Finite Group Databasebeta

← #27186969

Subgroups — #27186969

Order 12928.

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

normal subgroups only

This group has 584 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×#15692316C212yesmaximal, complemented
2C2×#15692315C212yesmaximal, complemented
3#20266147C212yesmaximal, complemented
4C22×#13108073C211yesmaximal, characteristic
5C2×#13108083C2214yes—
6#15692316—24no—
7#15692315—24no—
8C22⋊#11717581—24no—
9C2×#13108073—24no—
10C2×#13108073—24no—
11C2×#13108073—24no—
12C22×Dic202C2211yescharacteristic
13C22×C22⋊D101C411yescharacteristic
14C22×C22⋊D101C2211yescharacteristic
15C23×Dic101C411yescharacteristic
16C23×Dic101C2211yescharacteristic
17C22×C4×D101C411yescharacteristic
18C22×D4×C101C411yescharacteristic
19#13108083—24no—
20#13108083—24no—
21C2×#11717581—24no—
22C23×D101C2×C411yescharacteristic
23#13108073—232no—
24C2×Dic202—22no—
25C2×Dic202—22no—
26C2×Dic202D414yes—
27C2×C22⋊D101—22no—
28C2×C22⋊D101—22no—
29C2×C22⋊D101—22no—
30C2×C22⋊D101—22no—
31C2×C22⋊D101—22no—
32C2×C22⋊D101D414yes—
33C22×Dic101—22no—
34C22×Dic101—22no—
35C22×Dic101—22no—
36C22×Dic101—22no—
37C22×Dic101—22no—
38C22×Dic101C2×C411yescharacteristic
39C22×Dic101C2×C411yescharacteristic
40C22×Dic101D414yes—
41C22×Dic101C2311yescharacteristic
42C2×C4×D101—22no—
43C2×C4×D101—22no—
44C2×C4×D101—22no—
45C24×C101C2×C411yescharacteristic
46C24×C101C2×C411yescharacteristic
47C2×D4×C101—22no—
48C2×D4×C101—22no—
49C2×D4×C101—22no—
50C22×C4×C101C2×C411yescharacteristic
51C23×C101—21no—
52C23×C101—21no—
53C23×C101—21no—
54C23×C101—21no—
55C23×C101—42no—
56C23×C101—42no—
57C23×C101C22⋊C412yes—
58C23×C101C22⋊C412yes—
59C23×C101C22×C411yescharacteristic
60C22×D101—21no—
61C22×D101—21no—
62C22×D101—42no—
63C22×D101C22⋊C412yes—
64D4×C101—28no—
65C2×C4×C101—21no—
66C2×C4×C101—21no—
67C2×C4×C101C22⋊C412yes—
68C22⋊D101—28no—
69C22⋊D101—28no—
70C2×Dic101—28no—
71C2×Dic101—21no—
72C2×Dic101—21no—
73C2×Dic101—21no—
74C2×Dic101—21no—
75C2×Dic101—28no—
76C2×Dic101C22⋊C412yes—
77C2×Dic101C22⋊C412yes—
78C2×Dic101C2×D412yes—
79C2×Dic101C2×D412yes—
80C2×Dic101C2×D412yes—
81C4×D101—28no—
82Dic202—28no—
83#11717581—44no—
84C22×C101—22no—
85C22×C101—22no—
86C22×C101—22no—
87C22×C101—24no—
88C22×C101—42no—
89C22×C101—42no—
90C22×C101—42no—
91C22×C101—42no—
92C22×C101—42no—
93C22×C101C2×C22⋊C411yescharacteristic
94C22×C101C2×C22⋊C411yescharacteristic
95C22×C101C2×C22⋊C411yescharacteristic
96C2×D101—22no—
97C2×D101—42no—
98C2×D101—42no—
99C2×D101—42no—
100C4×C101—22no—
101Dic101—22no—
102Dic101—22no—
103Dic101#12014yes—
104C2×C101—21no—
105C2×C101—21no—
106C2×C101—42no—
107C2×C101—42no—
108C2×C101#37911yescharacteristic
109C2×C101#41312yes—
110D101—42no—
111#1785—1011nomaximal, Sylow
112C101#178511yescharacteristic, complemented, Sylow
113C22×#41—1011no—
114C2×#101—1012no—
115C2×#100—1012no—
116#407—1012no—
117C2×#41—2024no—
118C2×#41—2024no—
119C2×#41—2024no—
120C22×Q8—1011no—
121C22×D4—1011no—
122C22×D4—1011no—
123C22×D4#8793811yescharacteristic
124C23×C4—1011no—
125C23×C4—1011no—
126C23×C4—1011no—
127C2×M16—1014no—
128#101—2024no—
129#100—2024no—
130C22⋊C8—2024no—
131C24—1011no—
132C24C2×#8793811yescharacteristic
133C24C2×#8793811yescharacteristic
134#41—20232no—
135C2×Q8—1014no—
136C2×Q8—2022no—
137C2×Q8—2022no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—1014no—
142C2×D4—2022no—
143C2×D4—2022no—
144C2×D4—2022no—
145C2×D4—2022no—
146C2×D4—2022no—
147C22×C4—1011no—
148C22×C4—1014no—
149C22×C4—1011no—
150C22×C4—1011no—
151C22×C4—2022no—
152C22×C4—2022no—
153C22×C4—2022no—
154C22×C4—2022no—
155C22×C4—2022no—
156C22×C4—2022no—
157C22×C4—2022no—
158C22×C4—2022no—
159C22×C4C2×#8793811yescharacteristic
160M16—2024no—
161M16—2024no—
162C2×C8—2024no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—42no—
168C23—42no—
169C23—1012no—
170C23—4042no—
171C23—2021no—
172C23—2021no—
173C23C22⋊#8793812yes—
174C23C22⋊#8793812yes—
175C23C22×#8793811yescharacteristic
176Q8—2028no—
177D4—28no—
178D4—2028no—
179D4—2028no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—1012no—
183C2×C4—1012no—
184C2×C4—1012no—
185C2×C4—1012no—
186C2×C4—1012no—
187C2×C4—2028no—
188C2×C4—2028no—
189C2×C4—2028no—
190C2×C4—2021no—
191C2×C4—2021no—
192C2×C4—2021no—
193C2×C4—2021no—
194C2×C4C22⋊#8793812yes—
195C8—4044no—
196C22—22no—
197C22—22no—
198C22—22no—
199C22—24no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—42no—
205C22—4042no—
206C22—4042no—
207C22—4042no—
208C22—2022no—
209C22C2×C22⋊#8793811yescharacteristic
210C22C2×C22⋊#8793811yescharacteristic
211C22C2×C22⋊#8793811yescharacteristic
212C4—22no—
213C4—1014no—
214C4—2022no—
215C4—2022no—
216C2—21no—
217C2—21no—
218C2—42no—
219C2—42no—
220C2—4042no—
221C2#2026611811yescharacteristic
222C2#2026616412yes—
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
FittingC22×D4×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC22×#13108073C2
cosocleC22×Dic101C23
hypercenterC22×D4#87938
hypocenterC101#1785
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#15692316C2semidirect
2. C2×#15692315C2semidirect
3. #20266147C2semidirect
112. C101#1785semidirect

Maximal subgroups

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

subgroupindex
C2×#156923162
C2×#156923152
#202661472
C22×#131080732
#1785101