4×C181H(5,𝔽2)11yescharacteristic
191Dic181—22no—
192Dic181—41no—
193Dic181—41no—
194Dic181—41no—
195Dic181—41no—
196Dic181—41no—
197C2×C181—22no—
198C2×C181—41no—
199C2×C181D4211yescharacteristic
200C2×C181C2×#13711yescharacteristic
201C2×C181#57711yescharacteristic
202D181—42no—
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#461—1811no—
215#452—1811no—
216#451—1811no—
217#450—1811no—
218#448—1811no—
219#440—1811no—
220#435D18111yescharacteristic, complemented
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×SD16—1811no—
236C2×M16C2×D18111yescharacteristic, complemented
237C22×C8C2×D18111yescharacteristic
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⋊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#103—1811no—
260#103—1811no—
261#102—1811no—
262#102—1811no—
263C22⋊C8C2×D18111yescharacteristic
264C22⋊C8C2×D18111yescharacteristic, complemented
265#97C2×D18111yescharacteristic, complemented
266C24—3621no—
267#41—3622no—
268#41—7241no—
269C2×Q8—3622no—
270C2×Q8—3622no—
271C2×Q8—3621no—
272C2×Q8—3621no—
273C2×Q8—1811no—
274C2×Q8—1811no—
275C2×Q8—1811no—
276C2×Q8—1811no—
277C2×Q8—7241no—
278C2×Q8—7241no—
279C2×D4—3622no—
280C2×D4—3621no—
281C2×D4—3621no—
282C2×D4—3621no—
283C2×D4—1811no—
284C2×D4—1811no—
285C2×D4—7241no—
286C2×D4D72411yescharacteristic
287C22×C4—3621no—
288C22×C4—3621no—
289C22×C4—3621no—
290C22×C4—3621no—
291C22×C4C22×D18111yescharacteristic
292C22×C4C22×D18111yescharacteristic
293Q16—3622no—
294Q16—3622no—
295Q16—7241no—
296SD16—3622no—
297SD16—3622no—
298SD16—3622no—
299SD16—3622no—
300SD16—1814no—
301SD16—7241no—
302M16—22no—
303C2×C8—22no—
304C2×C8C22×D18111yescharacteristic
305C2×C8C22×D18111yescharacteristic
306C2×C8C22×D18111yescharacteristic
307C2×C8C22×D18111yescharacteristic
308C4⋊C4—3621no—
309C4⋊C4—3621no—
310C4⋊C4—3621no—
311C4⋊C4—3621no—
312C4⋊C4—3621no—
313C4⋊C4—3621no—
314C4⋊C4—1811no—
315C4⋊C4—1811no—
316C4⋊C4D72411yescharacteristic
317C22⋊C4—3621no—
318C22⋊C4—3621no—
319C22⋊C4—3621no—
320C22⋊C4—3621no—
321C22⋊C4—3621no—
322C22⋊C4—3621no—
323C22⋊C4—3621no—
324C22⋊C4D72411yescharacteristic
325C22⋊C4D72411yescharacteristic
326C42—3621no—
327C42C22×D18111yescharacteristic
328C23—3621no—
329C23—3621no—
330C23—14481no—
331C23—7242no—
332C23—7242no—
333C23C2×D72411yescharacteristic
334C23C2×D72411yescharacteristic
335Q8—3621no—
336Q8—3621no—
337Q8—3621no—
338Q8—3621no—
339Q8—3621no—
340Q8—3621no—
341Q8—3621no—
342Q8—1812no—
343Q8—7242no—
344Q8—7242no—
345D4—22no—
346D4—3621no—
347D4—3621no—
348D4—3621no—
349D4—14481no—
350D4—1812no—
351D4—7242no—
352D4—7242no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—3621no—
358C2×C4—3621no—
359C2×C4—3621no—
360C2×C4—3621no—
361C2×C4—3621no—
362C2×C4—3621no—
363C2×C4—14481no—
364C2×C4—7242no—
365C2×C4—7242no—
366C2×C4—7242no—
367C2×C4C2×D72411yescharacteristic
368C2×C4C2×D72411yescharacteristic
369C2×C4C2×D72411yescharacteristic
370C2×C4C2×D72411yescharacteristic
371C2×C4C23×D18111yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8D4×D18112yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—3622no—
382C22—14484no—
383C22—7241no—
384C22—7241no—
385C22—7241no—
386C22—7241no—
387C22—7241no—
388C22#1915384912yescomplemented
389C22C22×D72411yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—3622no—
395C4—7241no—
396C4—7241no—
397C4—7241no—
398C4—7241no—
399C4—7241no—
400C4C2×D4×D18111yescharacteristic
401C4#1915386211yescharacteristic
402C2—22no—
403C2—41no—
404C2—14481no—
405C2—7242no—
406C2D4×D72411yescharacteristic
407C2C2×#1915384911yescharacteristic
408C2#2604209111yescharacteristic
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#435×C181C2
socleC22×C181C22×D4
prime radical#26041925C2
hypercenter#435D181
hypocenterC181#3266
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#19153849C2semidirect
2. C22×#15668438C2semidirect
3. D4×Dic362C2semidirect
5. #26041673C2semidirect
6. #26041525C2semidirect
7. #26041520C2semidirect
8. #26041502C2semidirect
9. #26041485C2semidirect
10. #26041443C2semidirect
11. #26041437C2semidirect
12. #26041432C2semidirect
13. #26041420C2semidirect
14. #26041381C2semidirect
15. #435×C181C2semidirect
18. C22×Dic362C22semidirect
21. C2×Dic724C22semidirect
22. C2×Dic724C22semidirect
23. C2×Dic724C22semidirect
31. #19153816C22semidirect
37. #19153795C22semidirect
43. C4×Dic362C22semidirect
47. #19153750C22semidirect
48. #19153741C22semidirect
49. #19153741C22semidirect
50. #19153741C22semidirect
51. #19153741C22semidirect
54. C2×M16×C181C22semidirect
57. C4⋊C8×C181C22semidirect
59. C22⋊C8×C181C22semidirect
60. #97×C181C22semidirect
93. #15668438D4semidirect
176. C22×C181#137semidirect
204. C181#3266semidirect
220. #435D181semidirect
236. C2×M16C2×D181semidirect
256. C4⋊C8C2×D181semidirect
264. C22⋊C8C2×D181semidirect
265. #97C2×D181semidirect
388. C22#19153849semidirect

Maximal subgroups

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

subgroupindex
C2×#191538492
C22×#156684382
D4×Dic3622
#260419252
#260416732
#260415252
#260415202
#260415022
#260414852
#260414432
#260414372
#260414322
#260414202
#260413812
#435×C1812
#3266181