Finite Group Databasebeta

← #27186488

Subgroups — #27186488

Order 12928.

216 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 60 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#20266195C211yesmaximal, characteristic, complemented
2#20266181C211yesmaximal, characteristic, complemented
3Q8⋊#11717581C212yesmaximal, complemented
4D4⋊#11717581C212yesmaximal, complemented
5#20265706C211yesmaximal, characteristic
6C2×#13108084—21no—
7#15692322—21no—
8C4⋊#11717581—21no—
9C4⋊#11717581C2212yes—
10#15692312—21no—
11C4×#11717581C2212yes—
12#15692248—21no—
13#15692247C411yescharacteristic
14#15692237—21no—
15#15692231C411yescharacteristic
16#15692226C411yescharacteristic
17Q8×Dic101C2211yescharacteristic
18D4×Dic101C2211yescharacteristic
19#15692215—21no—
20#15692211—21no—
21C42×D101C2211yescharacteristic
22#124×C101C411yescharacteristic
23#13108084—42no—
24C2×#11717581—22no—
25C2×#11717581—22no—
26C2×C22⋊D101—21no—
27C22×Dic101—21no—
28C2×C4×D101C2×C411yescharacteristic
29C2×C4×D101D411yescharacteristic
30C2×C4×D101D411yescharacteristic
31C4⋊Dic101—21no—
32C4⋊Dic101C2×C411yescharacteristic
33C4⋊Dic101C2×C411yescharacteristic
34#13108053—21no—
35#13108053—21no—
36#13108053—21no—
37C22⋊Dic101—21no—
38C22⋊Dic101—21no—
39#13108051—21no—
40C4×Dic101—21no—
41C4×Dic101D411yescharacteristic
42C4×Dic101D411yescharacteristic
43C4×Dic101C2311yescharacteristic
44C2×Q8×C101C2×C411yescharacteristic
45C2×D4×C101C2×C411yescharacteristic
46C22⋊C4×C101—21no—
47C22⋊C4×C101—21no—
48C42×C101C2×C411yescharacteristic
49C23×C101—21no—
50C22×D101C22⋊C411yescharacteristic
51Q8×C101—22no—
52D4×C101—22no—
53C2×C4×C101—21no—
54C2×C4×C101C22⋊C411yescharacteristic
55C2×C4×C101C22⋊C411yescharacteristic
56C2×C4×C101C22×C411yescharacteristic
57C22⋊D101—42no—
58C2×Dic101—21no—
59C2×Dic101—21no—
60C2×Dic101—41no—
61C2×Dic101—41no—
62C2×Dic101C22⋊C411yescharacteristic
63C2×Dic101C2×D411yescharacteristic
64C2×Dic101C2×D411yescharacteristic
65C4×D101—22no—
66C4×D101—21no—
67C4×D101—21no—
68C4×D101—41no—
69#11717581—42no—
70#11717581—42no—
71C22×C101—41no—
72C22×C101—41no—
73C22×C101—41no—
74C22×C101C2×C22⋊C411yescharacteristic
75C2×D101—21no—
76C2×D101—21no—
77C2×D101—21no—
78C4×C101—21no—
79C4×C101—21no—
80C4×C101—41no—
81C4×C101#13112yes—
82Dic101—21no—
83Dic101—21no—
84Dic101—41no—
85Dic101—41no—
86Dic101AGL(1,ℤ/8)11yescharacteristic
87Dic101#13711yescharacteristic
88C2×C101—81no—
89C2×C101#40811yescharacteristic
90C2×C101#41711yescharacteristic
91C2×C101#42111yescharacteristic
92D101—41no—
93#1495—1011nomaximal, Sylow
94C101#149511yescharacteristic, complemented, Sylow
95#529—1011no—
96#431—1011no—
97#423—1011no—
98Q8⋊C8—1012no—
99D4⋊C8—1012no—
100C2×M16—2021no—
101#126—2021no—
102#125—1011no—
103#124#8793811yescharacteristic
104#123—2021no—
105C22⋊Q8—1011no—
106#121—1011no—
107C4×Q8—1011no—
108C4×D4—1011no—
109C4×D4—2021no—
110#117—2021no—
111C2×C42—1011no—
112C4⋊C8—1012no—
113C4⋊C8—2021no—
114C4⋊C8—2021no—
115C22⋊C8—2021no—
116C4×C8—1012no—
117C2×Q8C2×#8793811yescharacteristic
118C2×D4—2021no—
119C2×D4C2×#8793811yescharacteristic
120C22×C4—1011no—
121C22×C4—1011no—
122C22×C4—1011no—
123C22×C4—2021no—
124M16—4042no—
125C2×C8—2022no—
126C2×C8—2022no—
127C4⋊C4—1011no—
128C4⋊C4—1011no—
129C4⋊C4—2021no—
130C4⋊C4—2021no—
131C4⋊C4—2021no—
132C4⋊C4—2021no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—2021no—
136C22⋊C4—2021no—
137C22⋊C4—2021no—
138C42—1011no—
139C42—1011no—
140C42—1011no—
141C42—2021no—
142C42C2×#8793811yescharacteristic
143C23—21no—
144C23—1011no—
145Q8—22no—
146D4—22no—
147D4—4042no—
148C2×C4—21no—
149C2×C4—1011no—
150C2×C4—1011no—
151C2×C4—1011no—
152C2×C4—4041no—
153C2×C4—4041no—
154C2×C4—4041no—
155C2×C4—2021no—
156C2×C4—2021no—
157C2×C4—2021no—
158C2×C4—2021no—
159C2×C4—2022no—
160C2×C4C22⋊#8793811yescharacteristic
161C2×C4C22⋊#8793811yescharacteristic
162C2×C4C22×#8793811yescharacteristic
163C8—4042no—
164C8—4042no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—2021no—
169C22—2021no—
170C22—2021no—
171C22C2×C22⋊#8793811yescharacteristic
172C4—21no—
173C4—21no—
174C4—41no—
175C4—1011no—
176C4—1011no—
177C4—4041no—
178C4—4041no—
179C4—2021no—
180C4—2021no—
181C4#1569234212yes—
182C2—81no—
183C2—4041no—
184C2#2026615111yescharacteristic
185C2#2026616811yescharacteristic
186C2#2026617411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#124×C101C4
socleC22×C101C2×C22⋊C4
prime radical#15692226C4
cosocleC4×Dic101C23
hypercenter#124#87938
hypocenterC101#1495
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #20266195C2semidirect
2. #20266181C2semidirect
3. Q8⋊#11717581C2semidirect
4. D4⋊#11717581C2semidirect
94. C101#1495semidirect

Maximal subgroups

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

subgroupindex
#202661952
#202661812
Q8⋊#117175812
D4⋊#117175812
#202657062
#1495101