Finite Group Databasebeta

← #27180960

Subgroups — #27180960

Order 12928.

398 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265807C211yesmaximal, characteristic, complemented
2#20265649C211yesmaximal, characteristic
3#20265480C211yesmaximal, characteristic, complemented
4#20265475C211yesmaximal, characteristic, complemented
5#20265474C211yesmaximal, characteristic, complemented
6#20265467C211yesmaximal, characteristic, complemented
7#20265466C211yesmaximal, characteristic, complemented
8#20265458C211yesmaximal, characteristic, complemented
9#20265455C211yesmaximal, characteristic, complemented
10#20265216C211yesmaximal, characteristic, complemented
11#20265214C211yesmaximal, characteristic, complemented
12#20265202C211yesmaximal, characteristic, complemented
13#20265180C211yesmaximal, characteristic
14#102×D101C211yesmaximal, characteristic
15#478×C101C211yesmaximal, characteristic, complemented
16C2×D4×D101C2211yescharacteristic
17C2×#13108057C2211yescharacteristic
18C2×C8×D101C2211yescharacteristic
19#15692255C2211yescharacteristic, complemented
20#15692254—21no—
21#15692254C2211yescharacteristic, complemented
22#15692246C2211yescharacteristic, complemented
23#15692245—21no—
24#15692235—21no—
25#15692233—21no—
26C4×D404C2211yescharacteristic
27D4×Dic101C2211yescharacteristic
28#15692217C2211yescharacteristic
29C4×C22⋊D101—21no—
30#15692215—21no—
31#15692213C2211yescharacteristic
32#15692213C2211yescharacteristic
33#15692212C2211yescharacteristic
34C4⋊C4×D101C2211yescharacteristic
35C2×#13108051—21no—
36C22⋊C4×D101—21no—
37#15692189C2211yescharacteristic, complemented
38#15692189C2211yescharacteristic, complemented
39#15692188C2211yescharacteristic, complemented
40#15692186C2211yescharacteristic, complemented
41#15692186C2211yescharacteristic, complemented
42#15692186C2211yescharacteristic, complemented
43C4⋊#11717579C2211yescharacteristic, complemented
44#15692184C2211yescharacteristic, complemented
45#15692184C2211yescharacteristic, complemented
46#15692177C2211yescharacteristic
47#15692177C2211yescharacteristic
48#15692176C2211yescharacteristic
49#15692176C2211yescharacteristic
50#15692175C2211yescharacteristic
51#15692175C2211yescharacteristic
52#125×C101C2211yescharacteristic, complemented
53C4×D4×C101C2211yescharacteristic
54#107×C101C2211yescharacteristic, complemented
55#106×C101C2211yescharacteristic, complemented
56C4⋊C8×C101C2211yescharacteristic, complemented
57#102×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59C23×D101—21no—
60C2×Dic202—21no—
61C2×Dic202—21no—
62C2×D404C2311yescharacteristic
63C2×C22⋊D101—21no—
64D4×D101—22no—
65D4×D101Q814yescomplemented
66C22×Dic101—21no—
67C2×C4×D101—21no—
68C2×C4×D101—21no—
69C2×C4×D101—21no—
70C2×C4×D101C2311yescharacteristic
71#13108057—22no—
72C2×#11717579C2311yescharacteristic
73C2×#11717579C2311yescharacteristic
74C8×D101—22no—
75C4⋊Dic101—21no—
76C4⋊Dic101—21no—
77C4⋊Dic101C2311yescharacteristic
78C4⋊Dic101C2311yescharacteristic
79C4⋊Dic101C2311yescharacteristic
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053D411yescharacteristic
84#13108053D411yescharacteristic
85C22⋊Dic101—21no—
86#13108051—21no—
87#13108051—21no—
88#13108051—21no—
89#13108051—21no—
90#13108051D411yescharacteristic
91#13108051D411yescharacteristic
92C4×Dic101—21no—
93C4×Dic101—21no—
94C4×Dic101C2311yescharacteristic
95C2×D4×C101C2311yescharacteristic
96C22×C4×C101—21no—
97C2×C8×C101C2311yescharacteristic
98C2×C8×C101C2311yescharacteristic
99C4⋊C4×C101—21no—
100C4⋊C4×C101—21no—
101C4⋊C4×C101C2311yescharacteristic
102C4⋊C4×C101C2311yescharacteristic
103C4⋊C4×C101C2311yescharacteristic
104C22⋊C4×C101—21no—
105C42×C101C2311yescharacteristic
106C23×C101—21no—
107C22×D101—21no—
108C22×D101—24no—
109C22×D101—81no—
110C22×D101—81no—
111C22×D101C2×D411yescharacteristic
112D4×C101—21no—
113D4×C101C2×Q812yes—
114C2×C4×C101—21no—
115C2×C4×C101—21no—
116C2×C4×C101—21no—
117C2×C4×C101—81no—
118C2×C4×C101C2×D411yescharacteristic
119C2×C4×C101C2×D411yescharacteristic
120C2×C4×C101C2411yescharacteristic
121C22⋊D101—24no—
122C2×Dic101—21no—
123C2×Dic101—21no—
124C2×Dic101—21no—
125C2×Dic101—81no—
126C2×Dic101C2×D411yescharacteristic
127C2×Dic101C2×D411yescharacteristic
128C2×Dic101C2×D411yescharacteristic
129D404—21no—
130D404C2×Q812yes—
131C4×D101—81no—
132C4×D101—81no—
133C4×D101—81no—
134C4×D101C2×Q812yes—
135C4×D101#4112yes—
136Dic202—81no—
137Dic202—81no—
138C8×C101—21no—
139C8×C101—21no—
140#11717579—21no—
141#11717579—21no—
142C22×C101—22no—
143C22×C101—41no—
144C22×C101—41no—
145C22×C101C22×D411yescharacteristic
146C2×D101—21no—
147C2×D101—21no—
148C2×D101—22no—
149C2×D101—41no—
150C2×D101—41no—
151C2×D101—84no—
152C2×D101C22⋊Q812yes—
153C4×C101—21no—
154C4×C101—21no—
155C4×C101—41no—
156C4×C101—41no—
157C4×C101—41no—
158C4×C101C22×Q811yescharacteristic
159C4×C101C2×#4111yescharacteristic
160Dic101—21no—
161Dic101—21no—
162Dic101—41no—
163Dic101—41no—
164Dic101—41no—
165Dic101C22⋊Q812yes—
166C2×C101—42no—
167C2×C101C2×C22⋊Q811yescharacteristic
168C2×C101#57611yescharacteristic
169C2×C101#57711yescharacteristic
170D101—22no—
171D101—42no—
172#3081—1011nomaximal, Sylow
173C101#308111yescharacteristic, complemented, Sylow
174#544—1011no—
175#518—1011no—
176#478—1011no—
177#478—1011no—
178#478—1011no—
179#478D10111yescharacteristic, complemented
180#476—1011no—
181#476—1011no—
182#474—1011no—
183#474—1011no—
184#428—1011no—
185#427—1011no—
186#424—1011no—
187#418—1011no—
188C2×#102—1011no—
189C22×D4—1011no—
190C2×M16—1011no—
191C22×C8—1011no—
192C4⋊Q8—1011no—
193C4⋊Q8—1011no—
194C4⋊Q8—2021no—
195#125—1011no—
196#125—2021no—
197#125C2×D10111yescharacteristic, complemented
198C22⋊Q8—2021no—
199C22⋊Q8—2021no—
200C4×D4—1011no—
201C4×D4—1011no—
202C4×D4—1011no—
203C4×D4—2021no—
204C4×D4—2021no—
205C4×D4C2×D10111yescharacteristic
206#117—1011no—
207#117—1011no—
208#117—1011no—
209C2×C4⋊C4—1011no—
210C2×C22⋊C4—2021no—
211C2×C22⋊C4—2021no—
212#107—1011no—
213#107—1011no—
214#107—1011no—
215#107C2×D10111yescharacteristic, complemented
216#106—1011no—
217#106—1011no—
218#106—1011no—
219#106C2×D10111yescharacteristic, complemented
220C4⋊C8—1011no—
221C4⋊C8—1011no—
222C4⋊C8—1011no—
223C4⋊C8C2×D10111yescharacteristic, complemented
224#102—1011no—
225#102—1011no—
226#102—1011no—
227#102—1011no—
228#102—1011no—
229#102—1011no—
230#102C2×D10111yescharacteristic
231#102C2×D10111yescharacteristic
232C24—2021no—
233C2×Q8—2021no—
234C2×Q8—2021no—
235C2×D4—1014no—
236C2×D4—1011no—
237C2×D4—2022no—
238C2×D4—2021no—
239C2×D4C22×D10111yescharacteristic
240C22×C4—21no—
241C22×C4—1011no—
242C22×C4—2021no—
243C22×C4—2021no—
244C22×C4—2021no—
245C22×C4—2021no—
246M16—2022no—
247C2×C8—1011no—
248C2×C8—1011no—
249C2×C8—2022no—
250C2×C8C22×D10111yescharacteristic
251C2×C8C22×D10111yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—1011no—
255C4⋊C4—1011no—
256C4⋊C4—1011no—
257C4⋊C4—1011no—
258C4⋊C4—1011no—
259C4⋊C4—2021no—
260C4⋊C4—2021no—
261C4⋊C4—2021no—
262C4⋊C4—2021no—
263C4⋊C4—2021no—
264C4⋊C4C22×D10111yescharacteristic
265C4⋊C4C22×D10111yescharacteristic
266C4⋊C4C22×D10111yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—1011no—
269C22⋊C4—1011no—
270C22⋊C4—2021no—
271C22⋊C4—2021no—
272C22⋊C4—2021no—
273C22⋊C4—2021no—
274C22⋊C4—2021no—
275C42—1011no—
276C42—2021no—
277C42—2021no—
278C42C22×D10111yescharacteristic
279C23—21no—
280C23—8081no—
281C23—8081no—
282C23—1011no—
283C23—2024no—
284C23—2021no—
285Q8—8081no—
286Q8—8081no—
287D4—21no—
288D4—1012no—
289D4—2024no—
290D4—2021no—
291D4Q8×D10112yes—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—81no—
296C2×C4—8081no—
297C2×C4—8081no—
298C2×C4—8081no—
299C2×C4—8081no—
300C2×C4—1011no—
301C2×C4—1012no—
302C2×C4—1012no—
303C2×C4—1011no—
304C2×C4—1011no—
305C2×C4—2021no—
306C2×C4—2021no—
307C2×C4—2021no—
308C2×C4D4×D10111yescharacteristic
309C2×C4D4×D10111yescharacteristic
310C2×C4C23×D10111yescharacteristic
311C8—21no—
312C8—21no—
313C8—2021no—
314C8—2021no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—8084no—
319C22—1012no—
320C22—4041no—
321C22—4041no—
322C22—2022no—
323C22—2021no—
324C22—2021no—
325C22C2×D4×D10111yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—1012no—
332C4—4041no—
333C4—4041no—
334C4—4041no—
335C4—2021no—
336C4—2021no—
337C4C2×Q8×D10111yescharacteristic
338C4#41×D10111yescharacteristic
339C2—42no—
340C2—4042no—
341C2—2022no—
342C2C22⋊Q8×D10111yescharacteristic
343C2#2026597911yescharacteristic
344C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#478×C101C2
socleC22×C101C22×D4
prime radicalC2×D4×D101C22
hypercenter#478D101
hypocenterC101#3081
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #20265807C2semidirect
3. #20265480C2semidirect
4. #20265475C2semidirect
5. #20265474C2semidirect
6. #20265467C2semidirect
7. #20265466C2semidirect
8. #20265458C2semidirect
9. #20265455C2semidirect
10. #20265216C2semidirect
11. #20265214C2semidirect
12. #20265202C2semidirect
15. #478×C101C2semidirect
19. #15692255C22semidirect
21. #15692254C22semidirect
22. #15692246C22semidirect
37. #15692189C22semidirect
38. #15692189C22semidirect
39. #15692188C22semidirect
40. #15692186C22semidirect
41. #15692186C22semidirect
42. #15692186C22semidirect
43. C4⋊#11717579C22semidirect
44. #15692184C22semidirect
45. #15692184C22semidirect
52. #125×C101C22semidirect
54. #107×C101C22semidirect
55. #106×C101C22semidirect
56. C4⋊C8×C101C22semidirect
65. D4×D101Q8semidirect
173. C101#3081semidirect
179. #478D101semidirect
197. #125C2×D101semidirect
215. #107C2×D101semidirect
219. #106C2×D101semidirect
223. C4⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658072
#202656492
#202654802
#202654752
#202654742
#202654672
#202654662
#202654582
#202654552
#202652162
#202652142
#202652022
#202651802
#102×D1012
#478×C1012
#3081101