Finite Group Databasebeta

← #27186510

Subgroups — #27186510

Order 12928.

304 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C2×#15692312C211yesmaximal, characteristic, complemented
2C4×C22⋊#87938C211yesmaximal, characteristic, complemented
3#20266031C212yesmaximal, complemented
4#20266029C212yesmaximal, complemented
5#20265690C211yesmaximal, characteristic
6C22×#11717581—21no—
7C2×C22⋊#87938—21no—
8C2×C4×#87938—21no—
9#15692312—21no—
10#15692312C2212yes—
11(C2×#87938)⋊C4C2212yes—
12C2×#13108075C411yescharacteristic, complemented
13C22×C4×D101C2211yescharacteristic
14#15692240—21no—
15#15692231C2211yescharacteristic
16#15692230C2211yescharacteristic
17#15692229C411yescharacteristic, complemented
18#15692225—21no—
19#15692217—21no—
20#15692215—21no—
21#15692213C411yescharacteristic, complemented
22C22⋊Q8×C101C411yescharacteristic, complemented
23C22×#87938—22no—
24C2×#11717581—22no—
25C2×#11717581—42no—
26C22⋊#87938—24no—
27C4×#87938—42no—
28C23×D101D411yescharacteristic
29#13108075—42no—
30C2×D404—21no—
31C2×D404—21no—
32C2×D404C2×C411yescharacteristic
33C2×C22⋊D101—21no—
34C22×Dic101D411yescharacteristic
35C2×C4×D101—21no—
36C2×C4×D101—42no—
37C2×C4×D101C2×C411yescharacteristic
38C2×C4×D101D411yescharacteristic
39C2×C4×D101D411yescharacteristic
40C2×C4×D101C2311yescharacteristic
41C4⋊Dic101C2×C411yescharacteristic
42#13108053—21no—
43#13108051—21no—
44#13108051—21no—
45#13108051—21no—
46#13108051—21no—
47C4×Dic101—21no—
48C2×Q8×C101C2×C411yescharacteristic
49C22×C4×C101C2×C411yescharacteristic
50C4⋊C4×C101—21no—
51C4⋊C4×C101C2×C411yescharacteristic
52C22⋊C4×C101—21no—
53C23×C101C22⋊C411yescharacteristic
54C22×D101—22no—
55C22×D101—22no—
56C22×D101—21no—
57C22×D101—41no—
58C22×D101C22⋊C411yescharacteristic
59C22×D101C2×D411yescharacteristic
60C2×#87938—24no—
61C2×#87938—42no—
62C2×#87938—42no—
63Q8×C101—41no—
64C2×C4×C101—21no—
65C2×C4×C101—21no—
66C2×C4×C101—41no—
67C2×C4×C101C22⋊C411yescharacteristic
68C2×C4×C101C22×C411yescharacteristic
69C22⋊D101—42no—
70C2×Dic101—22no—
71C2×Dic101—21no—
72C2×Dic101C22⋊C411yescharacteristic
73C2×Dic101C2×D411yescharacteristic
74D404—41no—
75D404—42no—
76D404—42no—
77C4×D101—22no—
78C4×D101—22no—
79C4×D101—41no—
80C4×D101—41no—
81C4×D101—42no—
82#11717581—42no—
83C22×C101—21no—
84C22×C101—21no—
85C22×C101#10412yes—
86C22×C101C2×C22⋊C411yescharacteristic
87C2×D101—21no—
88C2×D101—21no—
89C2×D101—21no—
90C2×D101—21no—
91C2×D101—21no—
92C2×D101—44no—
93C2×D101—41no—
94C2×D101—41no—
95C2×D101—41no—
96C2×D101#13512yes—
97#87938—44no—
98C4×C101—21no—
99C4×C101—21no—
100C4×C101—41no—
101C4×C101—41no—
102Dic101—21no—
103Dic101—41no—
104Dic101#9912yes—
105C2×C101—22no—
106C2×C101C2×#9911yescharacteristic
107C2×C101#41611yescharacteristic
108C2×C101C2×#10411yescharacteristic
109D101—22no—
110D101—41no—
111D101—81no—
112#1501—1011nomaximal, Sylow
113C101#150111yescharacteristic, complemented, Sylow
114#525—1011no—
115C2×C22⋊C8—1011no—
116C4×C22⋊C4—1011no—
117#328—1012no—
118#327—1012no—
119C2×#41—1011no—
120C23×C4—1011no—
121C22×C8—2021no—
122#123—2021no—
123C22⋊Q8—1011no—
124C22⋊Q8#8793811yescharacteristic, complemented
125#121—1011no—
126#121—1011no—
127#120—2021no—
128C4×D4—2021no—
129C4×D4—2021no—
130#117—1011no—
131C2×C22⋊C4—2021no—
132C2×C42—2021no—
133C22⋊C8—1012no—
134C22⋊C8—2021no—
135#95—1012no—
136C24—1011no—
137#41—4042no—
138C2×Q8C2×#8793811yescharacteristic
139C2×D4—1011no—
140C2×D4—2021no—
141C2×D4—2021no—
142C2×D4—2021no—
143C22×C4—1011no—
144C22×C4—1011no—
145C22×C4—1011no—
146C22×C4—1011no—
147C22×C4—1011no—
148C22×C4—4042no—
149C22×C4—2021no—
150C22×C4—2022no—
151C22×C4C2×#8793811yescharacteristic
152C2×C8—4042no—
153C2×C8—2022no—
154C4⋊C4—21no—
155C4⋊C4—1011no—
156C4⋊C4—2021no—
157C4⋊C4C2×#8793811yescharacteristic
158C22⋊C4—21no—
159C22⋊C4—2021no—
160C22⋊C4—2021no—
161C22⋊C4—2021no—
162C22⋊C4—2024no—
163C22⋊C4—2021no—
164C42—4042no—
165C42—2021no—
166C23—1011no—
167C23—1011no—
168C23—4041no—
169C23—2022no—
170C23—2021no—
171C23—2022no—
172C23C22⋊#8793811yescharacteristic
173Q8—41no—
174D4—4041no—
175D4—4042no—
176D4—4042no—
177D4—4042no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—1011no—
182C2×C4—1011no—
183C2×C4—4041no—
184C2×C4—4041no—
185C2×C4—4042no—
186C2×C4—4042no—
187C2×C4—4042no—
188C2×C4—2022no—
189C2×C4—2022no—
190C2×C4—2022no—
191C2×C4—2024no—
192C2×C4—2021no—
193C2×C4C22⋊#8793811yescharacteristic
194C2×C4C22×#8793811yescharacteristic
195C8—4042no—
196C22—21no—
197C22—21no—
198C22—1012no—
199C22—4044no—
200C22—4041no—
201C22—4041no—
202C22—4041no—
203C22—2021no—
204C22—2021no—
205C22—2021no—
206C22—2021no—
207C22—2021no—
208C22#1569232012yes—
209C22C2×C22⋊#8793811yescharacteristic
210C4—21no—
211C4—21no—
212C4—41no—
213C4—41no—
214C4—1012no—
215C4—4044no—
216C4—4041no—
217C4—2021no—
218C2—22no—
219C2—8081no—
220C2—4041no—
221C2—2022no—
222C2D101⋊#9911yescharacteristic
223C2#2026616711yescharacteristic
224C2C2×#1569232011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC22⋊Q8×C101C4
socleC22×C101C2×C22⋊C4
prime radical#20265690C2
cosocleC2×C4×D101C23
hypercenterC22⋊Q8#87938
hypocenterC101#1501
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#15692312C2semidirect
2. C4×C22⋊#87938C2semidirect
3. #20266031C2semidirect
4. #20266029C2semidirect
12. C2×#13108075C4semidirect
17. #15692229C4semidirect
21. #15692213C4semidirect
22. C22⋊Q8×C101C4semidirect
113. C101#1501semidirect
124. C22⋊Q8#87938semidirect

Maximal subgroups

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

subgroupindex
C2×#156923122
C4×C22⋊#879382
#202660312
#202660292
#202656902
#1501101