Finite Group Databasebeta

← #27186432

Subgroups — #27186432

Order 12928.

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

normal subgroups only

This group has 312 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
1#20266196C211yesmaximal, characteristic, complemented
2#20266195C211yesmaximal, characteristic, complemented
3#20266025C212yesmaximal, complemented
4D4⋊#11717581C212yesmaximal, complemented
5C4×D4×D101C211yesmaximal, characteristic
6#15692322—21no—
7C4⋊#11717581C2212yescomplemented
8#15692312—21no—
9#15692312—21no—
10C4×#11717581—21no—
11C4×#11717581C2212yescomplemented
12C2×D4×D101C411yescharacteristic
13C22×C4×D101—21no—
14C4×D404C411yescharacteristic
15D4×Dic101C2211yescharacteristic
16#15692217C2211yescharacteristic
17C4×C22⋊D101—21no—
18#15692215—21no—
19C4⋊C4×D101C411yescharacteristic
20C22⋊C4×D101—21no—
21C42×D101C2211yescharacteristic
22C4×D4×C101C411yescharacteristic
23C2×#11717581—22no—
24C2×#11717581—22no—
25C23×D101—21no—
26C2×D404C2×C411yescharacteristic
27C2×C22⋊D101—21no—
28D4×D101—22no—
29D4×D101C814yescomplemented
30C22×Dic101—21no—
31C2×C4×D101—21no—
32C2×C4×D101—21no—
33C2×C4×D101—21no—
34C2×C4×D101—44no—
35C2×C4×D101C2×C411yescharacteristic
36C2×C4×D101D411yescharacteristic
37C2×C4×D101D411yescharacteristic
38C4⋊Dic101C2×C411yescharacteristic
39#13108053—21no—
40C22⋊Dic101—21no—
41#13108051—21no—
42#13108051—21no—
43C4×Dic101D411yescharacteristic
44C4×Dic101D411yescharacteristic
45C4×Dic101C2311yescharacteristic
46C2×D4×C101C2×C411yescharacteristic
47C22×C4×C101—21no—
48C4⋊C4×C101C2×C411yescharacteristic
49C22⋊C4×C101—21no—
50C42×C101C2×C411yescharacteristic
51C23×C101—21no—
52C22×D101—21no—
53C22×D101—24no—
54C22×D101—42no—
55C22×D101—42no—
56C22×D101C22⋊C411yescharacteristic
57D4×C101—21no—
58D4×C101C2×C812yes—
59C2×C4×C101—21no—
60C2×C4×C101—42no—
61C2×C4×C101C22⋊C411yescharacteristic
62C2×C4×C101C22⋊C411yescharacteristic
63C2×C4×C101C22×C411yescharacteristic
64C22⋊D101—24no—
65C2×Dic101—21no—
66C2×Dic101—42no—
67C2×Dic101C22⋊C411yescharacteristic
68C2×Dic101C2×D411yescharacteristic
69C2×Dic101C2×D411yescharacteristic
70D404—21no—
71D404C2×C812yes—
72C4×D101—22no—
73C4×D101—41no—
74C4×D101—42no—
75C4×D101—42no—
76C4×D101—42no—
77C4×D101C2×C812yes—
78C4×D101M1612yes—
79#11717581—42no—
80#11717581—42no—
81C22×C101—22no—
82C22×C101—41no—
83C22×C101—41no—
84C22×C101C2×C22⋊C411yescharacteristic
85C2×D101—21no—
86C2×D101—21no—
87C2×D101—22no—
88C2×D101—48no—
89C2×D101—41no—
90C2×D101—41no—
91C2×D101C22⋊C812yes—
92C4×C101—21no—
93C4×C101—21no—
94C4×C101—41no—
95C4×C101C22×C811yescharacteristic
96C4×C101C2×M1611yescharacteristic
97Dic101—21no—
98Dic101—41no—
99Dic101C22⋊C812yes—
100Dic101AGL(1,ℤ/8)12yes—
101C2×C101—42no—
102C2×C101C2×C22⋊C811yescharacteristic
103C2×C101#41811yescharacteristic
104C2×C101#42111yescharacteristic
105D101—22no—
106D101—42no—
107#1478—1011nomaximal, Sylow
108C101#147811yescharacteristic, complemented, Sylow
109C2×C4×D4—1011no—
110#431—1011no—
111#431—1011no—
112D4⋊C8—1012no—
113D4⋊C8—1012no—
114C22×D4—1011no—
115C23×C4—2021no—
116C4×D4—1011no—
117C4×D4—1011no—
118C4×D4—1011no—
119C4×D4—2021no—
120C4×D4—2021no—
121C4×D4#8793811yescharacteristic
122C2×C4⋊C4—1011no—
123C2×C22⋊C4—2021no—
124C2×C42—1011no—
125C4⋊C8—1012no—
126C4⋊C8—2021no—
127C22⋊C8—2021no—
128C22⋊C8—2021no—
129C4×C8—1012no—
130C4×C8—2021no—
131C24—2021no—
132C2×D4—1014no—
133C2×D4—1011no—
134C2×D4—2022no—
135C2×D4—2021no—
136C2×D4C2×#8793811yescharacteristic
137C22×C4—21no—
138C22×C4—1011no—
139C22×C4—1011no—
140C22×C4—1011no—
141C22×C4—4044no—
142C22×C4—2021no—
143C22×C4—2021no—
144C22×C4—2021no—
145C22×C4—2021no—
146C2×C8—2022no—
147C2×C8—2022no—
148C4⋊C4—1011no—
149C4⋊C4—2021no—
150C4⋊C4C2×#8793811yescharacteristic
151C22⋊C4—21no—
152C22⋊C4—2021no—
153C22⋊C4—2021no—
154C22⋊C4—2021no—
155C42—1011no—
156C42—1011no—
157C42—1011no—
158C42C2×#8793811yescharacteristic
159C23—21no—
160C23—1011no—
161C23—4042no—
162C23—4042no—
163C23—2024no—
164C23—2021no—
165D4—21no—
166D4—1012no—
167D4—2024no—
168D4—2021no—
169D4D101⋊C812yes—
170C2×C4—21no—
171C2×C4—42no—
172C2×C4—1011no—
173C2×C4—1011no—
174C2×C4—1012no—
175C2×C4—1012no—
176C2×C4—1011no—
177C2×C4—4042no—
178C2×C4—4041no—
179C2×C4—4042no—
180C2×C4—4042no—
181C2×C4—4042no—
182C2×C4—2022no—
183C2×C4—2021no—
184C2×C4C22⋊#8793811yescharacteristic
185C2×C4C22⋊#8793811yescharacteristic
186C2×C4C22×#8793811yescharacteristic
187C8—4042no—
188C8—4042no—
189C22—22no—
190C22—41no—
191C22—41no—
192C22—1012no—
193C22—4048no—
194C22—4041no—
195C22—4041no—
196C22—2022no—
197C22—2021no—
198C22—2021no—
199C22C2×C22⋊#8793811yescharacteristic
200C4—21no—
201C4—21no—
202C4—41no—
203C4—1012no—
204C4—1012no—
205C4—4041no—
206C4—2021no—
207C4C2×D101⋊C811yescharacteristic
208C4D101⋊M1611yescharacteristic
209C2—42no—
210C2—4042no—
211C2—2022no—
212C2D101⋊(C22⋊C8)11yescharacteristic
213C2#2026616911yescharacteristic
214C2#2026617411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC4×C101C22×C8
FrattiniC2×C4C22×#87938
FittingC4×D4×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC2×D4×D101C4
cosocleC4×Dic101C23
hypercenterC4×D4#87938
hypocenterC101#1478
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #20266196C2semidirect
2. #20266195C2semidirect
3. #20266025C2semidirect
4. D4⋊#11717581C2semidirect
7. C4⋊#11717581C22semidirect
11. C4×#11717581C22semidirect
29. D4×D101C8semidirect
108. C101#1478semidirect

Maximal subgroups

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

subgroupindex
#202661962
#202661952
#202660252
D4⋊#117175812
C4×D4×D1012
#1478101