Finite Group Databasebeta

← #27187309

Subgroups — #27187309

Order 12928.

260 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C4×#13108084C211yesmaximal, characteristic
2C4×C4⋊#87938C211yesmaximal, characteristic
3#20266047C212yesmaximal
4#20266047C212yesmaximal
5#20265733C211yesmaximal, characteristic
6C2×#13108084C2212yes—
7C2×C4⋊#87938—21no—
8C2×C4×#87938C2212yes—
9#15692310—21no—
10C4×#11717581—21no—
11(C2×#87938)⋊C4—21no—
12#15692254C411yescharacteristic, complemented
13#15692254C411yescharacteristic, complemented
14#15692247C411yescharacteristic, complemented
15#15692233—21no—
16#15692233—21no—
17#15692220—21no—
18#15692220—21no—
19#15692213C2211yescharacteristic
20#15692213C2211yescharacteristic
21C42×D101C2211yescharacteristic
22#125×C101C411yescharacteristic, complemented
23C22×#87938—22no—
24#13108084—22no—
25#13108084Q814yescomplemented
26C2×#11717581—22no—
27C4⋊#87938—24no—
28C4×#87938—24no—
29C2×Dic202—21no—
30C2×Dic202—21no—
31C2×C4×D101D411yescharacteristic
32C2×C4×D101D411yescharacteristic
33C2×C4×D101C2311yescharacteristic
34C4⋊Dic101C2×C411yescharacteristic
35C4⋊Dic101C2×C411yescharacteristic
36#13108053—21no—
37#13108053—21no—
38#13108053—21no—
39#13108053—21no—
40#13108051—21no—
41#13108051—21no—
42C4×Dic101—21no—
43C4×Dic101—21no—
44C4×Dic101C2×C411yescharacteristic
45C4×Dic101D411yescharacteristic
46C4×Dic101D411yescharacteristic
47C4⋊C4×C101—21no—
48C4⋊C4×C101—21no—
49C4⋊C4×C101C2×C411yescharacteristic
50C4⋊C4×C101C2×C411yescharacteristic
51C42×C101C2×C411yescharacteristic
52C22×D101C2×D411yescharacteristic
53C2×#87938—24no—
54C2×#87938—42no—
55C2×#87938—42no—
56C2×C4×C101—21no—
57C2×C4×C101—21no—
58C2×C4×C101C22⋊C411yescharacteristic
59C2×C4×C101C22⋊C411yescharacteristic
60C2×C4×C101C22×C411yescharacteristic
61C2×Dic101—21no—
62C2×Dic101—21no—
63C2×Dic101C22⋊C411yescharacteristic
64C2×Dic101C22⋊C411yescharacteristic
65C2×Dic101C2×D411yescharacteristic
66C4×D101—22no—
67C4×D101—41no—
68C4×D101C2×Q812yes—
69C4×D101#4112yes—
70Dic202—42no—
71Dic202—42no—
72#11717581—24no—
73C22×C101C2×C22⋊C411yescharacteristic
74C2×D101—21no—
75C2×D101—21no—
76C2×D101#12411yescharacteristic
77C2×D101C4⋊Q811yescharacteristic
78#87938—44no—
79C4×C101—21no—
80C4×C101—21no—
81C4×C101—41no—
82C4×C101—41no—
83C4×C101C4×Q812yes—
84Dic101—21no—
85Dic101—41no—
86Dic101—41no—
87Dic101#10412yes—
88Dic101C22⋊Q812yes—
89C2×C101#39111yescharacteristic
90C2×C101C2×#10411yescharacteristic
91C2×C101#42111yescharacteristic
92D101—21no—
93D101—21no—
94#1986—1011nomaximal, Sylow
95C101#198611yescharacteristic, complemented, Sylow
96#533—1011no—
97C4×M16—1011no—
98C4×C4⋊C4—1011no—
99#339—1012no—
100#339—1012no—
101C2×M16—1012no—
102C4⋊Q8—1011no—
103C4⋊Q8—1011no—
104#125—1011no—
105#125#8793811yescharacteristic, complemented
106C22⋊Q8—2021no—
107C22⋊Q8—2021no—
108C4×Q8—2021no—
109C4×Q8—2021no—
110#117—1011no—
111#117—1011no—
112C2×C4⋊C4—2021no—
113C2×C42—1011no—
114C2×C42—1012no—
115#97—2021no—
116C4×C8—2021no—
117#95—2021no—
118C2×Q8—2021no—
119C2×Q8—2021no—
120C22×C4—1011no—
121C22×C4—1011no—
122C22×C4—1011no—
123C22×C4—2022no—
124M16—1014no—
125M16—2022no—
126C2×C8—2022no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—1011no—
130C4⋊C4—1011no—
131C4⋊C4—2021no—
132C4⋊C4—2021no—
133C4⋊C4—2024no—
134C4⋊C4—2021no—
135C4⋊C4—2021no—
136C4⋊C4C2×#8793811yescharacteristic
137C4⋊C4C2×#8793811yescharacteristic
138C22⋊C4—2021no—
139C22⋊C4—2021no—
140C42—1011no—
141C42—1011no—
142C42—1011no—
143C42—2021no—
144C42—2021no—
145C42—2024no—
146C42C2×#8793811yescharacteristic
147C23—1011no—
148Q8—4042no—
149Q8—4042no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—1011no—
153C2×C4—1011no—
154C2×C4—1012no—
155C2×C4—1012no—
156C2×C4—1011no—
157C2×C4—4041no—
158C2×C4—4042no—
159C2×C4—4042no—
160C2×C4—2022no—
161C2×C4—2024no—
162C2×C4—2021no—
163C2×C4—2021no—
164C2×C4C22⋊#8793811yescharacteristic
165C2×C4C22⋊#8793811yescharacteristic
166C2×C4C22×#8793811yescharacteristic
167C8—2024no—
168C22—1011no—
169C22—1011no—
170C22—2021no—
171C22—2021no—
172C22C2×C22⋊#8793811yescharacteristic
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—1012no—
178C4—1012no—
179C4—4044no—
180C4—4041no—
181C4—4041no—
182C4—2021no—
183C4Q8×#8793812yes—
184C2—2021no—
185C2—2021no—
186C2(C2×#87938)⋊Q811yescharacteristic
187C2D101⋊#10411yescharacteristic
188C2#2026617411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#125×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC22×D101C2×D4
cosocleC2×C4×D101C23
hypercenter#125#87938
hypocenterC101#1986
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
12. #15692254C4semidirect
13. #15692254C4semidirect
14. #15692247C4semidirect
22. #125×C101C4semidirect
25. #13108084Q8semidirect
95. C101#1986semidirect
105. #125#87938semidirect

Maximal subgroups

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

subgroupindex
C4×#131080842
C4×C4⋊#879382
#202660472
#202657332
#1986101