Finite Group Databasebeta

← #27183854

Subgroups — #27183854

Order 12928.

372 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265901C211yesmaximal, characteristic, complemented
2D4×Dic202C211yesmaximal, characteristic, complemented
3#20265538C212yesmaximal, complemented
4#20265527C211yesmaximal, characteristic, complemented
5#20265470C212yesmaximal, complemented
6#20265458C211yesmaximal, characteristic, complemented
7#20265394C212yesmaximal
8#20265304C211yesmaximal, characteristic, complemented
9#20265278C212yesmaximal, complemented
10C4×#13108059C211yesmaximal, characteristic
11#461×C101C211yesmaximal, characteristic, complemented
12C22×Dic202—21no—
13C2×#13108059C2211yescharacteristic
14#15692255C2211yescharacteristic
15#15692254C2212yes—
16#15692250—22no—
17#15692244C2212yes—
18#15692235—21no—
19#15692234—21no—
20#15692232—22no—
21Q8×Dic101C2212yes—
22C4×Dic202C2211yescharacteristic
23C4×D404C2211yescharacteristic
24D4×Dic101C2212yes—
25#15692213—21no—
26#15692189C2212yes—
27#15692188C2211yescharacteristic
28#15692184C2212yes—
29#15692179C2212yes—
30#15692179C2212yes—
31#15692177C2212yes—
32#15692176C2211yescharacteristic
33#15692175C2211yescharacteristic
34#15692165C2212yes—
35C4×#11717579C2211yescharacteristic
36C2×SD16×C101C2212yes—
37C4⋊Q8×C101C2211yescharacteristic
38C4×D4×C101C2211yescharacteristic
39C4⋊C8×C101C2211yescharacteristic
40#103×C101C2212yes—
41C2×Dic202—21no—
42C2×Dic202—22no—
43C2×Dic202—44no—
44C2×Dic202D411yescharacteristic
45C2×D404C2311yescharacteristic
46C22×Dic101—22no—
47C2×C4×D101—21no—
48#13108059—22no—
49C2×#11717579C2311yescharacteristic
50C2×#11717579C2311yescharacteristic
51C4⋊Dic101—21no—
52C4⋊Dic101—22no—
53C4⋊Dic101D411yescharacteristic
54C4⋊Dic101C2311yescharacteristic
55C4⋊Dic101C2312yes—
56#13108053—22no—
57#13108053D412yes—
58C22⋊Dic101—22no—
59#13108051—21no—
60#13108051—22no—
61#13108051—22no—
62C4×Dic101—22no—
63C4×Dic101C2312yes—
64C2×Q8×C101C2312yes—
65C2×D4×C101C2311yescharacteristic
66C22×C4×C101—21no—
67SD16×C101—24no—
68C2×C8×C101C2312yes—
69C4⋊C4×C101—21no—
70C4⋊C4×C101—21no—
71C4⋊C4×C101C2311yescharacteristic
72C22⋊C4×C101—21no—
73C42×C101C2311yescharacteristic
74C23×C101—21no—
75C22×D101—21no—
76Q8×C101—22no—
77Q8×C101—22no—
78D4×C101—21no—
79D4×C101#4112yes—
80C2×C4×C101—21no—
81C2×C4×C101—22no—
82C2×C4×C101—42no—
83C2×C4×C101C2×D411yescharacteristic
84C2×C4×C101C2×D411yescharacteristic
85C2×C4×C101C2411yescharacteristic
86C2×Dic101—21no—
87C2×Dic101—22no—
88C2×Dic101—44no—
89C2×Dic101C2×D412yes—
90C2×Dic101C2×D412yes—
91D404—21no—
92D404—21no—
93C4×D101—42no—
94Dic202—22no—
95Dic202—44no—
96Dic202—42no—
97C8×C101—22no—
98#11717579—21no—
99#11717579—21no—
100C22×C101—22no—
101C22×C101—41no—
102C22×C101—41no—
103C22×C101C22×D411yescharacteristic
104C2×D101—41no—
105C2×D101—41no—
106C2×D101—41no—
107C4×C101—21no—
108C4×C101—42no—
109C4×C101—41no—
110C4×C101AGL(1,ℤ/8)12yes—
111C4×C101C2×#4111yescharacteristic
112C4×C1012−1+411yescharacteristic
113Dic101—22no—
114Dic101—22no—
115Dic101—42no—
116Dic101—41no—
117C2×C101—42no—
118C2×C101#54811yescharacteristic
119C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C101#57811yescharacteristic
121D101—81no—
122#3367—1011nomaximal, Sylow
123C101#336711yescharacteristic, complemented, Sylow
124#556—1011no—
125D4×Q8—1011no—
126#489—1012no—
127#487—1011no—
128#477—1012no—
129#474—1011no—
130#463—1012no—
131#461D10111yescharacteristic, complemented
132#446—1011no—
133#440—1012no—
134C4×D8—1011no—
135C22×Q8—2021no—
136C2×SD16C2×D10112yes—
137C2×D8—1011no—
138C4⋊Q8—1012no—
139C4⋊Q8—1011no—
140C4⋊Q8C2×D10111yescharacteristic
141#126—2022no—
142#124—1012no—
143C22⋊Q8—2022no—
144C22⋊Q8—2021no—
145C22⋊Q8—2021no—
146C4×Q8—1011no—
147C4×Q8—1012no—
148C4×D4—1012no—
149C4×D4—1011no—
150C4×D4C2×D10111yescharacteristic
151#117—2021no—
152#107—1012no—
153#107—1011no—
154C4⋊C8—1012no—
155C4⋊C8C2×D10111yescharacteristic
156#103—1012no—
157#103—1012no—
158#103C2×D10112yes—
159#102—1011no—
160#102—1012no—
161#102—1011no—
162#97—1012no—
163C4×C8—1011no—
164C2×Q8—1011no—
165C2×Q8—4044no—
166C2×Q8—2022no—
167C2×Q8—2021no—
168C2×Q8C22×D10112yes—
169C2×D4—1011no—
170C2×D4C22×D10111yescharacteristic
171C22×C4—21no—
172C22×C4—2022no—
173C22×C4—2021no—
174SD16—24no—
175D8—2022no—
176C2×C8—1011no—
177C2×C8—1011no—
178C2×C8C22×D10112yes—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—1012no—
182C4⋊C4—1011no—
183C4⋊C4—1012no—
184C4⋊C4—1011no—
185C4⋊C4—2022no—
186C4⋊C4—2022no—
187C4⋊C4—2021no—
188C4⋊C4C22×D10111yescharacteristic
189C22⋊C4—21no—
190C22⋊C4—2022no—
191C22⋊C4—2022no—
192C22⋊C4—2022no—
193C22⋊C4—2021no—
194C42—1012no—
195C42—2022no—
196C42C22×D10111yescharacteristic
197C23—21no—
198C23—2021no—
199Q8—22no—
200Q8—22no—
201Q8—4044no—
202Q8—4042no—
203Q8—2022no—
204D4—21no—
205D4—2021no—
206D4—2021no—
207D4#1310807312yes—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—1012no—
212C2×C4—1012no—
213C2×C4—4044no—
214C2×C4—4042no—
215C2×C4—2021no—
216C2×C4—2022no—
217C2×C4D4×D10111yescharacteristic
218C2×C4D4×D10111yescharacteristic
219C2×C4C23×D10111yescharacteristic
220C8—22no—
221C8—2021no—
222C8—2021no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—4041no—
227C22—4041no—
228C22—4041no—
229C22C2×D4×D10111yescharacteristic
230C4—21no—
231C4—42no—
232C4—41no—
233C4—4042no—
234C4—4041no—
235C4—2022no—
236C4—2022no—
237C4#1569228212yes—
238C4C2×#1310807311yescharacteristic
239C4#1569230211yescharacteristic
240C2—42no—
241C2—8081no—
242C2#2026583311yescharacteristic
243C2C2×#1569228211yescharacteristic
244C2#2026599311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#461×C101C2
socleC22×C101C22×D4
prime radicalC2×#13108059C22
hypercenter#461D101
hypocenterC101#3367
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #20265901C2semidirect
2. D4×Dic202C2semidirect
3. #20265538C2semidirect
4. #20265527C2semidirect
5. #20265470C2semidirect
6. #20265458C2semidirect
8. #20265304C2semidirect
9. #20265278C2semidirect
11. #461×C101C2semidirect
123. C101#3367semidirect
131. #461D101semidirect

Maximal subgroups

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

subgroupindex
#202659012
D4×Dic2022
#202655382
#202655272
#202654702
#202654582
#202653942
#202653042
#202652782
C4×#131080592
#461×C1012
#3367101