Finite Group Databasebeta

← #27183875

Subgroups — #27183875

Order 12928.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265907C211yesmaximal, characteristic, complemented
2#20265899C211yesmaximal, characteristic
3#20265539C211yesmaximal, characteristic
4#20265538C211yesmaximal, characteristic
5#20265526C211yesmaximal, characteristic, complemented
6#20265466C211yesmaximal, characteristic
7#20265461C211yesmaximal, characteristic, complemented
8#20265402C211yesmaximal, characteristic
9#20265400C211yesmaximal, characteristic
10#20265386C211yesmaximal, characteristic, complemented
11#20265306C211yesmaximal, characteristic, complemented
12#20265283C211yesmaximal, characteristic, complemented
13#20265282C211yesmaximal, characteristic, complemented
14#20265264C211yesmaximal, characteristic
15#475×C101C211yesmaximal, characteristic, complemented
16C2×Dic404C2211yescharacteristic
17C2×#13108064C2211yescharacteristic
18C2×#13108063C2211yescharacteristic
19#15692254—21no—
20#15692254C2211yescharacteristic
21#15692254C2211yescharacteristic
22#15692253—21no—
23#15692253—21no—
24#15692253C2211yescharacteristic
25#15692250—21no—
26#15692249—21no—
27#15692244C2211yescharacteristic
28#15692243C2211yescharacteristic
29#15692233—21no—
30C4×Dic202C2211yescharacteristic
31C4×Dic202C2211yescharacteristic
32#15692220—21no—
33#15692220C2211yescharacteristic
34#15692220C2211yescharacteristic
35#15692220C2211yescharacteristic
36#15692217C2211yescharacteristic
37#15692212—21no—
38#15692186C2211yescharacteristic
39#15692184C2211yescharacteristic
40#15692184C2211yescharacteristic
41#15692180C2211yescharacteristic
42#15692179C2211yescharacteristic
43#15692178C2211yescharacteristic
44#15692178C2211yescharacteristic
45#15692178C2211yescharacteristic
46#15692177C2211yescharacteristic
47#15692175C2211yescharacteristic
48#15692175C2211yescharacteristic
49#15692166C2211yescharacteristic
50#15692165C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C4⋊Q8×C101C2211yescharacteristic
53C4×Q8×C101C2211yescharacteristic
54#106×C101C2211yescharacteristic
55#106×C101C2211yescharacteristic
56C4⋊C8×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58#103×C101C2211yescharacteristic
59C2×Dic202—21no—
60C2×Dic202—21no—
61C2×Dic202D411yescharacteristic
62C2×Dic202C2311yescharacteristic
63C2×Dic202C2311yescharacteristic
64C2×D404C2311yescharacteristic
65C2×C4×D101—21no—
66Dic404—22no—
67#13108064—22no—
68#13108063—22no—
69C2×#11717579C2311yescharacteristic
70C2×#11717579C2311yescharacteristic
71C4⋊Dic101D411yescharacteristic
72C4⋊Dic101C2311yescharacteristic
73#13108053—21no—
74#13108053—21no—
75#13108053—21no—
76#13108053—21no—
77#13108053—21no—
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053D411yescharacteristic
82#13108053D411yescharacteristic
83#13108051—21no—
84#13108051—21no—
85#13108051—21no—
86#13108051—21no—
87#13108051—21no—
88C4×Dic101—21no—
89C4×Dic101—21no—
90C4×Dic101—21no—
91C4×Dic101—21no—
92C4×Dic101C2311yescharacteristic
93C4×Dic101C2311yescharacteristic
94C2×Q8×C101—21no—
95C2×Q8×C101C2311yescharacteristic
96C2×C8×C101C2311yescharacteristic
97C2×C8×C101C2311yescharacteristic
98C4⋊C4×C101—21no—
99C4⋊C4×C101—21no—
100C4⋊C4×C101C2311yescharacteristic
101C4⋊C4×C101C2311yescharacteristic
102C4⋊C4×C101C2311yescharacteristic
103C42×C101—21no—
104C42×C101C2311yescharacteristic
105C22×D101—21no—
106Q8×C101—21no—
107Q8×C101—21no—
108Q8×C101—81no—
109C2×C4×C101—21no—
110C2×C4×C101—21no—
111C2×C4×C101—21no—
112C2×C4×C101—21no—
113C2×C4×C101C2×D411yescharacteristic
114C2×C4×C101C2×D411yescharacteristic
115C2×C4×C101C2411yescharacteristic
116C2×Dic101—21no—
117C2×Dic101—21no—
118C2×Dic101—21no—
119C2×Dic101C2×D411yescharacteristic
120C2×Dic101C2×D411yescharacteristic
121C2×Dic101C2×D411yescharacteristic
122C2×Dic101C2×D411yescharacteristic
123D404—21no—
124D404—21no—
125C4×D101—42no—
126Dic202—21no—
127Dic202—22no—
128Dic202—21no—
129Dic202—21no—
130Dic202—81no—
131Dic202—81no—
132Dic202#4112yes—
133C8×C101—21no—
134C8×C101—21no—
135#11717579—21no—
136#11717579—21no—
137C22×C101C22×D411yescharacteristic
138C2×D101—41no—
139C2×D101—41no—
140C2×D101—41no—
141C4×C101—21no—
142C4×C101—21no—
143C4×C101—41no—
144C4×C101—41no—
145C4×C101—41no—
146C4×C101—41no—
147C4×C101C2×#4111yescharacteristic
148C4×C1012−1+411yescharacteristic
149Dic101—22no—
150Dic101—21no—
151Dic101—21no—
152Dic101—21no—
153Dic101—41no—
154Dic101—41no—
155Dic101#13712yes—
156C2×C101#54811yescharacteristic
157C2×C101C2×#13711yescharacteristic
158C2×C101#57711yescharacteristic
159D101—81no—
160#3369—1011nomaximal, Sylow
161C101#336911yescharacteristic, complemented, Sylow
162Q82—1011no—
163#556—1011no—
164#489—1011no—
165#489—1011no—
166#487—1011no—
167#476—1011no—
168#475—1011no—
169#475D10111yescharacteristic, complemented
170#464—1011no—
171#464—1011no—
172#462—1011no—
173#446—1011no—
174#441—1011no—
175#441—1011no—
176C4×SD16—1011no—
177C2×Q16—1011no—
178C2×Q16—1011no—
179C2×SD16—1011no—
180C4⋊Q8—1011no—
181C4⋊Q8—1011no—
182C4⋊Q8—1011no—
183C4⋊Q8—2021no—
184C4⋊Q8—2021no—
185C4⋊Q8—2021no—
186C4⋊Q8C2×D10111yescharacteristic
187#126—2021no—
188#126—2021no—
189#124—1011no—
190#124—1011no—
191C22⋊Q8—2021no—
192C4×Q8—1011no—
193C4×Q8—1011no—
194C4×Q8—1011no—
195C4×Q8—1011no—
196C4×Q8—1011no—
197C4×Q8—2021no—
198C4×Q8C2×D10111yescharacteristic
199C4×D4—1011no—
200#117—2021no—
201#106—1011no—
202#106C2×D10111yescharacteristic
203#106C2×D10111yescharacteristic
204C4⋊C8—1011no—
205C4⋊C8—1011no—
206C4⋊C8C2×D10111yescharacteristic
207#103—1011no—
208#103—1011no—
209#103—1011no—
210#103—1011no—
211#103—1011no—
212#103C2×D10111yescharacteristic
213#103C2×D10111yescharacteristic
214#102—1011no—
215#102—1011no—
216#102—1011no—
217#97—1011no—
218#97—1011no—
219C4×C8—1011no—
220C2×Q8—21no—
221C2×Q8—1011no—
222C2×Q8—1011no—
223C2×Q8—1011no—
224C2×Q8—2021no—
225C2×Q8—2021no—
226C2×Q8C22×D10111yescharacteristic
227C2×D4—1011no—
228C22×C4—2021no—
229Q16—2022no—
230Q16—2022no—
231SD16—2022no—
232C2×C8—1011no—
233C2×C8—1011no—
234C2×C8C22×D10111yescharacteristic
235C2×C8C22×D10111yescharacteristic
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—1011no—
239C4⋊C4—1011no—
240C4⋊C4—1011no—
241C4⋊C4—1011no—
242C4⋊C4—2021no—
243C4⋊C4—2021no—
244C4⋊C4—2021no—
245C4⋊C4—2021no—
246C4⋊C4—2021no—
247C4⋊C4—2021no—
248C4⋊C4—2021no—
249C4⋊C4—2021no—
250C4⋊C4C22×D10111yescharacteristic
251C4⋊C4C22×D10111yescharacteristic
252C4⋊C4C22×D10111yescharacteristic
253C22⋊C4—2021no—
254C22⋊C4—2021no—
255C22⋊C4—2021no—
256C22⋊C4—2021no—
257C22⋊C4—2021no—
258C42—21no—
259C42—1011no—
260C42—1011no—
261C42—2021no—
262C42—2021no—
263C42—2021no—
264C42—2021no—
265C42C22×D10111yescharacteristic
266C23—2021no—
267Q8—21no—
268Q8—21no—
269Q8—81no—
270Q8—8081no—
271Q8—8081no—
272Q8—1012no—
273Q8—2022no—
274Q8—2021no—
275Q8—2021no—
276Q8—2021no—
277D4—2021no—
278D4—2021no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—1011no—
284C2×C4—1011no—
285C2×C4—1011no—
286C2×C4—1011no—
287C2×C4—4042no—
288C2×C4—2021no—
289C2×C4—2021no—
290C2×C4—2021no—
291C2×C4D4×D10111yescharacteristic
292C2×C4D4×D10111yescharacteristic
293C2×C4C23×D10111yescharacteristic
294C8—21no—
295C8—21no—
296C8—2021no—
297C8—2021no—
298C22—4041no—
299C22—4041no—
300C22—4041no—
301C22C2×D4×D10111yescharacteristic
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—1012no—
309C4—4041no—
310C4—4041no—
311C4—2022no—
312C4—2021no—
313C4—2021no—
314C4—2021no—
315C4#41×D10111yescharacteristic
316C4#1569230311yescharacteristic
317C2—8081no—
318C2#2026583711yescharacteristic
319C2#137×D10111yescharacteristic
320C2#2026598311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#475×C101C2
socleC22×C101C22×D4
prime radicalC2×D404C23
hypercenter#475D101
hypocenterC101#3369
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. #20265907C2semidirect
5. #20265526C2semidirect
7. #20265461C2semidirect
10. #20265386C2semidirect
11. #20265306C2semidirect
12. #20265283C2semidirect
13. #20265282C2semidirect
15. #475×C101C2semidirect
161. C101#3369semidirect
169. #475D101semidirect

Maximal subgroups

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

subgroupindex
#202659072
#202658992
#202655392
#202655382
#202655262
#202654662
#202654612
#202654022
#202654002
#202653862
#202653062
#202652832
#202652822
#202652642
#475×C1012
#3369101