Finite Group Databasebeta

← #27169690

Subgroups — #27169690

Order 12928.

264 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 69 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#20265041C211yesmaximal, 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×Dic101C2211yescharacteristic
19#15692184—21no—
20C22⋊#11717579C2211yescharacteristic
21C8×Dic101—21no—
22#15692162—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—21no—
42#13108053—42no—
43C22⋊Dic101—21no—
44#13108051—21no—
45#13108051—21no—
46C4×Dic101C2×C411yescharacteristic
47C4×Dic101C2×C411yescharacteristic
48C4×Dic101D411yescharacteristic
49C4×Dic101D411yescharacteristic
50C2×D4×C101—21no—
51C2×D4×C101C2×C411yescharacteristic
52C22×C4×C101—21no—
53C22×C4×C101C2311yescharacteristic
54C2×C8×C101—21no—
55C4⋊C4×C101C2×C411yescharacteristic
56C22⋊C4×C101—21no—
57C23×C101—21no—
58C23×C101C2×D411yescharacteristic
59C22×D101—21no—
60D4×C101—41no—
61D4×C101—41no—
62D4×C101—41no—
63C2×C4×C101—21no—
64C2×C4×C101—41no—
65C2×C4×C101—41no—
66C2×C4×C101—41no—
67C2×C4×C101—41no—
68C2×C4×C101C22×C411yescharacteristic
69C2×C4×C101C2×D411yescharacteristic
70C22⋊D101—42no—
71C22⋊D101—41no—
72C22⋊D101—41no—
73C22⋊D101—42no—
74C2×Dic101—21no—
75C2×Dic101—41no—
76C2×Dic101—42no—
77C2×Dic101—41no—
78C2×Dic101—41no—
79C2×Dic101—41no—
80C2×Dic101C22⋊C411yescharacteristic
81C2×Dic101C22⋊C411yescharacteristic
82C2×Dic101C22⋊C411yescharacteristic
83C2×Dic101C22⋊C411yescharacteristic
84C2×Dic101D812yes—
85C2×Dic101SD1612yes—
86D404—41no—
87C4×D101—42no—
88Dic202—42no—
89C8×C101—41no—
90#11717579—41no—
91C22×C101—21no—
92C22×C101—21no—
93C22×C101—41no—
94C22×C101—41no—
95C22×C101—41no—
96C22×C101C2×C22⋊C411yescharacteristic
97C22×C101C2×D811yescharacteristic
98C22×C101C2×SD1611yescharacteristic
99C2×D101—41no—
100C2×D101—41no—
101C2×D101—41no—
102C4×C101—21no—
103C4×C101—21no—
104C4×C101—41no—
105C4×C101—81no—
106Dic101—21no—
107Dic101—21no—
108Dic101—41no—
109Dic101—81no—
110Dic101#10212yes—
111Dic101#10212yes—
112C2×C101—21no—
113C2×C101—21no—
114C2×C101—81no—
115C2×C101#41011yescharacteristic
116C2×C101C2×#10211yescharacteristic
117C2×C101#42111yescharacteristic
118D101—81no—
119#1517—1011nomaximal, Sylow
120C101#151711yescharacteristic, complemented, Sylow
121#532—1011no—
122#431—1011no—
123#382—1011no—
124#327—1011no—
125#327—1011no—
126#327—1011no—
127#327D10111yescharacteristic, complemented
128C2×#41—2021no—
129C4⋊Q8—1011no—
130#127—1011no—
131#124—2021no—
132#121—1011no—
133#121—2021no—
134#121C2×D10111yescharacteristic
135C4×D4—1011no—
136C4×D4—1011no—
137C4×D4—2021no—
138C2×C4⋊C4—2021no—
139C2×C42—1011no—
140C4⋊C8—2021no—
141C22⋊C8—1011no—
142C22⋊C8C2×D10111yescharacteristic
143C4×C8—2021no—
144#95—1011no—
145#95—2021no—
146#95C2×D10111yescharacteristic
147#41—4042no—
148#41—4042no—
149C2×Q8—2021no—
150C2×D4—21no—
151C2×D4—1011no—
152C2×D4—2021no—
153C2×D4—2021no—
154C2×D4—2021no—
155C2×D4C4×D10111yescharacteristic
156C22×C4—21no—
157C22×C4—1011no—
158C22×C4—1011no—
159C22×C4—2021no—
160C22×C4—2021no—
161C22×C4—2021no—
162C22×C4C22×D10111yescharacteristic
163C2×C8—21no—
164C2×C8—2021no—
165C4⋊C4—1011no—
166C4⋊C4—4042no—
167C4⋊C4—2021no—
168C4⋊C4C4×D10111yescharacteristic
169C22⋊C4—21no—
170C22⋊C4—2021no—
171C22⋊C4—2021no—
172C22⋊C4—2021no—
173C42—1011no—
174C42—1011no—
175C42—1011no—
176C42—1011no—
177C23—21no—
178C23—2021no—
179C23D4×D10111yescharacteristic
180Q8—4042no—
181D4—41no—
182D4—41no—
183D4—41no—
184D4—4041no—
185D4—4042no—
186D4—4041no—
187D4—4041no—
188D4—4042no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—1011no—
195C2×C4—1011no—
196C2×C4—1012no—
197C2×C4—1012no—
198C2×C4—1011no—
199C2×C4—1011no—
200C2×C4—4042no—
201C2×C4—4041no—
202C2×C4—4041no—
203C2×C4—4041no—
204C2×C4—4042no—
205C2×C4—4041no—
206C2×C4—2021no—
207C2×C4C2×C4×D10111yescharacteristic
208C2×C4D4×D10111yescharacteristic
209C8—41no—
210C8—4041no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—4041no—
217C22—4041no—
218C22—4041no—
219C22C22⋊C4×D10111yescharacteristic
220C22D8×D10111yescharacteristic
221C22SD16×D10111yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—81no—
226C4—8081no—
227C4—1012no—
228C4—1012no—
229C4—4041no—
230C4—2021no—
231C4—2021no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—8081no—
236C2#2026514911yescharacteristic
237C2#102×D10111yescharacteristic
238C2#2026519211yescharacteristic
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#1517
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. #20265041C2semidirect
4. #20264912C2semidirect
5. #20264911C2semidirect
6. #20264910C2semidirect
7. #327×C101C2semidirect
9. #15692254C4semidirect
10. #15692251C4semidirect
14. D4×Dic101C4semidirect
15. #15692217C4semidirect
120. C101#1517semidirect
127. #327D101semidirect

Maximal subgroups

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

subgroupindex
#202657222
#202652272
#202650412
#202649122
#202649112
#202649102
#327×C1012
#1517101