Finite Group Databasebeta

← #27186991

Subgroups — #27186991

Order 12928.

312 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 92 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×#15692322C211yesmaximal, characteristic, complemented
2C2×C4⋊#11717581C211yesmaximal, characteristic, complemented
3C2×#15692312C212yesmaximal
4#20266043C212yesmaximal, complemented
5C2×#15692213C211yesmaximal, characteristic
6C22×#11717581C2212yes—
7C22×#11717581C2212yes—
8#15692322—22no—
9C4⋊#11717581—22no—
10#15692312—24no—
11C22×C4×D101C2211yescharacteristic
12#15692213—22no—
13#15692213—22no—
14C2×C4⋊Dic101C411yescharacteristic
15C2×#13108051C411yescharacteristic
16C2×#13108051C411yescharacteristic
17C2×C4×Dic101C2211yescharacteristic
18C2×C4×Dic101C2211yescharacteristic
19C2×C4⋊C4×C101C411yescharacteristic
20C2×#11717581—22no—
21C2×#11717581—22no—
22C2×#11717581—22no—
23C2×#11717581—22no—
24C2×#11717581—22no—
25C2×#11717581—22no—
26C23×D101C2×C411yescharacteristic
27C22×Dic101C2×C411yescharacteristic
28C22×Dic101C2×C411yescharacteristic
29C22×Dic101C2311yescharacteristic
30C2×C4×D101—21no—
31C2×C4×D101—21no—
32C2×C4×D101—21no—
33C2×C4×D101—21no—
34C2×C4×D101D412yes—
35C2×C4×D101Q812yes—
36C4⋊Dic101—22no—
37#13108051—22no—
38#13108051—22no—
39C4×Dic101—21no—
40C4×Dic101—21no—
41C4×Dic101D412yes—
42C4×Dic101D412yes—
43C22×C4×C101C2×C411yescharacteristic
44C22×C4×C101C2×C411yescharacteristic
45C22×C4×C101C2×C411yescharacteristic
46C4⋊C4×C101—22no—
47C23×C101C22×C411yescharacteristic
48C22×D101—21no—
49C22×D101—21no—
50C22×D101—22no—
51C22×D101—22no—
52C22×D101C4⋊C412yes—
53C2×C4×C101—21no—
54C2×C4×C101—21no—
55C2×C4×C101—22no—
56C2×C4×C101—21no—
57C2×C4×C101—22no—
58C2×C4×C101C22⋊C412yes—
59C2×C4×C101C4⋊C412yes—
60C2×Dic101—22no—
61C2×Dic101—21no—
62C2×Dic101—22no—
63C2×Dic101C22⋊C412yes—
64C2×Dic101C2×D411yescharacteristic
65C2×Dic101C2×D411yescharacteristic
66C2×Dic101C2×D411yescharacteristic
67C2×Dic101C2×Q811yescharacteristic
68C2×Dic101#4111yescharacteristic
69C2×Dic101#4111yescharacteristic
70C4×D101—22no—
71C4×D101—22no—
72C4×D101—22no—
73C4×D101—22no—
74#11717581—42no—
75#11717581—42no—
76C22×C101C2×C22⋊C411yescharacteristic
77C22×C101C2×C4⋊C411yescharacteristic
78C22×C101#11711yescharacteristic
79C22×C101#13112yes—
80C22×C101#13112yes—
81C2×D101—22no—
82C2×D101—24no—
83C2×D101—41no—
84C2×D101—41no—
85C2×D101—42no—
86C4×C101—22no—
87C4×C101—22no—
88C4×C101—41no—
89Dic101—22no—
90Dic101—41no—
91Dic101#12111yescharacteristic
92Dic101#12111yescharacteristic
93Dic101C22⋊Q811yescharacteristic
94Dic101C22⋊Q811yescharacteristic
95C2×C101#38011yescharacteristic
96C2×C101#40811yescharacteristic
97C2×C101#40811yescharacteristic
98C2×C101#42412yes—
99C2×C101#43312yes—
100D101—42no—
101#1798—1011nomaximal, Sylow
102C101#179811yescharacteristic, complemented, Sylow
103C2×#117—1011no—
104C2×C4⋊C8—1011no—
105C2×C4⋊C8—1011no—
106C2×C22⋊C8—1012no—
107#336—1012no—
108C23×C4—1011no—
109C22×C8—1012no—
110C22×C8—1012no—
111#117—2022no—
112#117—2022no—
113C2×C4⋊C4—1011no—
114C2×C4⋊C4#8793811yescharacteristic
115C2×C22⋊C4—1011no—
116C2×C22⋊C4—1011no—
117C2×C42—1011no—
118C2×C42—1011no—
119C4⋊C8—2022no—
120C4⋊C8—2022no—
121C22⋊C8—2024no—
122C24—1011no—
123C22×C4—1011no—
124C22×C4—1011no—
125C22×C4—1012no—
126C22×C4—1012no—
127C22×C4—1011no—
128C22×C4—2021no—
129C22×C4—2021no—
130C22×C4—2021no—
131C22×C4—2021no—
132C22×C4C2×#8793811yescharacteristic
133C22×C4C2×#8793811yescharacteristic
134C22×C4C2×#8793811yescharacteristic
135C2×C8—2022no—
136C2×C8—2022no—
137C2×C8—2022no—
138C2×C8—2022no—
139C2×C8—2022no—
140C2×C8—2022no—
141C4⋊C4—22no—
142C4⋊C4—2022no—
143C22⋊C4—2022no—
144C22⋊C4—2022no—
145C42—1012no—
146C42—1012no—
147C42—2021no—
148C42—2021no—
149C23—1012no—
150C23—2021no—
151C23—2021no—
152C23—2022no—
153C23—2022no—
154C23C22×#8793811yescharacteristic
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—1011no—
161C2×C4—1011no—
162C2×C4—1011no—
163C2×C4—1011no—
164C2×C4—1011no—
165C2×C4—1011no—
166C2×C4—1012no—
167C2×C4—2022no—
168C2×C4—2022no—
169C2×C4—2022no—
170C2×C4—2022no—
171C2×C4—2021no—
172C2×C4—2022no—
173C2×C4—2022no—
174C2×C4C22⋊#8793812yes—
175C2×C4C4⋊#8793812yes—
176C8—4042no—
177C8—4042no—
178C22—4041no—
179C22—4041no—
180C22—4042no—
181C22—2022no—
182C22—2024no—
183C22C2×C22⋊#8793811yescharacteristic
184C22C2×C4⋊#8793811yescharacteristic
185C22#1569233411yescharacteristic
186C22#1569234212yes—
187C22#1569234312yes—
188C4—22no—
189C4—22no—
190C4—41no—
191C4—1011no—
192C4—1011no—
193C4—1011no—
194C4—1011no—
195C4—4041no—
196C4—2022no—
197C2—4042no—
198C2#2026611911yescharacteristic
199C2#2026615011yescharacteristic
200C2#2026615211yescharacteristic
201C2#2026618412yes—
202C2#2026620212yes—
subgroupG/H
center, FrattiniC23C22×#87938
commutator, socleC23×C101C22×C4
FittingC2×C4⋊C4×C101C4
prime radicalC23×D101C2×C4
cosocleC22×Dic101C23
hypercenterC2×C4⋊C4#87938
hypocenterC101#1798
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#15692322C2semidirect
2. C2×C4⋊#11717581C2semidirect
4. #20266043C2semidirect
102. C101#1798semidirect

Maximal subgroups

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

subgroupindex
C2×#156923222
C2×C4⋊#117175812
C2×#156923122
#202660432
C2×#156922132
#1798101