Finite Group Databasebeta

← #27186961

Subgroups — #27186961

Order 12928.

612 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 86 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×#15692314C212yesmaximal, complemented
2C2×#15692313C212yesmaximal, complemented
3#20266120C212yesmaximal, complemented
4C22×#13108074C211yesmaximal, characteristic
5C2×C22⋊#87938C2214yes—
6#15692314—24no—
7#15692313—24no—
8(C2×#87938)⋊C4—24no—
9C2×#13108074—24no—
10C2×#13108074—24no—
11C2×#13108074—24no—
12C22×Dic202C411yescharacteristic, complemented
13C22×D404C2211yescharacteristic
14C22×C22⋊D101C411yescharacteristic, complemented
15C22×C22⋊D101C2211yescharacteristic
16C22×C4×D101C411yescharacteristic, complemented
17C22×C4×D101C2211yescharacteristic
18C23×C4×C101C411yescharacteristic, complemented
19C22×#87938—24no—
20C22⋊#87938—24no—
21C22⋊#87938—24no—
22C23×D101C2×C411yescharacteristic
23C23×D101C2311yescharacteristic
24#13108074—232no—
25C2×Dic202—22no—
26C2×Dic202—22no—
27C2×Dic202—22no—
28C2×D404—22no—
29C2×D404—22no—
30C2×D404D414yes—
31C2×C22⋊D101—22no—
32C2×C22⋊D101—22no—
33C2×C22⋊D101—22no—
34C2×C22⋊D101—22no—
35C2×C22⋊D101—22no—
36C2×C22⋊D101D414yes—
37C22×Dic101C2×C411yescharacteristic
38C22×Dic101C2×C411yescharacteristic
39C2×C4×D101—22no—
40C2×C4×D101—22no—
41C2×C4×D101—22no—
42C2×C4×D101—22no—
43C2×C4×D101—22no—
44C2×C4×D101D414yes—
45C24×C101C2×C411yescharacteristic
46C22×C4×C101—22no—
47C22×C4×C101—22no—
48C22×C4×C101—22no—
49C22×C4×C101C2×C411yescharacteristic
50C22×C4×C101C2×C411yescharacteristic
51C23×C101—21no—
52C23×C101—21no—
53C23×C101—42no—
54C23×C101C22⋊C412yes—
55C23×C101C22×C411yescharacteristic
56C22×D101—21no—
57C22×D101—21no—
58C22×D101—42no—
59C22×D101—42no—
60C22×D101C22⋊C412yes—
61C22×D101C2×D412yes—
62C22×D101C2×D412yes—
63C22×D101C2×D412yes—
64C2×#87938—48no—
65C2×#87938—44no—
66C2×C4×C101—28no—
67C2×C4×C101—21no—
68C2×C4×C101—21no—
69C2×C4×C101—21no—
70C2×C4×C101—21no—
71C2×C4×C101C22⋊C412yes—
72C2×C4×C101C22⋊C412yes—
73C22⋊D101—28no—
74C22⋊D101—28no—
75C2×Dic101—21no—
76C2×Dic101—21no—
77C2×Dic101—21no—
78C2×Dic101—21no—
79C2×Dic101C22⋊C412yes—
80C2×Dic101C22⋊C412yes—
81D404—28no—
82C4×D101—28no—
83C4×D101—28no—
84Dic202—28no—
85C22×C101—22no—
86C22×C101—22no—
87C22×C101—42no—
88C22×C101—42no—
89C22×C101—42no—
90C22×C101C2×C22⋊C411yescharacteristic
91C22×C101C2×C22⋊C411yescharacteristic
92C22×C101C2×C22⋊C411yescharacteristic
93C2×D101—24no—
94C2×D101—22no—
95C2×D101—42no—
96C2×D101—42no—
97C2×D101—42no—
98C2×D101—42no—
99C2×D101—42no—
100C2×D101#12014yes—
101#87938—84no—
102C4×C101—22no—
103C4×C101—22no—
104Dic101—22no—
105Dic101—22no—
106C2×C101—21no—
107C2×C101—21no—
108C2×C101—42no—
109C2×C101#37911yescharacteristic
110C2×C101#41012yes—
111D101—24no—
112D101—42no—
113#1782—1011nomaximal, Sylow
114C101#178211yescharacteristic, complemented, Sylow
115C22×#41—1011no—
116C2×#99—1012no—
117C2×#99—1012no—
118#381—1012no—
119C2×#41—2024no—
120C2×#41—2024no—
121C2×#41—2024no—
122C22×Q8—1011no—
123C22×D4—1011no—
124C22×D4—1011no—
125C22×D4—1011no—
126C23×C4—1011no—
127C23×C4—1011no—
128C23×C4#8793811yescharacteristic, complemented
129C2×C22⋊C4—1014no—
130#99—2024no—
131#99—2024no—
132#95—2024no—
133C24—1011no—
134C24—1011no—
135C24C2×#8793811yescharacteristic
136#41—20232no—
137C2×Q8—2022no—
138C2×Q8—2022no—
139C2×Q8—2022no—
140C2×D4—1014no—
141C2×D4—1014no—
142C2×D4—2022no—
143C2×D4—2022no—
144C2×D4—2022no—
145C2×D4—2022no—
146C2×D4—2022no—
147C2×D4—2022no—
148C2×D4—2022no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—1011no—
153C22×C4—1011no—
154C22×C4—1014no—
155C22×C4—2022no—
156C22×C4—2022no—
157C22×C4—2022no—
158C22×C4—2022no—
159C22×C4—2024no—
160C22×C4—2022no—
161C22×C4C2×#8793811yescharacteristic
162C22×C4C2×#8793811yescharacteristic
163C22⋊C4—2024no—
164C22⋊C4—2024no—
165C23—21no—
166C23—21no—
167C23—42no—
168C23—1012no—
169C23—1012no—
170C23—1012no—
171C23—1012no—
172C23—4042no—
173C23—4042no—
174C23—2021no—
175C23—2021no—
176C23C22⋊#8793812yes—
177C23C22×#8793811yescharacteristic
178Q8—2028no—
179D4—2028no—
180D4—2028no—
181D4—2028no—
182C2×C4—28no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—1012no—
188C2×C4—1012no—
189C2×C4—4044no—
190C2×C4—4048no—
191C2×C4—2028no—
192C2×C4—2028no—
193C2×C4—2021no—
194C2×C4—2021no—
195C2×C4—2021no—
196C2×C4—2021no—
197C2×C4C22⋊#8793812yes—
198C2×C4C22⋊#8793812yes—
199C22—22no—
200C22—22no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—1014no—
205C22—4042no—
206C22—4042no—
207C22—4042no—
208C22—4042no—
209C22—4042no—
210C22—2022no—
211C22—2024no—
212C22C2×C22⋊#8793811yescharacteristic
213C22C2×C22⋊#8793811yescharacteristic
214C22C2×C22⋊#8793811yescharacteristic
215C4—22no—
216C4—22no—
217C4—8084no—
218C4—2022no—
219C4—2022no—
220C2—21no—
221C2—21no—
222C2—42no—
223C2—4042no—
224C2—2024no—
225C2#2026611811yescharacteristic
226C2#2026615612yes—
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
FittingC23×C4×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC22×#13108074C2
cosocleC23×D101C23
hypercenterC23×C4#87938
hypocenterC101#1782
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C2×#15692314C2semidirect
2. C2×#15692313C2semidirect
3. #20266120C2semidirect
12. C22×Dic202C4semidirect
14. C22×C22⋊D101C4semidirect
16. C22×C4×D101C4semidirect
18. C23×C4×C101C4semidirect
114. C101#1782semidirect
128. C23×C4#87938semidirect

Maximal subgroups

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

subgroupindex
C2×#156923142
C2×#156923132
#202661202
C22×#131080742
#1782101