Finite Group Databasebeta

← #27169533

Subgroups — #27169533

Order 12928.

216 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 73 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#20265731C211yesmaximal, characteristic
2#20265228C211yesmaximal, characteristic
3C4⋊C8×D101C211yesmaximal, characteristic
4#20264909C211yesmaximal, characteristic, complemented
5Q8⋊#11717579C211yesmaximal, characteristic, complemented
6#20264907C211yesmaximal, characteristic, complemented
7Q8⋊C8×C101C211yesmaximal, characteristic, complemented
8C2×C8×D101—21no—
9#15692254C411yescharacteristic
10#15692253C411yescharacteristic
11#15692245—21no—
12#15692235C411yescharacteristic
13#15692233—21no—
14#15692231C411yescharacteristic
15C4×Dic202—21no—
16C4×Dic202C2211yescharacteristic
17#15692212—21no—
18C42×D101C2211yescharacteristic
19C4⋊#11717579C2211yescharacteristic
20#15692184—21no—
21#15692184—21no—
22#15692167—21no—
23C4×#11717579C2211yescharacteristic
24C4×Q8×C101C2211yescharacteristic
25C4⋊C8×C101C2211yescharacteristic
26C4×C8×C101C2211yescharacteristic
27C2×Dic202—21no—
28C2×Dic202C2×C411yescharacteristic
29C2×C4×D101C2×C411yescharacteristic
30C2×C4×D101D411yescharacteristic
31C2×C4×D101D411yescharacteristic
32C2×#11717579—21no—
33C2×#11717579—21no—
34C8×D101—42no—
35C4⋊Dic101—21no—
36C4⋊Dic101C2×C411yescharacteristic
37#13108053—21no—
38#13108053—21no—
39#13108053—21no—
40#13108053—21no—
41#13108053—21no—
42#13108051—21no—
43#13108051—21no—
44C4×Dic101—21no—
45C4×Dic101C2×C411yescharacteristic
46C4×Dic101D411yescharacteristic
47C4×Dic101D411yescharacteristic
48C2×Q8×C101C2×C411yescharacteristic
49C2×C8×C101—21no—
50C2×C8×C101—21no—
51C4⋊C4×C101—21no—
52C4⋊C4×C101C2×C411yescharacteristic
53C42×C101—21no—
54C42×C101C2311yescharacteristic
55C22×D101C22⋊C411yescharacteristic
56Q8×C101—21no—
57Q8×C101—21no—
58C2×C4×C101—21no—
59C2×C4×C101—21no—
60C2×C4×C101C22×C411yescharacteristic
61C2×C4×C101C2×D411yescharacteristic
62C2×C4×C101C2×D411yescharacteristic
63C2×Dic101—21no—
64C2×Dic101—21no—
65C2×Dic101C22⋊C411yescharacteristic
66C2×Dic101C22⋊C411yescharacteristic
67C2×Dic101C22⋊C411yescharacteristic
68C4×D101—21no—
69C4×D101—21no—
70C4×D101—41no—
71C4×D101SD1612yes—
72C4×D101Q1612yes—
73Dic202—21no—
74Dic202—21no—
75Dic202—42no—
76C8×C101—41no—
77C8×C101—41no—
78#11717579—41no—
79#11717579—41no—
80C22×C101C2×C22⋊C411yescharacteristic
81C2×D101—21no—
82C2×D101—21no—
83C2×D101#10312yes—
84C4×C101—21no—
85C4×C101—41no—
86C4×C101—41no—
87C4×C101#13111yescharacteristic
88C4×C101#13111yescharacteristic
89C4×C101C2×SD1611yescharacteristic
90C4×C101C2×Q1611yescharacteristic
91Dic101—21no—
92Dic101—21no—
93Dic101—41no—
94Dic101—41no—
95Dic101#10312yes—
96C2×C101#40811yescharacteristic
97C2×C101C2×#10311yescharacteristic
98C2×C101#42111yescharacteristic
99D101—22no—
100#1481—1011nomaximal, Sylow
101C101#148111yescharacteristic, complemented, Sylow
102#533—1011no—
103#431—1011no—
104C2×C4⋊C8—1011no—
105Q8⋊C8—1011no—
106Q8⋊C8—1011no—
107Q8⋊C8—1011no—
108Q8⋊C8D10111yescharacteristic, complemented
109C22×C8—2021no—
110C4⋊Q8—1011no—
111C4⋊Q8—1011no—
112#125—2021no—
113C22⋊Q8—1011no—
114C22⋊Q8—1011no—
115C22⋊Q8—2021no—
116C4×Q8—1011no—
117C4×Q8—2021no—
118C4×Q8C2×D10111yescharacteristic
119#117—2021no—
120C2×C42—1011no—
121C4⋊C8—1011no—
122C4⋊C8—2021no—
123C4⋊C8—2021no—
124C4⋊C8C2×D10111yescharacteristic
125C22⋊C8—2021no—
126C4×C8—1011no—
127C4×C8C2×D10111yescharacteristic
128C2×Q8—1011no—
129C2×Q8—2021no—
130C2×Q8C4×D10111yescharacteristic
131C22×C4—1011no—
132C22×C4—1011no—
133C22×C4—1011no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—4042no—
137C2×C8—2021no—
138C2×C8—2021no—
139C4⋊C4—21no—
140C4⋊C4—1011no—
141C4⋊C4—2021no—
142C4⋊C4—2021no—
143C4⋊C4—2021no—
144C4⋊C4—2021no—
145C4⋊C4—2021no—
146C4⋊C4—2021no—
147C4⋊C4C4×D10111yescharacteristic
148C22⋊C4—2021no—
149C22⋊C4—2021no—
150C42—21no—
151C42—1011no—
152C42—1011no—
153C42—1011no—
154C42—2021no—
155C42C22×D10111yescharacteristic
156C23—1011no—
157Q8—21no—
158Q8—21no—
159Q8—4042no—
160Q8—2021no—
161Q8—2021no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—1011no—
165C2×C4—1012no—
166C2×C4—1012no—
167C2×C4—1011no—
168C2×C4—1011no—
169C2×C4—4041no—
170C2×C4—2021no—
171C2×C4—2021no—
172C2×C4—2021no—
173C2×C4—2021no—
174C2×C4C2×C4×D10111yescharacteristic
175C2×C4D4×D10111yescharacteristic
176C2×C4D4×D10111yescharacteristic
177C8—41no—
178C8—41no—
179C8—4041no—
180C8—4041no—
181C22—1012no—
182C22—2021no—
183C22—2021no—
184C22C22⋊C4×D10111yescharacteristic
185C4—21no—
186C4—41no—
187C4—41no—
188C4—1012no—
189C4—4041no—
190C4—4041no—
191C4—2021no—
192C4—2021no—
193C4#1569226111yescharacteristic
194C4#1569226311yescharacteristic
195C4SD16×D10111yescharacteristic
196C4Q16×D10111yescharacteristic
197C2—2022no—
198C2#2026514211yescharacteristic
199C2#103×D10111yescharacteristic
200C2#2026519211yescharacteristic
subgroupG/H
centerC2×C4D4×D101
commutatorC2×C4×C101C22×C4
FrattiniC42C22×D101
FittingQ8⋊C8×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC22×D101C22⋊C4
cosocleC42×C101C23
hypercenterQ8⋊C8D101
hypocenterC101#1481
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
4. #20264909C2semidirect
5. Q8⋊#11717579C2semidirect
6. #20264907C2semidirect
7. Q8⋊C8×C101C2semidirect
101. C101#1481semidirect
108. Q8⋊C8D101semidirect

Maximal subgroups

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

subgroupindex
#202657312
#202652282
C4⋊C8×D1012
#202649092
Q8⋊#117175792
#202649072
Q8⋊C8×C1012
#1481101