Finite Group Databasebeta

← #27187118

Subgroups — #27187118

Order 12928.

360 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 78 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×#15692334C211yesmaximal, characteristic
2C2×#15692318C212yesmaximal, complemented
3D101⋊(C22⋊C8)C211yesmaximal, characteristic
4#20266050C212yesmaximal, complemented
5C22⋊Q8×D101C211yesmaximal, characteristic
6C2×D101⋊C8C2212yes—
7#15692334—24no—
8C2×C4⋊#87938C2212yes—
9C2×C22⋊#87938—21no—
10C2×C4×#87938—21no—
11#15692318—24no—
12#15692312—21no—
13C22⋊#11717581—21no—
14C2×Q8×D101C2211yescharacteristic
15C22×C4×D101C2211yescharacteristic
16#15692235C411yescharacteristic, complemented
17#15692234C411yescharacteristic, complemented
18#15692233—21no—
19#15692232—21no—
20#15692231C411yescharacteristic, complemented
21C4⋊C4×D101—21no—
22C4⋊C4×D101C2211yescharacteristic
23C22⋊C4×D101—21no—
24C22⋊Q8×C101C411yescharacteristic, complemented
25C22×#87938—22no—
26D101⋊C8—22no—
27D101⋊C8—22no—
28C2×#11717581—22no—
29C4⋊#87938—22no—
30C4⋊#87938D414yes—
31C22⋊#87938—24no—
32C4×#87938—24no—
33C23×D101D411yescharacteristic
34C2×Dic202—21no—
35C2×Dic202C2×C411yescharacteristic
36Q8×D101—22no—
37Q8×D101—22no—
38C22×Dic101D411yescharacteristic
39C2×C4×D101—22no—
40C2×C4×D101—22no—
41C2×C4×D101—21no—
42C2×C4×D101—21no—
43C2×C4×D101C2×C411yescharacteristic
44C2×C4×D101D411yescharacteristic
45C2×C4×D101D411yescharacteristic
46C2×C4×D101C2311yescharacteristic
47C4⋊Dic101—21no—
48C4⋊Dic101C2×C411yescharacteristic
49#13108053—21no—
50#13108053—21no—
51#13108053—21no—
52C22⋊Dic101—21no—
53#13108051—21no—
54#13108051—21no—
55C2×Q8×C101C2×C411yescharacteristic
56C22×C4×C101C2×C411yescharacteristic
57C4⋊C4×C101—21no—
58C4⋊C4×C101C2×C411yescharacteristic
59C22⋊C4×C101—21no—
60C23×C101C22⋊C411yescharacteristic
61C22×D101—21no—
62C22×D101—21no—
63C22×D101—22no—
64C22×D101—22no—
65C22×D101C22⋊C411yescharacteristic
66C22×D101C2×D411yescharacteristic
67C2×#87938—24no—
68C2×#87938—42no—
69C2×#87938—42no—
70Q8×C101—22no—
71C2×C4×C101—22no—
72C2×C4×C101—21no—
73C2×C4×C101—21no—
74C2×C4×C101C22⋊C411yescharacteristic
75C2×C4×C101C22×C411yescharacteristic
76C2×Dic101—22no—
77C2×Dic101—21no—
78C2×Dic101—21no—
79C2×Dic101C22⋊C411yescharacteristic
80C2×Dic101C2×D411yescharacteristic
81C4×D101—21no—
82C4×D101—21no—
83C4×D101—22no—
84C4×D101—22no—
85C4×D101—41no—
86C4×D101—41no—
87C4×D101—42no—
88C4×D101C2×D412yes—
89C4×D101#4112yes—
90Dic202—22no—
91Dic202—42no—
92#11717581—42no—
93C22×C101—21no—
94C22×C101—21no—
95C22×C101—21no—
96C22×C101C2×C22⋊C411yescharacteristic
97C2×D101—21no—
98C2×D101—21no—
99C2×D101—21no—
100C2×D101—42no—
101C2×D101—42no—
102C2×D101#12011yescharacteristic
103C2×D101#12311yescharacteristic
104C2×D101#13512yes—
105C2×D101#13712yes—
106#87938—44no—
107C4×C101—21no—
108C4×C101—41no—
109C4×C101—41no—
110C4×C101C4×D412yes—
111Dic101—21no—
112Dic101—41no—
113Dic101—41no—
114Dic101#12112yes—
115C2×C101—41no—
116C2×C101#38611yescharacteristic
117C2×C101#41611yescharacteristic
118C2×C101#41911yescharacteristic
119D101—41no—
120D101#45212yes—
121D101#48512yes—
122#1873—1011nomaximal, Sylow
123C101#187311yescharacteristic, complemented, Sylow
124C2×C22⋊Q8—1011no—
125C2×#117—1011no—
126C2×#103—1012no—
127C2×C22⋊C8—1011no—
128#340—1012no—
129C22×Q8—1011no—
130C23×C4—1011no—
131C22×C8—1012no—
132C22⋊Q8—1011no—
133C22⋊Q8—1011no—
134C22⋊Q8—1011no—
135C22⋊Q8—2021no—
136C22⋊Q8—2021no—
137C22⋊Q8#8793811yescharacteristic, complemented
138#117—2024no—
139C2×C4⋊C4—1012no—
140C2×C4⋊C4—1011no—
141C2×C4⋊C4—2021no—
142C2×C22⋊C4—2021no—
143C2×C22⋊C4—2021no—
144C2×C42—2021no—
145#103—2024no—
146C22⋊C8—2021no—
147C22⋊C8—2021no—
148C24—1011no—
149C2×Q8—1011no—
150C2×Q8—2021no—
151C2×Q8—2022no—
152C2×Q8—2022no—
153C2×Q8C2×#8793811yescharacteristic
154C22×C4—1011no—
155C22×C4—1011no—
156C22×C4—1011no—
157C22×C4—1011no—
158C22×C4—1011no—
159C22×C4—2022no—
160C22×C4—2022no—
161C22×C4—2021no—
162C22×C4—2021no—
163C22×C4—2022no—
164C22×C4C2×#8793811yescharacteristic
165C2×C8—2022no—
166C2×C8—2022no—
167C2×C8—2022no—
168C4⋊C4—21no—
169C4⋊C4—1014no—
170C4⋊C4—1011no—
171C4⋊C4—2021no—
172C4⋊C4—2021no—
173C4⋊C4—2021no—
174C4⋊C4—2022no—
175C4⋊C4—2021no—
176C4⋊C4C2×#8793811yescharacteristic
177C22⋊C4—21no—
178C22⋊C4—2021no—
179C22⋊C4—2021no—
180C22⋊C4—2024no—
181C22⋊C4—2021no—
182C42—2024no—
183C23—1011no—
184C23—1011no—
185C23—2021no—
186C23—2021no—
187C23—2022no—
188C23—2022no—
189C23C22⋊#8793811yescharacteristic
190Q8—22no—
191Q8—4042no—
192Q8—2022no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—1011no—
197C2×C4—1012no—
198C2×C4—1012no—
199C2×C4—1011no—
200C2×C4—4041no—
201C2×C4—4041no—
202C2×C4—4042no—
203C2×C4—4042no—
204C2×C4—4042no—
205C2×C4—2022no—
206C2×C4—2021no—
207C2×C4—2021no—
208C2×C4—2022no—
209C2×C4—2024no—
210C2×C4—2021no—
211C2×C4—2021no—
212C2×C4—2022no—
213C2×C4C22⋊#8793811yescharacteristic
214C2×C4C22×#8793811yescharacteristic
215C8—4042no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—1011no—
220C22—1011no—
221C22—1012no—
222C22—1012no—
223C22—4042no—
224C22—4042no—
225C22—2021no—
226C22—2021no—
227C22—2021no—
228C22C2×C22⋊#8793811yescharacteristic
229C4—21no—
230C4—41no—
231C4—41no—
232C4—1012no—
233C4—4044no—
234C4—4041no—
235C4—4041no—
236C4—2021no—
237C4D4×#8793812yes—
238C2—41no—
239C2—1012no—
240C2—1012no—
241C2—4041no—
242C2#2026612511yescharacteristic
243C2#2026616711yescharacteristic
244C2#2026617011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC22⋊Q8×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC23×D101D4
cosocleC2×C4×D101C23
hypercenterC22⋊Q8#87938
hypocenterC101#1873
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#15692318C2semidirect
4. #20266050C2semidirect
16. #15692235C4semidirect
17. #15692234C4semidirect
20. #15692231C4semidirect
24. C22⋊Q8×C101C4semidirect
123. C101#1873semidirect
137. C22⋊Q8#87938semidirect

Maximal subgroups

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

subgroupindex
C2×#156923342
C2×#156923182
D101⋊(C22⋊C8)2
#202660502
C22⋊Q8×D1012
#1873101