Finite Group Databasebeta

← #27187111

Subgroups — #27187111

Order 12928.

408 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C22×#13108083C211yesmaximal, characteristic, complemented
2C2×#15692312C212yesmaximal, complemented
3C2×C22⋊#11717581C211yesmaximal, characteristic, complemented
4#20266043C212yesmaximal, complemented
5C2×#15692215C211yesmaximal, characteristic
6C2×#13108083—24no—
7C2×#13108083—22no—
8C22×#11717581C2212yes—
9C22×#11717581C2212yes—
10#15692312—24no—
11C22⋊#11717581—22no—
12C22×C22⋊D101C411yescharacteristic
13C23×Dic101C2211yescharacteristic
14C22×C4×D101C2211yescharacteristic
15#15692215—22no—
16#15692215—22no—
17C2×#13108053C411yescharacteristic
18C2×#13108051C411yescharacteristic
19C2×C4×Dic101C2211yescharacteristic
20C2×C22⋊C4×C101C411yescharacteristic
21#13108083—28no—
22C2×#11717581—22no—
23C2×#11717581—22no—
24C2×#11717581—22no—
25C2×#11717581—22no—
26C2×#11717581—22no—
27C2×#11717581D414yescomplemented
28C23×D101C2×C411yescharacteristic
29C2×C22⋊D101—22no—
30C2×C22⋊D101—22no—
31C2×C22⋊D101—22no—
32C22×Dic101—22no—
33C22×Dic101—22no—
34C22×Dic101C2×C411yescharacteristic
35C22×Dic101C2×C411yescharacteristic
36C22×Dic101D412yes—
37C22×Dic101D412yes—
38C22×Dic101C2311yescharacteristic
39C2×C4×D101—22no—
40C2×C4×D101—21no—
41C2×C4×D101—21no—
42C2×C4×D101—22no—
43#13108053—22no—
44#13108051—22no—
45C4×Dic101—21no—
46C4×Dic101—21no—
47C24×C101C2×C411yescharacteristic
48C22×C4×C101C2×C411yescharacteristic
49C22×C4×C101C2×C411yescharacteristic
50C22⋊C4×C101—22no—
51C23×C101—22no—
52C23×C101—21no—
53C23×C101—21no—
54C23×C101—22no—
55C23×C101C22⋊C412yes—
56C23×C101C22×C411yescharacteristic
57C22×D101—21no—
58C22×D101—22no—
59C22×D101—21no—
60C22×D101—22no—
61C22×D101M1612yescomplemented
62C2×C4×C101—21no—
63C2×C4×C101—21no—
64C2×C4×C101—22no—
65C2×C4×C101—22no—
66C22⋊D101—28no—
67C2×Dic101—21no—
68C2×Dic101—22no—
69C2×Dic101—21no—
70C2×Dic101—22no—
71C2×Dic101—24no—
72C2×Dic101—22no—
73C2×Dic101C22⋊C412yes—
74C2×Dic101M1612yescomplemented
75C2×Dic101C2×D412yes—
76C2×Dic101C2×D411yescharacteristic
77C2×Dic101C2×D411yescharacteristic
78C2×Dic101#4112yes—
79C4×D101—41no—
80C4×D101—42no—
81C4×D101—41no—
82#11717581—24no—
83#11717581—42no—
84C22×C101—22no—
85C22×C101—22no—
86C22×C101—22no—
87C22×C101—42no—
88C22×C101—42no—
89C22×C101C2×C22⋊C411yescharacteristic
90C22×C101C4×D412yes—
91C22×C101C2×M1611yescharacteristic
92C22×C101C2×M1611yescharacteristic
93C22×C101#13112yes—
94C2×D101—22no—
95C2×D101—24no—
96C2×D101—41no—
97C2×D101—41no—
98C2×D101—42no—
99C4×C101—41no—
100C4×C101—41no—
101Dic101—22no—
102Dic101—22no—
103Dic101#12011yescharacteristic
104Dic101#12112yes—
105Dic101#12311yescharacteristic
106C2×C101—42no—
107C2×C101#38611yescharacteristic
108C2×C101#40711yescharacteristic
109C2×C101#40811yescharacteristic
110C2×C101#43512yes—
111C2×C101#43612yes—
112D101—42no—
113#1870—1011nomaximal, Sylow
114C101#187011yescharacteristic, complemented, Sylow
115C22×M16—1011no—
116C2×C4×D4—1011no—
117C2×C22⋊C8—1011no—
118C2×C22⋊C8—1012no—
119#336—1012no—
120C22×D4—1011no—
121C23×C4—1011no—
122C23×C4—1011no—
123C2×M16—2024no—
124C2×M16—2022no—
125C22×C8—1012no—
126C22×C8—1012no—
127C4×D4—2022no—
128C4×D4—2022no—
129C2×C4⋊C4—1011no—
130C2×C22⋊C4—1011no—
131C2×C22⋊C4#8793811yescharacteristic
132C2×C42—1011no—
133C22⋊C8—2024no—
134C22⋊C8—2022no—
135C24—1011no—
136C24C2×#8793811yescharacteristic
137C2×D4—2022no—
138C2×D4—2022no—
139C2×D4—2022no—
140C22×C4—1011no—
141C22×C4—1012no—
142C22×C4—1012no—
143C22×C4—1011no—
144C22×C4—1011no—
145C22×C4—2022no—
146C22×C4—2021no—
147C22×C4—2021no—
148C22×C4—2022no—
149C22×C4—2022no—
150C22×C4—2022no—
151C22×C4C2×#8793811yescharacteristic
152C22×C4C2×#8793811yescharacteristic
153M16—2028no—
154C2×C8—1014no—
155C2×C8—2022no—
156C2×C8—2022no—
157C2×C8—2022no—
158C2×C8—2022no—
159C2×C8—2022no—
160C4⋊C4—2022no—
161C22⋊C4—22no—
162C22⋊C4—2022no—
163C42—2021no—
164C42—2021no—
165C23—22no—
166C23—21no—
167C23—21no—
168C23—22no—
169C23—1012no—
170C23—2021no—
171C23—2022no—
172C23—2021no—
173C23—2022no—
174C23C22⋊#8793812yes—
175C23C22×#8793811yescharacteristic
176D4—2028no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—1011no—
182C2×C4—1011no—
183C2×C4—1012no—
184C2×C4—1012no—
185C2×C4—1012no—
186C2×C4—1012no—
187C2×C4—4042no—
188C2×C4—4041no—
189C2×C4—4041no—
190C2×C4—2022no—
191C2×C4—2024no—
192C2×C4—2022no—
193C2×C4—2021no—
194C2×C4—2022no—
195C2×C4—2021no—
196C8—4042no—
197C8—2024no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—42no—
202C22—42no—
203C22—4041no—
204C22—4041no—
205C22—4042no—
206C22—2022no—
207C22—2024no—
208C22C2×C22⋊#8793811yescharacteristic
209C22D4×#8793812yes—
210C22D101⋊M1611yescharacteristic
211C22D101⋊M1611yescharacteristic
212C22#1569234212yes—
213C4—41no—
214C4—41no—
215C4—1012no—
216C4—1011no—
217C4—1011no—
218C4—2022no—
219C4—2022no—
220C2—42no—
221C2—4042no—
222C2#2026612511yescharacteristic
223C2#2026614911yescharacteristic
224C2#2026615011yescharacteristic
225C2#2026620912yes—
226C2#2026621012yes—
subgroupG/H
center, FrattiniC23C22×#87938
commutator, socleC23×C101C22×C4
FittingC2×C22⋊C4×C101C4
prime radicalC22×C22⋊D101C4
cosocleC22×Dic101C23
hypercenterC2×C22⋊C4#87938
hypocenterC101#1870
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#13108083C2semidirect
2. C2×#15692312C2semidirect
3. C2×C22⋊#11717581C2semidirect
4. #20266043C2semidirect
27. C2×#11717581D4semidirect
61. C22×D101M16semidirect
74. C2×Dic101M16semidirect
114. C101#1870semidirect

Maximal subgroups

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

subgroupindex
C22×#131080832
C2×#156923122
C2×C22⋊#117175812
#202660432
C2×#156922152
#1870101