Finite Group Databasebeta

← #27186447

Subgroups — #27186447

Order 12928.

234 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 64 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×C4⋊#11717581C211yesmaximal, characteristic, complemented
2C4×#13108083C211yesmaximal, characteristic, complemented
3#20266027C212yesmaximal, complemented
4D4⋊#11717581C212yesmaximal, complemented
5#20265708C211yesmaximal, characteristic
6C2×#13108083—21no—
7C22×#11717581—21no—
8C4⋊#11717581—21no—
9C4⋊#11717581C2212yes—
10#15692310—21no—
11C4×#11717581C2212yes—
12#15692248—21no—
13#15692246C411yescharacteristic
14#15692242C411yescharacteristic
15#15692236—21no—
16#15692234C411yescharacteristic
17#15692220C2211yescharacteristic
18D4×Dic101—21no—
19D4×Dic101C2211yescharacteristic
20#15692211—21no—
21C2×C4×Dic101C2211yescharacteristic
22#121×C101C411yescharacteristic
23#13108083—24no—
24C2×#11717581—22no—
25C2×#11717581—22no—
26C2×#11717581—42no—
27C2×Dic202C2×C411yescharacteristic
28C22×Dic101—21no—
29C22×Dic101D411yescharacteristic
30C22×Dic101D411yescharacteristic
31C4⋊Dic101—21no—
32C4⋊Dic101C2×C411yescharacteristic
33#13108053—21no—
34#13108053—21no—
35#13108053—21no—
36C22⋊Dic101—21no—
37C22⋊Dic101—21no—
38C22⋊Dic101—21no—
39C22⋊Dic101—21no—
40C4×Dic101—21no—
41C4×Dic101C2×C411yescharacteristic
42C4×Dic101D411yescharacteristic
43C4×Dic101D411yescharacteristic
44C4×Dic101C2311yescharacteristic
45C2×D4×C101—21no—
46C2×D4×C101C2×C411yescharacteristic
47C22×C4×C101C2×C411yescharacteristic
48C4⋊C4×C101C2×C411yescharacteristic
49C22⋊C4×C101—21no—
50C23×C101—21no—
51C23×C101C22⋊C411yescharacteristic
52D4×C101—22no—
53D4×C101—42no—
54C2×C4×C101—22no—
55C2×C4×C101—21no—
56C2×C4×C101C22⋊C411yescharacteristic
57C2×C4×C101C22×C411yescharacteristic
58C2×Dic101—22no—
59C2×Dic101—21no—
60C2×Dic101—21no—
61C2×Dic101—21no—
62C2×Dic101—21no—
63C2×Dic101—41no—
64C2×Dic101—41no—
65C2×Dic101C22⋊C411yescharacteristic
66C2×Dic101C22⋊C411yescharacteristic
67C2×Dic101C2×D411yescharacteristic
68C2×Dic101C2×D411yescharacteristic
69Dic202—22no—
70#11717581—24no—
71#11717581—42no—
72C22×C101—21no—
73C22×C101—21no—
74C22×C101—41no—
75C22×C101—41no—
76C22×C101—41no—
77C22×C101#10412yes—
78C22×C101C2×C22⋊C411yescharacteristic
79C4×C101—21no—
80C4×C101—41no—
81C4×C101#13112yes—
82Dic101—21no—
83Dic101—21no—
84Dic101—21no—
85Dic101—41no—
86Dic101—41no—
87Dic101AGL(1,ℤ/8)11yescharacteristic
88Dic101#13711yescharacteristic
89C2×C101—22no—
90C2×C101—81no—
91C2×C101#40811yescharacteristic
92C2×C101#41711yescharacteristic
93C2×C101C2×#10411yescharacteristic
94#1483—1011nomaximal, Sylow
95C101#148311yescharacteristic, complemented, Sylow
96#529—1011no—
97C2×C4⋊C8—1011no—
98C4×M16—1011no—
99Q8⋊C8—1012no—
100D4⋊C8—1012no—
101C2×M16—2021no—
102C22×C8—2021no—
103#126—2021no—
104#125—1011no—
105#124—1011no—
106#123—2021no—
107C22⋊Q8—1011no—
108#121#8793811yescharacteristic
109C4×Q8—1011no—
110C4×D4—1011no—
111C4×D4—2021no—
112#117—2021no—
113C2×C42—1011no—
114C4⋊C8—1012no—
115C4⋊C8—2021no—
116#97—2021no—
117C4×C8—1012no—
118C2×Q8—1011no—
119C2×D4—21no—
120C2×D4C2×#8793811yescharacteristic
121C22×C4—1011no—
122C22×C4—1011no—
123C22×C4—2021no—
124C22×C4C2×#8793811yescharacteristic
125M16—2024no—
126C2×C8—4042no—
127C2×C8—2022no—
128C2×C8—2022no—
129C4⋊C4—1011no—
130C4⋊C4—2021no—
131C4⋊C4—2021no—
132C4⋊C4—2021no—
133C4⋊C4—2021no—
134C4⋊C4C2×#8793811yescharacteristic
135C22⋊C4—21no—
136C22⋊C4—2021no—
137C22⋊C4—2021no—
138C22⋊C4—2021no—
139C22⋊C4—2021no—
140C42—1011no—
141C42—1011no—
142C42—1011no—
143C42—1011no—
144C42—2021no—
145C23—21no—
146C23C22⋊#8793811yescharacteristic
147Q8—2022no—
148D4—22no—
149D4—42no—
150C2×C4—22no—
151C2×C4—21no—
152C2×C4—1011no—
153C2×C4—1011no—
154C2×C4—1011no—
155C2×C4—1011no—
156C2×C4—4041no—
157C2×C4—4041no—
158C2×C4—2021no—
159C2×C4—2021no—
160C2×C4—2021no—
161C2×C4—2021no—
162C2×C4—2022no—
163C2×C4C22⋊#8793811yescharacteristic
164C2×C4C22×#8793811yescharacteristic
165C8—4042no—
166C8—2024no—
167C22—21no—
168C22—21no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22#1569231912yes—
173C22C2×C22⋊#8793811yescharacteristic
174C4—21no—
175C4—41no—
176C4—1011no—
177C4—1011no—
178C4—4041no—
179C4—4041no—
180C4—2021no—
181C4—2021no—
182C4—2021no—
183C4#1569234212yes—
184C2—22no—
185C2—81no—
186C2#2026615011yescharacteristic
187C2#2026616811yescharacteristic
188C2C2×#1569231911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting, prime radical#121×C101C4
socleC22×C101C2×C22⋊C4
cosocleC4×Dic101C23
hypercenter#121#87938
hypocenterC101#1483
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊#11717581C2semidirect
2. C4×#13108083C2semidirect
3. #20266027C2semidirect
4. D4⋊#11717581C2semidirect
95. C101#1483semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#117175812
C4×#131080832
#202660272
D4⋊#117175812
#202657082
#1483101