Finite Group Databasebeta

← #27182662

Subgroups — #27182662

Order 12928.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692283C211yesmaximal, characteristic, complemented
2C2×#15692277C211yesmaximal, characteristic, complemented
3#20265832C211yesmaximal, characteristic, complemented
4#20265823C211yesmaximal, characteristic, complemented
5#20265571C211yesmaximal, characteristic, complemented
6#20265435C211yesmaximal, characteristic, complemented
7#20265406C211yesmaximal, characteristic, complemented
8#20265393C211yesmaximal, characteristic, complemented
9#20265383C211yesmaximal, characteristic, complemented
10#20265341C211yesmaximal, characteristic, complemented
11#20265341C211yesmaximal, characteristic, complemented
12#20265330C211yesmaximal, characteristic, complemented
13#20265324C211yesmaximal, characteristic, complemented
14#20265290C211yesmaximal, characteristic, complemented
15#435×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C2×#13108074C2211yescharacteristic
18C2×#13108074C2211yescharacteristic, complemented
19C22×D404C2211yescharacteristic
20#15692283—24no—
21#15692277—24no—
22C2×Dic404C2211yescharacteristic, complemented
23C2×Dic404C2211yescharacteristic, complemented
24C2×#13108063C2211yescharacteristic
25C2×#13108063C2211yescharacteristic
26C2×#13108063C2211yescharacteristic
27C2×#13108063C2211yescharacteristic
28C2×#13108063C2211yescharacteristic, complemented
29C2×D808C2211yescharacteristic
30C2×D808C2211yescharacteristic
31#15692255C2211yescharacteristic, complemented
32#15692243C2211yescharacteristic
33#15692240—21no—
34#15692237—21no—
35#15692237—21no—
36#15692234C2211yescharacteristic
37#15692234C2211yescharacteristic
38#15692234C2211yescharacteristic, complemented
39#15692229C2211yescharacteristic
40#15692228—21no—
41#15692226—21no—
42#15692223—21no—
43C4×D404C2211yescharacteristic
44C4×D404C2211yescharacteristic, complemented
45C4⋊C4×D101—21no—
46C22⋊C4×D101—21no—
47#15692187C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692180C2211yescharacteristic, complemented
50#15692180C2211yescharacteristic, complemented
51#15692177C2211yescharacteristic
52#15692177C2211yescharacteristic
53#15692177C2211yescharacteristic, complemented
54C2×M16×C101C2211yescharacteristic
55C22×C8×C101C2211yescharacteristic, complemented
56C4×D4×C101C2211yescharacteristic
57C4⋊C8×C101C2211yescharacteristic, complemented
58C22⋊C8×C101C2211yescharacteristic
59C22⋊C8×C101C2211yescharacteristic, complemented
60#97×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62#13108074—22no—
63#13108074—22no—
64#13108074—22no—
65#13108074—41no—
66#13108074—41no—
67#13108074—41no—
68C2×Dic202C2311yescharacteristic
69C2×Dic202C2311yescharacteristic
70C2×Dic202C2311yescharacteristic
71C2×D404—22no—
72C2×D404—21no—
73C2×D404—41no—
74C2×D404C2311yescharacteristic
75C2×D404C2311yescharacteristic
76C2×D404C2311yescharacteristic
77C2×C22⋊D101—21no—
78C2×C22⋊D101—21no—
79C2×C22⋊D101—21no—
80C2×C22⋊D101—21no—
81C2×C4×D101—21no—
82C2×C4×D101—21no—
83C2×C4×D101—21no—
84C2×C4×D101—21no—
85C2×C4×D101—21no—
86Dic404—22no—
87Dic404—41no—
88#13108063—22no—
89#13108063—22no—
90#13108063—22no—
91#13108063—22no—
92#13108063—41no—
93D808—22no—
94D808D414yescomplemented
95C4⋊Dic101—21no—
96C4⋊Dic101—21no—
97C4⋊Dic101C2311yescharacteristic
98C4⋊Dic101C2311yescharacteristic
99#13108053—21no—
100#13108053—21no—
101#13108053—21no—
102C22⋊Dic101—21no—
103C22⋊Dic101—21no—
104C22⋊Dic101—21no—
105#13108051—21no—
106#13108051—21no—
107#13108051—21no—
108#13108051—21no—
109#13108051—21no—
110C2×D4×C101D411yescharacteristic
111C22×C4×C101C2311yescharacteristic
112C22×C4×C101C2311yescharacteristic
113M16×C101—22no—
114C2×C8×C101—22no—
115C2×C8×C101C2311yescharacteristic
116C2×C8×C101C2311yescharacteristic
117C2×C8×C101C2311yescharacteristic
118C2×C8×C101C2311yescharacteristic
119C4⋊C4×C101D411yescharacteristic
120C22⋊C4×C101D411yescharacteristic
121C22⋊C4×C101D411yescharacteristic
122C42×C101C2311yescharacteristic
123C23×C101C2×D411yescharacteristic
124C23×C101C2×D411yescharacteristic
125C22×D101—21no—
126C22×D101—21no—
127C22×D101—21no—
128C22×D101—42no—
129C22×D101—42no—
130C22×D101—81no—
131D4×C101—41no—
132C2×C4×C101—21no—
133C2×C4×C101—21no—
134C2×C4×C101—21no—
135C2×C4×C101—21no—
136C2×C4×C101C2×D411yescharacteristic
137C2×C4×C101C2×D411yescharacteristic
138C2×C4×C101C2×D411yescharacteristic
139C2×C4×C101C2×D411yescharacteristic
140C2×C4×C101C2411yescharacteristic
141C22⋊D101—42no—
142C22⋊D101—42no—
143C22⋊D101—42no—
144C22⋊D101—81no—
145C2×Dic101—21no—
146C2×Dic101—21no—
147C2×Dic101—21no—
148C2×Dic101—21no—
149C2×Dic101—21no—
150D404—21no—
151D404—21no—
152D404—21no—
153D404—21no—
154D404—42no—
155D404C2×D412yes—
156D404C2×D412yes—
157C4×D101—42no—
158C4×D101—42no—
159C4×D101—42no—
160C4×D101—42no—
161C4×D101—81no—
162Dic202—21no—
163Dic202—21no—
164Dic202—21no—
165Dic202—21no—
166Dic202—21no—
167Dic202—21no—
168C8×C101—21no—
169C8×C101—21no—
170C8×C101—21no—
171C8×C101C2×D412yes—
172C22×C101—21no—
173C22×C101—21no—
174C22×C101—21no—
175C22×C101—21no—
176C22×C101—21no—
177C22×C101—21no—
178C22×C101C22×D411yescharacteristic
179C2×D101—22no—
180C2×D101—22no—
181C2×D101—41no—
182C2×D101—41no—
183C2×D101—41no—
184C2×D101—41no—
185C2×D101—41no—
186C2×D101—41no—
187C2×D101—41no—
188C2×D101—84no—
189C4×C101—21no—
190C4×C101—21no—
191C4×C101—21no—
192C4×C101—21no—
193C4×C101C22×D411yescharacteristic
194C4×C101H(5,𝔽2)11yescharacteristic
195Dic101—41no—
196Dic101—41no—
197Dic101—41no—
198Dic101—41no—
199Dic101—41no—
200C2×C101—41no—
201C2×C101—41no—
202C2×C101D4211yescharacteristic
203C2×C101#57511yescharacteristic
204C2×C101#57711yescharacteristic
205D101—42no—
206D101—42no—
207D101—81no—
208#3267—1011nomaximal, Sylow
209C101#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—1011no—
211C2×#135—1011no—
212#547—1011no—
213#546—1011no—
214#497—1011no—
215#470—1011no—
216#465—1011no—
217#463—1011no—
218#461—1011no—
219#452—1011no—
220#452—1011no—
221#450—1011no—
222#449—1011no—
223#442—1011no—
224#435D10111yescharacteristic, complemented
225C2×#41—1011no—
226C2×#41—1011no—
227C2×#41—1011no—
228C22×D4—1011no—
229AGL(1,ℤ/8)—2024no—
230#135—2024no—
231C2×Q16—1011no—
232C2×Q16—1011no—
233C2×SD16—1011no—
234C2×SD16—1011no—
235C2×SD16—1011no—
236C2×SD16—1011no—
237C2×SD16—1011no—
238C2×D8—1011no—
239C2×D8—1011no—
240C2×M16C2×D10111yescharacteristic
241C22×C8C2×D10111yescharacteristic, complemented
242C4⋊Q8—1011no—
243#124—1011no—
244#123—2021no—
245#123—2021no—
246#123—2021no—
247C22⋊Q8—1011no—
248C22⋊Q8—1011no—
249C22⋊Q8—1011no—
250#121—1011no—
251#121—2021no—
252#121—2021no—
253#120—2021no—
254C4×D4—1011no—
255C4×D4—1011no—
256C4×D4C2×D10111yescharacteristic
257C2×C4⋊C4—2021no—
258C2×C22⋊C4—2021no—
259#106—1011no—
260C4⋊C8C2×D10111yescharacteristic, complemented
261#103—1011no—
262#103—1011no—
263#103—1011no—
264#102—1011no—
265#102—1011no—
266#102—1011no—
267C22⋊C8C2×D10111yescharacteristic
268C22⋊C8C2×D10111yescharacteristic, complemented
269#97C2×D10111yescharacteristic, complemented
270C24—2021no—
271#41—4041no—
272#41—4041no—
273#41—4041no—
274#41—2022no—
275#41—2022no—
276#41—2022no—
277C2×Q8—1011no—
278C2×Q8—1011no—
279C2×Q8—1011no—
280C2×D4—1011no—
281C2×D4—1011no—
282C2×D4—1011no—
283C2×D4—4041no—
284C2×D4—2022no—
285C2×D4—2021no—
286C2×D4—2021no—
287C2×D4—2021no—
288C2×D4—2021no—
289C2×D4—2021no—
290C2×D4D40411yescharacteristic
291C22×C4—2021no—
292C22×C4—2021no—
293C22×C4—2021no—
294C22×C4—2021no—
295C22×C4—2021no—
296C22×C4C22×D10111yescharacteristic
297C22×C4C22×D10111yescharacteristic
298Q16—4041no—
299Q16—2022no—
300SD16—4041no—
301SD16—2022no—
302SD16—2022no—
303SD16—2022no—
304SD16—2022no—
305D8—1014no—
306D8—2022no—
307M16—22no—
308C2×C8—22no—
309C2×C8C22×D10111yescharacteristic
310C2×C8C22×D10111yescharacteristic
311C2×C8C22×D10111yescharacteristic
312C2×C8C22×D10111yescharacteristic
313C4⋊C4—1011no—
314C4⋊C4—1011no—
315C4⋊C4—2021no—
316C4⋊C4—2021no—
317C4⋊C4—2021no—
318C4⋊C4—2021no—
319C4⋊C4—2021no—
320C4⋊C4D40411yescharacteristic
321C22⋊C4—2021no—
322C22⋊C4—2021no—
323C22⋊C4—2021no—
324C22⋊C4—2021no—
325C22⋊C4—2021no—
326C22⋊C4—2021no—
327C22⋊C4—2021no—
328C22⋊C4—2021no—
329C22⋊C4D40411yescharacteristic
330C22⋊C4D40411yescharacteristic
331C42C22×D10111yescharacteristic
332C23—8081no—
333C23—4042no—
334C23—4042no—
335C23—2021no—
336C23—2021no—
337C23—2021no—
338C23C2×D40411yescharacteristic
339C23C2×D40411yescharacteristic
340Q8—2021no—
341Q8—2021no—
342Q8—2021no—
343Q8—2021no—
344Q8—2021no—
345Q8—2021no—
346D4—41no—
347D4—8081no—
348D4—1012no—
349D4—1012no—
350D4—4042no—
351D4—4042no—
352D4—4042no—
353D4—4042no—
354D4—2021no—
355D4—2021no—
356D4—2021no—
357D4—2021no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—8081no—
363C2×C4—4042no—
364C2×C4—4042no—
365C2×C4—4042no—
366C2×C4—4042no—
367C2×C4—2021no—
368C2×C4—2021no—
369C2×C4—2021no—
370C2×C4—2021no—
371C2×C4—2021no—
372C2×C4C2×D40411yescharacteristic
373C2×C4C2×D40411yescharacteristic
374C2×C4C2×D40411yescharacteristic
375C2×C4C2×D40411yescharacteristic
376C2×C4C23×D10111yescharacteristic
377C8—21no—
378C8—21no—
379C8—21no—
380C8D4×D10112yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—8084no—
388C22—4041no—
389C22—4041no—
390C22—4041no—
391C22—4041no—
392C22—4041no—
393C22—4041no—
394C22—4041no—
395C22—2022no—
396C22—2022no—
397C22C22×D40411yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—4041no—
403C4—4041no—
404C4—4041no—
405C4—4041no—
406C4—4041no—
407C4C2×D4×D10111yescharacteristic
408C4#1569230111yescharacteristic
409C2—41no—
410C2—41no—
411C2—8081no—
412C2—4042no—
413C2—4042no—
414C2D4×D40411yescharacteristic
415C2#2026597711yescharacteristic
416C2#2026598911yescharacteristic
subgroupG/H
centerC22C22×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#435×C101C2
socleC22×C101C22×D4
hypercenter#435D101
hypocenterC101#3267
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#15692283C2semidirect
2. C2×#15692277C2semidirect
3. #20265832C2semidirect
4. #20265823C2semidirect
5. #20265571C2semidirect
6. #20265435C2semidirect
7. #20265406C2semidirect
8. #20265393C2semidirect
9. #20265383C2semidirect
10. #20265341C2semidirect
11. #20265341C2semidirect
12. #20265330C2semidirect
13. #20265324C2semidirect
14. #20265290C2semidirect
15. #435×C101C2semidirect
18. C2×#13108074C22semidirect
22. C2×Dic404C22semidirect
23. C2×Dic404C22semidirect
28. C2×#13108063C22semidirect
31. #15692255C22semidirect
38. #15692234C22semidirect
44. C4×D404C22semidirect
47. #15692187C22semidirect
48. #15692180C22semidirect
49. #15692180C22semidirect
50. #15692180C22semidirect
53. #15692177C22semidirect
55. C22×C8×C101C22semidirect
57. C4⋊C8×C101C22semidirect
59. C22⋊C8×C101C22semidirect
60. #97×C101C22semidirect
94. D808D4semidirect
209. C101#3267semidirect
224. #435D101semidirect
241. C22×C8C2×D101semidirect
260. C4⋊C8C2×D101semidirect
268. C22⋊C8C2×D101semidirect
269. #97C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922832
C2×#156922772
#202658322
#202658232
#202655712
#202654352
#202654062
#202653932
#202653832
#202653412
#202653302
#202653242
#202652902
#435×C1012
#3267101