Finite Group Databasebeta

← #27169625

Subgroups — #27169625

Order 12928.

284 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 71 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#20265722C211yesmaximal, characteristic
2#20265227C211yesmaximal, characteristic, complemented
3#20265028C211yesmaximal, characteristic, complemented
4#20264912C211yesmaximal, characteristic, complemented
5#20264911C211yesmaximal, characteristic, complemented
6#20264910C211yesmaximal, characteristic, complemented
7#327×C101C211yesmaximal, characteristic, complemented
8C2×#13108073—21no—
9#15692254C411yescharacteristic, complemented
10#15692251C411yescharacteristic, complemented
11#15692241—21no—
12#15692229C2211yescharacteristic
13#15692227—21no—
14D4×Dic101C411yescharacteristic, complemented
15#15692217C411yescharacteristic, complemented
16#15692215—21no—
17C2×#13108053—21no—
18C2×C4×Dic101—21no—
19C2×C4×Dic101C2211yescharacteristic
20#15692184—21no—
21C22⋊#11717579C2211yescharacteristic
22C8×Dic101—21no—
23#15692162C2211yescharacteristic
24#121×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic
26#95×C101C2211yescharacteristic
27#13108073—42no—
28#13108073—42no—
29C2×Dic202—21no—
30C2×D404C2×C411yescharacteristic
31C2×C22⋊D101—21no—
32C2×C22⋊D101—21no—
33C2×C22⋊D101—21no—
34C22×Dic101—21no—
35C22×Dic101—21no—
36C22×Dic101D411yescharacteristic
37C22×Dic101D411yescharacteristic
38C2×C4×D101—21no—
39C2×#11717579—21no—
40C4⋊Dic101C2×C411yescharacteristic
41#13108053—24no—
42#13108053—21no—
43C22⋊Dic101—21no—
44#13108051—21no—
45#13108051—21no—
46C4×Dic101—42no—
47C4×Dic101C2×C411yescharacteristic
48C4×Dic101C2×C411yescharacteristic
49C4×Dic101D411yescharacteristic
50C4×Dic101D411yescharacteristic
51C2×D4×C101—21no—
52C2×D4×C101C2×C411yescharacteristic
53C22×C4×C101—21no—
54C22×C4×C101C2311yescharacteristic
55C2×C8×C101—21no—
56C4⋊C4×C101C2×C411yescharacteristic
57C22⋊C4×C101—21no—
58C23×C101—21no—
59C23×C101C2×D411yescharacteristic
60C22×D101—21no—
61D4×C101—41no—
62D4×C101—41no—
63D4×C101—41no—
64C2×C4×C101—22no—
65C2×C4×C101—21no—
66C2×C4×C101—41no—
67C2×C4×C101—41no—
68C2×C4×C101—41no—
69C2×C4×C101C22×C411yescharacteristic
70C2×C4×C101C2×D411yescharacteristic
71C22⋊D101—41no—
72C22⋊D101—41no—
73C22⋊D101—42no—
74C22⋊D101—42no—
75C2×Dic101—22no—
76C2×Dic101—22no—
77C2×Dic101—21no—
78C2×Dic101—41no—
79C2×Dic101—41no—
80C2×Dic101—42no—
81C2×Dic101C22⋊C411yescharacteristic
82C2×Dic101C22⋊C411yescharacteristic
83C2×Dic101C22⋊C411yescharacteristic
84C2×Dic101C22⋊C411yescharacteristic
85C2×Dic101D812yes—
86C2×Dic101SD1612yes—
87D404—41no—
88C4×D101—42no—
89Dic202—42no—
90C8×C101—41no—
91#11717579—41no—
92C22×C101—21no—
93C22×C101—21no—
94C22×C101—41no—
95C22×C101—41no—
96C22×C101—41no—
97C22×C101C2×C22⋊C411yescharacteristic
98C22×C101C2×D811yescharacteristic
99C22×C101C2×SD1611yescharacteristic
100C2×D101—41no—
101C2×D101—41no—
102C2×D101—41no—
103C4×C101—21no—
104C4×C101—21no—
105C4×C101—42no—
106C4×C101—41no—
107Dic101—21no—
108Dic101—42no—
109Dic101—41no—
110Dic101#10212yes—
111Dic101#10212yes—
112Dic101#10412yes—
113C2×C101—21no—
114C2×C101—21no—
115C2×C101—81no—
116C2×C101#41011yescharacteristic
117C2×C101C2×#10211yescharacteristic
118C2×C101C2×#10411yescharacteristic
119D101—81no—
120#1502—1011nomaximal, Sylow
121C101#150211yescharacteristic, complemented, Sylow
122#532—1011no—
123#431—1011no—
124C4×C4⋊C4—1011no—
125#327—1011no—
126#327—1011no—
127#327—1011no—
128#327D10111yescharacteristic, complemented
129C2×#41—2021no—
130C4⋊Q8—1011no—
131#127—1011no—
132#124—2021no—
133#121—1011no—
134#121—2021no—
135#121C2×D10111yescharacteristic
136C4×D4—1011no—
137C4×D4—1011no—
138C4×D4—2021no—
139C2×C4⋊C4—2021no—
140C2×C42—1011no—
141C2×C42—2021no—
142C4⋊C8—2021no—
143C22⋊C8—1011no—
144C22⋊C8C2×D10111yescharacteristic
145C4×C8—2021no—
146#95—1011no—
147#95C2×D10111yescharacteristic
148#41—4042no—
149#41—4042no—
150C2×Q8—2021no—
151C2×D4—21no—
152C2×D4—1011no—
153C2×D4—2021no—
154C2×D4—2021no—
155C2×D4—2021no—
156C2×D4C4×D10111yescharacteristic
157C22×C4—21no—
158C22×C4—1011no—
159C22×C4—1011no—
160C22×C4—2021no—
161C22×C4—2021no—
162C22×C4—2021no—
163C22×C4C22×D10111yescharacteristic
164C2×C8—21no—
165C2×C8—2021no—
166C4⋊C4—1011no—
167C4⋊C4—2021no—
168C4⋊C4—2024no—
169C4⋊C4C4×D10111yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—2021no—
172C22⋊C4—2021no—
173C22⋊C4—2021no—
174C42—1011no—
175C42—1011no—
176C42—1011no—
177C42—1011no—
178C42—4042no—
179C23—21no—
180C23—2021no—
181C23D4×D10111yescharacteristic
182Q8—4042no—
183D4—41no—
184D4—41no—
185D4—41no—
186D4—4041no—
187D4—4042no—
188D4—4042no—
189D4—4041no—
190D4—4041no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—1011no—
197C2×C4—1011no—
198C2×C4—1011no—
199C2×C4—1012no—
200C2×C4—1012no—
201C2×C4—1011no—
202C2×C4—4042no—
203C2×C4—4042no—
204C2×C4—4041no—
205C2×C4—4041no—
206C2×C4—2022no—
207C2×C4—2022no—
208C2×C4—2021no—
209C2×C4C2×C4×D10111yescharacteristic
210C2×C4D4×D10111yescharacteristic
211C8—41no—
212C8—4041no—
213C22—21no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—4041no—
219C22—4041no—
220C22—4041no—
221C22C22⋊C4×D10111yescharacteristic
222C22D8×D10111yescharacteristic
223C22SD16×D10111yescharacteristic
224C4—21no—
225C4—21no—
226C4—42no—
227C4—41no—
228C4—1012no—
229C4—1012no—
230C4—1012no—
231C4—4042no—
232C4—4041no—
233C4—2021no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—8081no—
238C2#2026514911yescharacteristic
239C2#102×D10111yescharacteristic
240C2#104×D10111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#327×C101C2
socleC22×C101C2×C22⋊C4
prime radical#20265722C2
cosocleC22×C4×C101C23
hypercenter#327D101
hypocenterC101#1502
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #20265227C2semidirect
3. #20265028C2semidirect
4. #20264912C2semidirect
5. #20264911C2semidirect
6. #20264910C2semidirect
7. #327×C101C2semidirect
9. #15692254C4semidirect
10. #15692251C4semidirect
14. D4×Dic101C4semidirect
15. #15692217C4semidirect
121. C101#1502semidirect
128. #327D101semidirect

Maximal subgroups

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

subgroupindex
#202657222
#202652272
#202650282
#202649122
#202649112
#202649102
#327×C1012
#1502101