Finite Group Databasebeta

← #27183785

Subgroups — #27183785

Order 12928.

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

normal subgroups only

This group has 350 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
2#20265884C211yesmaximal, characteristic, complemented
3#20265533C212yesmaximal, complemented
4#20265527C211yesmaximal, characteristic, complemented
5#20265482C212yesmaximal, complemented
6#20265468C211yesmaximal, characteristic, complemented
7#20265394C212yesmaximal
8#20265301C211yesmaximal, characteristic, complemented
9SD16×Dic101C212yesmaximal, complemented
10C4×#13108059C211yesmaximal, characteristic
11#461×C101C211yesmaximal, characteristic, complemented
12C2×#13108059C2211yescharacteristic
13#15692255C2211yescharacteristic
14#15692250—22no—
15#15692246C2212yes—
16#15692244C2212yes—
17#15692235—21no—
18#15692234—21no—
19#15692232—22no—
20Q8×Dic101C2212yes—
21C4×Dic202C2211yescharacteristic
22C4×D404C2211yescharacteristic
23D4×Dic101C2212yes—
24#15692213—21no—
25C2×C4⋊Dic101—21no—
26#15692188C2211yescharacteristic
27#15692187C2212yes—
28#15692184C2212yes—
29#15692179C2212yes—
30#15692179C2212yes—
31#15692177C2212yes—
32#15692176C2211yescharacteristic
33#15692175C2211yescharacteristic
34C4×#11717579C2211yescharacteristic
35C8×Dic101C2212yes—
36C2×SD16×C101C2212yes—
37C4⋊Q8×C101C2211yescharacteristic
38C4×D4×C101C2211yescharacteristic
39C4⋊C8×C101C2211yescharacteristic
40#103×C101C2212yes—
41C2×Dic202—21no—
42C2×Dic202D411yescharacteristic
43C2×D404C2311yescharacteristic
44C22×Dic101—22no—
45C2×C4×D101—21no—
46#13108059—22no—
47C2×#11717579C2311yescharacteristic
48C2×#11717579C2311yescharacteristic
49C4⋊Dic101—21no—
50C4⋊Dic101—22no—
51C4⋊Dic101—22no—
52C4⋊Dic101D411yescharacteristic
53C4⋊Dic101C2311yescharacteristic
54C4⋊Dic101C2312yes—
55#13108053—22no—
56#13108053D412yes—
57C22⋊Dic101—22no—
58#13108051—21no—
59#13108051—22no—
60#13108051—22no—
61C4×Dic101—22no—
62C4×Dic101C2312yes—
63C2×Q8×C101C2312yes—
64C2×D4×C101C2311yescharacteristic
65C22×C4×C101—21no—
66SD16×C101—24no—
67C2×C8×C101C2312yes—
68C4⋊C4×C101—21no—
69C4⋊C4×C101—21no—
70C4⋊C4×C101C2311yescharacteristic
71C22⋊C4×C101—21no—
72C42×C101C2311yescharacteristic
73C23×C101—21no—
74C22×D101—21no—
75Q8×C101—22no—
76Q8×C101—22no—
77D4×C101—21no—
78D4×C101#4112yes—
79C2×C4×C101—21no—
80C2×C4×C101—22no—
81C2×C4×C101—42no—
82C2×C4×C101C2×D411yescharacteristic
83C2×C4×C101C2×D411yescharacteristic
84C2×C4×C101C2411yescharacteristic
85C2×Dic101—21no—
86C2×Dic101—22no—
87C2×Dic101—82no—
88C2×Dic101C2×D412yes—
89C2×Dic101C2×D412yes—
90D404—21no—
91D404—21no—
92C4×D101—42no—
93Dic202—22no—
94Dic202—81no—
95C8×C101—22no—
96#11717579—21no—
97#11717579—21no—
98C22×C101—22no—
99C22×C101—41no—
100C22×C101—41no—
101C22×C101C22×D411yescharacteristic
102C2×D101—41no—
103C2×D101—41no—
104C2×D101—41no—
105C4×C101—21no—
106C4×C101—42no—
107C4×C101—41no—
108C4×C101#13512yes—
109C4×C101C2×#4111yescharacteristic
110C4×C1012−1+411yescharacteristic
111Dic101—22no—
112Dic101—22no—
113Dic101—41no—
114Dic101—42no—
115C2×C101—42no—
116C2×C101#54811yescharacteristic
117C2×C101C2×#13511yescharacteristic
118C2×C101#57711yescharacteristic
119D101—81no—
120#3360—1011nomaximal, Sylow
121C101#336011yescharacteristic, complemented, Sylow
122#556—1011no—
123#554—1011no—
124#488—1012no—
125#487—1011no—
126#479—1012no—
127#476—1011no—
128#463—1012no—
129#461D10111yescharacteristic, complemented
130C8×Q8—1011no—
131C4×SD16—1012no—
132C4×D8—1011no—
133C2×SD16C2×D10112yes—
134C2×D8—1011no—
135C4⋊Q8—1011no—
136C4⋊Q8C2×D10111yescharacteristic
137#126—2022no—
138#125—1012no—
139#124—1012no—
140C22⋊Q8—2022no—
141C22⋊Q8—2021no—
142C22⋊Q8—2021no—
143C4×Q8—1011no—
144C4×Q8—1012no—
145C4×D4—1012no—
146C4×D4—1011no—
147C4×D4C2×D10111yescharacteristic
148#117—2021no—
149C2×C4⋊C4—2021no—
150#107—1011no—
151#106—1012no—
152C4⋊C8—1012no—
153C4⋊C8C2×D10111yescharacteristic
154#103—1012no—
155#103—1012no—
156#103C2×D10112yes—
157#102—1011no—
158#102—1012no—
159#102—1011no—
160C4×C8—1012no—
161C4×C8—1011no—
162C2×Q8—1011no—
163C2×Q8—2021no—
164C2×Q8C22×D10112yes—
165C2×D4—1011no—
166C2×D4C22×D10111yescharacteristic
167C22×C4—21no—
168C22×C4—2022no—
169C22×C4—2021no—
170SD16—24no—
171D8—2022no—
172C2×C8—1011no—
173C2×C8—1011no—
174C2×C8C22×D10112yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—1012no—
178C4⋊C4—1011no—
179C4⋊C4—1012no—
180C4⋊C4—1011no—
181C4⋊C4—2022no—
182C4⋊C4—2022no—
183C4⋊C4—2022no—
184C4⋊C4—2021no—
185C4⋊C4C22×D10111yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—2022no—
188C22⋊C4—2022no—
189C22⋊C4—2022no—
190C22⋊C4—2021no—
191C42—1012no—
192C42—2022no—
193C42C22×D10111yescharacteristic
194C23—21no—
195C23—2021no—
196Q8—22no—
197Q8—22no—
198Q8—8081no—
199Q8—2022no—
200D4—21no—
201D4—2021no—
202D4—2021no—
203D4#1310807312yes—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—8082no—
208C2×C4—1012no—
209C2×C4—1012no—
210C2×C4—4042no—
211C2×C4—2021no—
212C2×C4—2022no—
213C2×C4D4×D10111yescharacteristic
214C2×C4D4×D10111yescharacteristic
215C2×C4C23×D10111yescharacteristic
216C8—22no—
217C8—2021no—
218C8—2021no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—4041no—
223C22—4041no—
224C22—4041no—
225C22C2×D4×D10111yescharacteristic
226C4—21no—
227C4—42no—
228C4—41no—
229C4—4042no—
230C4—4041no—
231C4—2022no—
232C4—2022no—
233C4#1569227612yes—
234C4C2×#1310807311yescharacteristic
235C4#1569230211yescharacteristic
236C2—42no—
237C2—8081no—
238C2#2026583311yescharacteristic
239C2C2×#1569227611yescharacteristic
240C2#2026598611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#461×C101C2
socleC22×C101C22×D4
prime radicalC2×#13108059C22
hypercenter#461D101
hypocenterC101#3360
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. #20265884C2semidirect
3. #20265533C2semidirect
4. #20265527C2semidirect
5. #20265482C2semidirect
6. #20265468C2semidirect
8. #20265301C2semidirect
9. SD16×Dic101C2semidirect
11. #461×C101C2semidirect
121. C101#3360semidirect
129. #461D101semidirect

Maximal subgroups

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

subgroupindex
#202659012
#202658842
#202655332
#202655272
#202654822
#202654682
#202653942
#202653012
SD16×Dic1012
C4×#131080592
#461×C1012
#3360101