Finite Group Databasebeta

← #27169539

Subgroups — #27169539

Order 12928.

212 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 65 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#20265714C211yesmaximal, characteristic
2#20265229C211yesmaximal, characteristic, complemented
3#20265228C211yesmaximal, characteristic, complemented
4Q8⋊#11717579C211yesmaximal, characteristic, complemented
5#20264906C211yesmaximal, characteristic, complemented
6#20264904C211yesmaximal, characteristic, complemented
7Q8⋊C8×C101C211yesmaximal, characteristic, complemented
8#15692250—21no—
9#15692246C411yescharacteristic
10#15692244C411yescharacteristic
11#15692240—21no—
12#15692231C411yescharacteristic
13#15692230C411yescharacteristic
14C4×D404—21no—
15C4×D404C2211yescharacteristic
16#15692212—21no—
17C42×D101C2211yescharacteristic
18C4⋊#11717579C2211yescharacteristic
19#15692184—21no—
20#15692167—21no—
21#15692167—21no—
22C4×#11717579C2211yescharacteristic
23C8×Dic101—21no—
24C4×Q8×C101C2211yescharacteristic
25C4⋊C8×C101C2211yescharacteristic
26C4×C8×C101C2211yescharacteristic
27C2×D404—21no—
28C2×D404C2×C411yescharacteristic
29C2×C4×D101—21no—
30C2×C4×D101C2×C411yescharacteristic
31C2×C4×D101D411yescharacteristic
32C2×C4×D101D411yescharacteristic
33C2×#11717579—21no—
34C2×#11717579—21no—
35C4⋊Dic101—21no—
36C4⋊Dic101C2×C411yescharacteristic
37#13108053—21no—
38#13108053—21no—
39#13108051—21no—
40#13108051—21no—
41#13108051—21no—
42#13108051—21no—
43#13108051—21no—
44C4×Dic101C2×C411yescharacteristic
45C4×Dic101D411yescharacteristic
46C4×Dic101D411yescharacteristic
47C2×Q8×C101C2×C411yescharacteristic
48C2×C8×C101—21no—
49C2×C8×C101—21no—
50C4⋊C4×C101—21no—
51C4⋊C4×C101C2×C411yescharacteristic
52C42×C101—21no—
53C42×C101C2311yescharacteristic
54C22×D101—21no—
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×Dic101C22⋊C411yescharacteristic
65C2×Dic101C22⋊C411yescharacteristic
66C2×Dic101C22⋊C411yescharacteristic
67D404—21no—
68D404—21no—
69D404—42no—
70C4×D101—21no—
71C4×D101—21no—
72C4×D101—21no—
73C4×D101—21no—
74C4×D101—41no—
75C4×D101—41no—
76C4×D101—41no—
77C8×C101—41no—
78C8×C101—41no—
79#11717579—41no—
80#11717579—41no—
81C22×C101C2×C22⋊C411yescharacteristic
82C2×D101—21no—
83C2×D101—21no—
84C2×D101—21no—
85C2×D101—41no—
86C2×D101—41no—
87C2×D101—41no—
88C4×C101—21no—
89C4×C101—41no—
90C4×C101—41no—
91C4×C101#13111yescharacteristic
92C4×C101#13111yescharacteristic
93C4×C101#13511yescharacteristic
94C4×C101#13511yescharacteristic
95Dic101—21no—
96Dic101—21no—
97Dic101—21no—
98Dic101—41no—
99C2×C101#40811yescharacteristic
100C2×C101#41611yescharacteristic
101C2×C101#42111yescharacteristic
102D101—41no—
103D101—81no—
104#1482—1011nomaximal, Sylow
105C101#148211yescharacteristic, complemented, Sylow
106#529—1011no—
107#431—1011no—
108#431—1011no—
109Q8⋊C8—1011no—
110Q8⋊C8D10111yescharacteristic, complemented
111D4⋊C8—1011no—
112D4⋊C8—1011no—
113#126—2021no—
114#125—1011no—
115#124—1011no—
116#123—2021no—
117C22⋊Q8—1011no—
118#121—1011no—
119C4×Q8C2×D10111yescharacteristic
120C4×D4—1011no—
121C4×D4—2021no—
122#117—2021no—
123C2×C42—1011no—
124C4⋊C8—1011no—
125C4⋊C8—2021no—
126C4⋊C8C2×D10111yescharacteristic
127C22⋊C8—2021no—
128C22⋊C8—2021no—
129C4×C8—1011no—
130C4×C8—2021no—
131C4×C8C2×D10111yescharacteristic
132C2×Q8C4×D10111yescharacteristic
133C2×D4—1011no—
134C2×D4—2021no—
135C22×C4—1011no—
136C22×C4—1011no—
137C22×C4—1011no—
138C22×C4—2021no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—2021no—
142C2×C8—2021no—
143C4⋊C4—21no—
144C4⋊C4—1011no—
145C4⋊C4—2021no—
146C4⋊C4—2021no—
147C4⋊C4—2021no—
148C4⋊C4C4×D10111yescharacteristic
149C22⋊C4—2021no—
150C22⋊C4—2021no—
151C22⋊C4—2021no—
152C22⋊C4—2021no—
153C22⋊C4—2021no—
154C42—21no—
155C42—1011no—
156C42—1011no—
157C42—1011no—
158C42C22×D10111yescharacteristic
159C23—1011no—
160C23—2021no—
161Q8—21no—
162Q8—21no—
163D4—4042no—
164D4—2021no—
165D4—2021no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—1011no—
169C2×C4—1011no—
170C2×C4—1011no—
171C2×C4—4041no—
172C2×C4—4041no—
173C2×C4—4041no—
174C2×C4—2021no—
175C2×C4—2021no—
176C2×C4—2021no—
177C2×C4—2021no—
178C2×C4—2021no—
179C2×C4C2×C4×D10111yescharacteristic
180C2×C4D4×D10111yescharacteristic
181C2×C4D4×D10111yescharacteristic
182C8—41no—
183C8—41no—
184C8—4041no—
185C8—4041no—
186C22—4041no—
187C22—4041no—
188C22—4041no—
189C22—2021no—
190C22—2021no—
191C22—2021no—
192C22C22⋊C4×D10111yescharacteristic
193C4—21no—
194C4—41no—
195C4—41no—
196C4—4041no—
197C4—2021no—
198C4—2021no—
199C4—2021no—
200C4#1569226111yescharacteristic
201C4#1569226311yescharacteristic
202C4#1569227611yescharacteristic
203C4#1569227811yescharacteristic
204C2—8081no—
205C2—4041no—
206C2#2026514211yescharacteristic
207C2#2026517211yescharacteristic
208C2#2026519211yescharacteristic
subgroupG/H
centerC2×C4D4×D101
commutatorC2×C4×C101C22×C4
FrattiniC42C22×D101
FittingQ8⋊C8×C101C2
socleC22×C101C2×C22⋊C4
prime radical#15692230C4
cosocleC42×C101C23
hypercenterQ8⋊C8D101
hypocenterC101#1482
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
2. #20265229C2semidirect
3. #20265228C2semidirect
4. Q8⋊#11717579C2semidirect
5. #20264906C2semidirect
6. #20264904C2semidirect
7. Q8⋊C8×C101C2semidirect
105. C101#1482semidirect
110. Q8⋊C8D101semidirect

Maximal subgroups

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

subgroupindex
#202657142
#202652292
#202652282
Q8⋊#117175792
#202649062
#202649042
Q8⋊C8×C1012
#1482101