Finite Group Databasebeta

← #27169596

Subgroups — #27169596

Order 12928.

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

normal subgroups only

This group has 216 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#20265228C211yesmaximal, characteristic, complemented
3#20265199C211yesmaximal, characteristic, complemented
4Q8⋊#11717579C211yesmaximal, characteristic, complemented
5#20264906C211yesmaximal, characteristic, complemented
6#20264904C211yesmaximal, characteristic, complemented
7Q8⋊C8×C101C211yesmaximal, characteristic, complemented
8C2×#13108057—21no—
9#15692250—21no—
10#15692246C411yescharacteristic
11#15692244C411yescharacteristic
12#15692240—21no—
13#15692231C411yescharacteristic
14#15692230C411yescharacteristic
15C4×D404—21no—
16C4×D404C2211yescharacteristic
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×D404—21no—
28C2×D404C2×C411yescharacteristic
29C2×C4×D101—21no—
30C2×C4×D101C2×C411yescharacteristic
31C2×C4×D101D411yescharacteristic
32C2×C4×D101D411yescharacteristic
33#13108057—42no—
34C2×#11717579—21no—
35C2×#11717579—21no—
36C4⋊Dic101—21no—
37C4⋊Dic101C2×C411yescharacteristic
38#13108053—21no—
39#13108053—21no—
40#13108051—21no—
41#13108051—21no—
42#13108051—21no—
43#13108051—21no—
44#13108051—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×D101—21no—
56C22×D101C22⋊C411yescharacteristic
57Q8×C101—21no—
58Q8×C101—21no—
59C2×C4×C101—21no—
60C2×C4×C101—21no—
61C2×C4×C101C22×C411yescharacteristic
62C2×C4×C101C2×D411yescharacteristic
63C2×C4×C101C2×D411yescharacteristic
64C2×Dic101—21no—
65C2×Dic101C22⋊C411yescharacteristic
66C2×Dic101C22⋊C411yescharacteristic
67C2×Dic101C22⋊C411yescharacteristic
68D404—21no—
69D404—21no—
70D404—42no—
71C4×D101—21no—
72C4×D101—21no—
73C4×D101—21no—
74C4×D101—21no—
75C4×D101—41no—
76C4×D101—41no—
77C4×D101—41no—
78C8×C101—41no—
79C8×C101—41no—
80#11717579—41no—
81#11717579—41no—
82C22×C101C2×C22⋊C411yescharacteristic
83C2×D101—21no—
84C2×D101—21no—
85C2×D101—21no—
86C2×D101—41no—
87C2×D101—41no—
88C2×D101—41no—
89C4×C101—21no—
90C4×C101—41no—
91C4×C101—41no—
92C4×C101#13111yescharacteristic
93C4×C101#13111yescharacteristic
94C4×C101AGL(1,ℤ/8)11yescharacteristic
95C4×C101#13711yescharacteristic
96Dic101—21no—
97Dic101—21no—
98Dic101—21no—
99Dic101—41no—
100C2×C101#40811yescharacteristic
101C2×C101#41711yescharacteristic
102C2×C101#42111yescharacteristic
103D101—41no—
104D101—81no—
105#1495—1011nomaximal, Sylow
106C101#149511yescharacteristic, complemented, Sylow
107#529—1011no—
108#431—1011no—
109#423—1011no—
110Q8⋊C8—1011no—
111Q8⋊C8D10111yescharacteristic, complemented
112D4⋊C8—1011no—
113D4⋊C8—1011no—
114C2×M16—2021no—
115#126—2021no—
116#125—1011no—
117#124—1011no—
118#123—2021no—
119C22⋊Q8—1011no—
120#121—1011no—
121C4×Q8C2×D10111yescharacteristic
122C4×D4—1011no—
123C4×D4—2021no—
124#117—2021no—
125C2×C42—1011no—
126C4⋊C8—1011no—
127C4⋊C8—2021no—
128C4⋊C8—2021no—
129C4⋊C8C2×D10111yescharacteristic
130C22⋊C8—2021no—
131C4×C8—1011no—
132C4×C8C2×D10111yescharacteristic
133C2×Q8C4×D10111yescharacteristic
134C2×D4—1011no—
135C2×D4—2021no—
136C22×C4—1011no—
137C22×C4—1011no—
138C22×C4—1011no—
139C22×C4—2021no—
140M16—4042no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—2021no—
144C2×C8—2021no—
145C4⋊C4—21no—
146C4⋊C4—1011no—
147C4⋊C4—2021no—
148C4⋊C4—2021no—
149C4⋊C4—2021no—
150C4⋊C4C4×D10111yescharacteristic
151C22⋊C4—2021no—
152C22⋊C4—2021no—
153C22⋊C4—2021no—
154C22⋊C4—2021no—
155C22⋊C4—2021no—
156C42—21no—
157C42—1011no—
158C42—1011no—
159C42—1011no—
160C42C22×D10111yescharacteristic
161C23—1011no—
162C23—2021no—
163Q8—21no—
164Q8—21no—
165D4—4042no—
166D4—2021no—
167D4—2021no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—1011no—
171C2×C4—1011no—
172C2×C4—1011no—
173C2×C4—4041no—
174C2×C4—4041no—
175C2×C4—4041no—
176C2×C4—2021no—
177C2×C4—2021no—
178C2×C4—2021no—
179C2×C4—2021no—
180C2×C4—2021no—
181C2×C4C2×C4×D10111yescharacteristic
182C2×C4D4×D10111yescharacteristic
183C2×C4D4×D10111yescharacteristic
184C8—41no—
185C8—41no—
186C8—4041no—
187C8—4041no—
188C22—4041no—
189C22—4041no—
190C22—4041no—
191C22—2021no—
192C22—2021no—
193C22—2021no—
194C22C22⋊C4×D10111yescharacteristic
195C4—21no—
196C4—41no—
197C4—41no—
198C4—4041no—
199C4—2021no—
200C4—2021no—
201C4—2021no—
202C4#1569226111yescharacteristic
203C4#1569226311yescharacteristic
204C4#1569228211yescharacteristic
205C4#1569228711yescharacteristic
206C2—8081no—
207C2—4041no—
208C2#2026514211yescharacteristic
209C2#2026517711yescharacteristic
210C2#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#1495
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. #20265228C2semidirect
3. #20265199C2semidirect
4. Q8⋊#11717579C2semidirect
5. #20264906C2semidirect
6. #20264904C2semidirect
7. Q8⋊C8×C101C2semidirect
106. C101#1495semidirect
111. Q8⋊C8D101semidirect

Maximal subgroups

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

subgroupindex
#202657142
#202652282
#202651992
Q8⋊#117175792
#202649062
#202649042
Q8⋊C8×C1012
#1495101