Finite Group Databasebeta

← #27172816

Subgroups — #27172816

Order 12928.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#20265728C211yesmaximal, characteristic
2#20265160C212yesmaximal, complemented
3#20264976C211yesmaximal, characteristic, complemented
4#20264973C212yesmaximal, complemented
5#356×C101C211yesmaximal, characteristic, complemented
6C2×#13108075C411yescharacteristic
7C2×#13108075C2211yescharacteristic
8M16×D101—22no—
9#15692252C411yescharacteristic
10#15692244C411yescharacteristic
11#15692244C2211yescharacteristic
12#15692230—22no—
13#15692217—22no—
14C42×D101C411yescharacteristic
15#15692174C2212yes—
16#15692171—22no—
17C4⋊Q8×C101C2211yescharacteristic
18#101×C101C2212yes—
19#13108075—41no—
20#13108075—41no—
21#13108075—41no—
22#13108075—41no—
23C2×D404—21no—
24C2×D404—22no—
25C2×D404C2×C411yescharacteristic
26C2×D404C2×C411yescharacteristic
27C2×D404D412yes—
28C2×C4×D101—21no—
29C2×C4×D101—21no—
30C2×C4×D101C2×C411yescharacteristic
31C2×C4×D101D412yes—
32#13108058—22no—
33#13108057—42no—
34C8×D101—42no—
35#13108051—22no—
36#13108051—22no—
37C4×Dic101—21no—
38C4×Dic101C2×C411yescharacteristic
39C2×Q8×C101C2×C411yescharacteristic
40C2×Q8×C101C2311yescharacteristic
41M16×C101—22no—
42C4⋊C4×C101—22no—
43C42×C101C2×C411yescharacteristic
44C22×D101—21no—
45C22×D101C22⋊C412yes—
46C22×D101C22⋊C411yescharacteristic
47Q8×C101—41no—
48Q8×C101—41no—
49C2×C4×C101—21no—
50C2×C4×C101—21no—
51C2×C4×C101C22×C411yescharacteristic
52C2×C4×C101C2×D412yes—
53C2×Dic101—21no—
54C2×Dic101C22⋊C411yescharacteristic
55D404—41no—
56D404—42no—
57D404—41no—
58D404—44no—
59D404—42no—
60C4×D101—22no—
61C4×D101—22no—
62C4×D101—21no—
63C4×D101—21no—
64C4×D101—42no—
65C4×D101—41no—
66C4×D101—41no—
67C8×C101—42no—
68#11717579—42no—
69C22×C101C2×C22⋊C411yescharacteristic
70C2×D101—21no—
71C2×D101—22no—
72C2×D101—41no—
73C2×D101—42no—
74C2×D101—42no—
75C2×D101—41no—
76C4×C101—22no—
77C4×C101—22no—
78C4×C101—21no—
79C4×C101—41no—
80Dic101—41no—
81Dic101#9911yescharacteristic
82Dic101#9911yescharacteristic
83C2×C101—21no—
84C2×C101C2×#9911yescharacteristic
85D101—41no—
86D101—42no—
87D101—81no—
88#2132—1011nomaximal, Sylow
89C101#213211yescharacteristic, complemented, Sylow
90#532—1011no—
91#413—1012no—
92#356—1011no—
93#356D10111yescharacteristic, complemented
94#355—1012no—
95C2×#41—1011no—
96C2×#41—1011no—
97C2×M16—2022no—
98C4⋊Q8C2×D10111yescharacteristic
99#127—1011no—
100#124—1011no—
101#124—1011no—
102#121—2022no—
103C4×D4—2022no—
104C2×C42—1011no—
105#101—1012no—
106#101C2×D10112yes—
107#100—2022no—
108#41—4041no—
109#41—4041no—
110#41—4041no—
111#41—4041no—
112C2×Q8C4×D10111yescharacteristic
113C2×Q8C22×D10111yescharacteristic
114C2×D4—1011no—
115C2×D4—1011no—
116C2×D4—1012no—
117C2×D4—2022no—
118C2×D4—2021no—
119C22×C4—1011no—
120C22×C4—1012no—
121C22×C4—2021no—
122C22×C4—2021no—
123M16—22no—
124M16—4042no—
125M16—2022no—
126C2×C8—4042no—
127C4⋊C4—22no—
128C22⋊C4—2022no—
129C22⋊C4—2022no—
130C42—1011no—
131C42—2021no—
132C42C4×D10111yescharacteristic
133C23—1012no—
134C23—1011no—
135C23—2021no—
136Q8—41no—
137Q8—41no—
138D4—4041no—
139D4—4044no—
140D4—4042no—
141D4—4041no—
142D4—4042no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—1011no—
146C2×C4—4042no—
147C2×C4—4041no—
148C2×C4—4041no—
149C2×C4—2022no—
150C2×C4—2022no—
151C2×C4—2021no—
152C2×C4—2021no—
153C2×C4—2021no—
154C2×C4C2×C4×D10111yescharacteristic
155C2×C4D4×D10112yes—
156C8—42no—
157C8—4042no—
158C22—4042no—
159C22—4041no—
160C22—4042no—
161C22—4041no—
162C22—2021no—
163C22—2022no—
164C22C22⋊C4×D10111yescharacteristic
165C4—22no—
166C4—22no—
167C4—21no—
168C4—41no—
169C4—1011no—
170C4—1011no—
171C4—4041no—
172C2—21no—
173C2—8081no—
174C2—4041no—
175C2—4042no—
176C2#99×D10111yescharacteristic
subgroupG/H
centerC2#99×D101
commutatorC2×C4×C101C22×C4
FrattiniC2×Q8C22×D101
Fitting#356×C101C2
socleC2×C101C2×#99
prime radical#20265728C2
cosocleC2×Q8×C101C23
hypercenter#356D101
hypocenterC101#2132
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
2. #20265160C2semidirect
3. #20264976C2semidirect
4. #20264973C2semidirect
5. #356×C101C2semidirect
89. C101#2132semidirect
93. #356D101semidirect

Maximal subgroups

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

subgroupindex
#202657282
#202651602
#202649762
#202649732
#356×C1012
#2132101