Finite Group Databasebeta

← #27187255

Subgroups — #27187255

Order 12928.

396 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 104 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×#15692312C212yesmaximal, complemented
3C2×C22⋊#11717581C212yesmaximal, complemented
4C2×C4×#11717581C211yesmaximal, characteristic, complemented
5C2×#15692215C211yesmaximal, characteristic
6C22×#11717581C2212yes—
7C22×#11717581C2212yes—
8#15692322—22no—
9#15692312—24no—
10C22⋊#11717581—24no—
11C4×#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
21C2×#11717581—22no—
22C2×#11717581—22no—
23C2×#11717581—22no—
24C2×#11717581—22no—
25C2×#11717581—22no—
26C2×#11717581D414yescomplemented
27C23×D101C2×C411yescharacteristic
28C2×C22⋊D101—22no—
29C2×C22⋊D101—22no—
30C2×C22⋊D101C814yescomplemented
31C22×Dic101—22no—
32C22×Dic101—21no—
33C22×Dic101—21no—
34C22×Dic101—22no—
35C22×Dic101C2×C411yescharacteristic
36C22×Dic101C2×C411yescharacteristic
37C22×Dic101C2311yescharacteristic
38C2×C4×D101—22no—
39C2×C4×D101—21no—
40C2×C4×D101—21no—
41C2×C4×D101—22no—
42#13108053—22no—
43#13108051—22no—
44C4×Dic101D412yes—
45C4×Dic101D412yes—
46C24×C101C2×C411yescharacteristic
47C22×C4×C101C2×C411yescharacteristic
48C22×C4×C101C2×C411yescharacteristic
49C22⋊C4×C101—22no—
50C23×C101—21no—
51C23×C101—22no—
52C23×C101—21no—
53C23×C101—22no—
54C23×C101C2×C812yes—
55C23×C101C22×C411yescharacteristic
56C22×D101—21no—
57C22×D101—22no—
58C22×D101—21no—
59C22×D101—22no—
60C22×D101C2×C812yes—
61C2×C4×C101—22no—
62C2×C4×C101—21no—
63C2×C4×C101—22no—
64C2×C4×C101C22⋊C412yes—
65C22⋊D101—28no—
66C2×Dic101—21no—
67C2×Dic101—22no—
68C2×Dic101—42no—
69C2×Dic101—41no—
70C2×Dic101—41no—
71C2×Dic101C22⋊C412yes—
72C2×Dic101C2×C812yes—
73C2×Dic101M1612yes—
74C2×Dic101C2×D411yescharacteristic
75C2×Dic101C2×D412yes—
76C2×Dic101C2×D411yescharacteristic
77C2×Dic101#4112yes—
78C4×D101—42no—
79C4×D101—41no—
80C4×D101—41no—
81#11717581—24no—
82#11717581—42no—
83C22×C101—22no—
84C22×C101—24no—
85C22×C101—41no—
86C22×C101—41no—
87C22×C101—42no—
88C22×C101C2×C22⋊C411yescharacteristic
89C22×C101C4×D412yes—
90C22×C101C22×C811yescharacteristic
91C22×C101C2×M1611yescharacteristic
92C22×C101#13112yes—
93C2×D101—22no—
94C2×D101—24no—
95C2×D101—41no—
96C2×D101—41no—
97C2×D101—42no—
98C4×C101—22no—
99C4×C101—41no—
100Dic101—41no—
101Dic101C22⋊C814yescomplemented
102Dic101#12112yes—
103Dic101#12411yescharacteristic
104Dic101#12711yescharacteristic
105C2×C101—42no—
106C2×C101#39011yescharacteristic
107C2×C101C2×C22⋊C811yescharacteristic
108C2×C101#40811yescharacteristic
109C2×C101C8×D412yes—
110C2×C101#43612yes—
111D101—42no—
112#1954—1011nomaximal, Sylow
113C101#195411yescharacteristic, complemented, Sylow
114C2×C4×D4—1011no—
115C2×C4⋊C8—1011no—
116C2×C22⋊C8—1012no—
117C2×C22⋊C8—1012no—
118C2×C4×C8—1011no—
119C22×D4—1011no—
120C23×C4—1011no—
121C23×C4—1011no—
122C22×C8—1012no—
123C22×C8—1012no—
124C4×D4—2022no—
125C4×D4—2022no—
126C2×C4⋊C4—1011no—
127C2×C22⋊C4—1011no—
128C2×C22⋊C4#8793811yescharacteristic
129C2×C42—1011no—
130C4⋊C8—2022no—
131C22⋊C8—2024no—
132C22⋊C8—2024no—
133C4×C8—2022no—
134C24—1011no—
135C24C2×#8793811yescharacteristic
136C2×D4—1014no—
137C2×D4—2022no—
138C2×D4—2022no—
139C22×C4—1011no—
140C22×C4—1011no—
141C22×C4—1011no—
142C22×C4—2022no—
143C22×C4—2021no—
144C22×C4—2021no—
145C22×C4—2022no—
146C22×C4—2022no—
147C22×C4—2021no—
148C22×C4—2021no—
149C22×C4—2022no—
150C22×C4C2×#8793811yescharacteristic
151C22×C4C2×#8793811yescharacteristic
152C2×C8—1014no—
153C2×C8—2022no—
154C2×C8—2022no—
155C2×C8—2022no—
156C2×C8—2022no—
157C2×C8—2022no—
158C4⋊C4—2022no—
159C22⋊C4—22no—
160C22⋊C4—2022no—
161C42—1012no—
162C42—1012no—
163C23—21no—
164C23—22no—
165C23—21no—
166C23—22no—
167C23—1012no—
168C23—2021no—
169C23—2022no—
170C23—2021no—
171C23—2022no—
172C23D101⋊C812yes—
173C23C22×#8793811yescharacteristic
174D4—2028no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—1012no—
179C2×C4—1011no—
180C2×C4—1011no—
181C2×C4—1012no—
182C2×C4—1012no—
183C2×C4—1012no—
184C2×C4—1012no—
185C2×C4—4041no—
186C2×C4—4042no—
187C2×C4—4041no—
188C2×C4—4041no—
189C2×C4—4042no—
190C2×C4—4041no—
191C2×C4—2021no—
192C2×C4—2022no—
193C2×C4C22⋊#8793812yes—
194C8—4042no—
195C8—2024no—
196C22—22no—
197C22—24no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22—4041no—
202C22—4041no—
203C22—4042no—
204C22—2022no—
205C22—2024no—
206C22C2×C22⋊#8793811yescharacteristic
207C22D4×#8793812yes—
208C22C2×D101⋊C811yescharacteristic
209C22D101⋊M1611yescharacteristic
210C22#1569234212yes—
211C4—22no—
212C4—41no—
213C4—1011no—
214C4—1012no—
215C4—1011no—
216C4—1014no—
217C4—4041no—
218C2—42no—
219C2—4042no—
220C2#2026612911yescharacteristic
221C2D101⋊(C22⋊C8)11yescharacteristic
222C2#2026615011yescharacteristic
223C2#2026620412yes—
224C2#2026621012yes—
subgroupG/H
center, FrattiniC23C22×#87938
commutatorC22×C101C22×C8
FittingC2×C22⋊C4×C101C4
socleC23×C101C22×C4
prime radicalC22×C22⋊D101C4
cosocleC22×Dic101C23
hypercenterC2×C22⋊C4#87938
hypocenterC101#1954
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#15692322C2semidirect
2. C2×#15692312C2semidirect
3. C2×C22⋊#11717581C2semidirect
4. C2×C4×#11717581C2semidirect
26. C2×#11717581D4semidirect
30. C2×C22⋊D101C8semidirect
101. Dic101C22⋊C8semidirect
113. C101#1954semidirect

Maximal subgroups

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

subgroupindex
C2×#156923222
C2×#156923122
C2×C22⋊#117175812
C2×C4×#117175812
C2×#156922152
#1954101