Finite Group Databasebeta

← #27169544

Subgroups — #27169544

Order 12928.

234 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 67 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
2C4⋊C8×D101C211yesmaximal, characteristic, complemented
3C4×#13108057C211yesmaximal, characteristic, complemented
4Q8⋊#11717579C211yesmaximal, characteristic, complemented
5#20264906C211yesmaximal, characteristic, complemented
6#20264904C211yesmaximal, characteristic, complemented
7Q8⋊C8×C101C211yesmaximal, characteristic, complemented
8C2×#13108057—21no—
9C2×C8×D101—21no—
10#15692250—21no—
11#15692246C411yescharacteristic
12#15692244C411yescharacteristic
13#15692240—21no—
14#15692231C411yescharacteristic
15#15692230C411yescharacteristic
16C4×D404—21no—
17C4×D404C2211yescharacteristic
18#15692212—21no—
19C42×D101C2211yescharacteristic
20C4⋊#11717579C2211yescharacteristic
21#15692184—21no—
22#15692165—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—24no—
34C2×#11717579—21no—
35C2×#11717579—21no—
36C8×D101—42no—
37C4⋊Dic101—21no—
38C4⋊Dic101C2×C411yescharacteristic
39#13108053—21no—
40#13108053—21no—
41#13108051—21no—
42#13108051—21no—
43#13108051—21no—
44#13108051—21no—
45#13108051—21no—
46C4×Dic101C2×C411yescharacteristic
47C4×Dic101D411yescharacteristic
48C4×Dic101D411yescharacteristic
49C2×Q8×C101C2×C411yescharacteristic
50C2×C8×C101—21no—
51C2×C8×C101—21no—
52C4⋊C4×C101—21no—
53C4⋊C4×C101C2×C411yescharacteristic
54C42×C101—21no—
55C42×C101C2311yescharacteristic
56C22×D101—21no—
57C22×D101C22⋊C411yescharacteristic
58Q8×C101—21no—
59Q8×C101—21no—
60C2×C4×C101—21no—
61C2×C4×C101—21no—
62C2×C4×C101C22×C411yescharacteristic
63C2×C4×C101C2×D411yescharacteristic
64C2×C4×C101C2×D411yescharacteristic
65C2×Dic101—21no—
66C2×Dic101C22⋊C411yescharacteristic
67C2×Dic101C22⋊C411yescharacteristic
68C2×Dic101C22⋊C411yescharacteristic
69D404—21no—
70D404—21no—
71D404—42no—
72C4×D101—22no—
73C4×D101—21no—
74C4×D101—21no—
75C4×D101—21no—
76C4×D101—21no—
77C4×D101—41no—
78C4×D101—41no—
79C8×C101—22no—
80C8×C101—41no—
81#11717579—22no—
82#11717579—41no—
83C22×C101C2×C22⋊C411yescharacteristic
84C2×D101—21no—
85C2×D101—21no—
86C2×D101—41no—
87C2×D101—41no—
88C2×D101—41no—
89C2×D101#10412yes—
90C4×C101—21no—
91C4×C101—41no—
92C4×C101—41no—
93C4×C101#13111yescharacteristic
94C4×C101#13111yescharacteristic
95C4×C101AGL(1,ℤ/8)11yescharacteristic
96C4×C101#13711yescharacteristic
97Dic101—21no—
98Dic101—21no—
99Dic101—21no—
100Dic101—41no—
101C2×C101#40811yescharacteristic
102C2×C101#41711yescharacteristic
103C2×C101C2×#10411yescharacteristic
104D101—22no—
105D101—81no—
106#1483—1011nomaximal, Sylow
107C101#148311yescharacteristic, complemented, Sylow
108#529—1011no—
109C2×C4⋊C8—1011no—
110C4×M16—1011no—
111Q8⋊C8—1011no—
112Q8⋊C8D10111yescharacteristic, complemented
113D4⋊C8—1011no—
114D4⋊C8—1011no—
115C2×M16—2021no—
116C22×C8—2021no—
117#126—2021no—
118#125—1011no—
119#124—1011no—
120#123—2021no—
121C22⋊Q8—1011no—
122#121—1011no—
123C4×Q8C2×D10111yescharacteristic
124C4×D4—1011no—
125C4×D4—2021no—
126#117—2021no—
127C2×C42—1011no—
128C4⋊C8—1011no—
129C4⋊C8—2021no—
130C4⋊C8C2×D10111yescharacteristic
131#97—2021no—
132C4×C8—1011no—
133C4×C8C2×D10111yescharacteristic
134C2×Q8C4×D10111yescharacteristic
135C2×D4—1011no—
136C2×D4—2021no—
137C22×C4—1011no—
138C22×C4—1011no—
139C22×C4—1011no—
140C22×C4—2021no—
141M16—2024no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—4042no—
145C2×C8—2021no—
146C2×C8—2021no—
147C4⋊C4—21no—
148C4⋊C4—1011no—
149C4⋊C4—2021no—
150C4⋊C4—2021no—
151C4⋊C4—2021no—
152C4⋊C4C4×D10111yescharacteristic
153C22⋊C4—2021no—
154C22⋊C4—2021no—
155C22⋊C4—2021no—
156C22⋊C4—2021no—
157C22⋊C4—2021no—
158C42—21no—
159C42—1011no—
160C42—1011no—
161C42—1011no—
162C42C22×D10111yescharacteristic
163C23—1011no—
164C23—2021no—
165Q8—21no—
166Q8—21no—
167D4—4042no—
168D4—2021no—
169D4—2021no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—1011no—
173C2×C4—1011no—
174C2×C4—1011no—
175C2×C4—4041no—
176C2×C4—4041no—
177C2×C4—2022no—
178C2×C4—2021no—
179C2×C4—2021no—
180C2×C4—2021no—
181C2×C4—2021no—
182C2×C4—2021no—
183C2×C4C2×C4×D10111yescharacteristic
184C2×C4D4×D10111yescharacteristic
185C2×C4D4×D10111yescharacteristic
186C8—22no—
187C8—41no—
188C8—4041no—
189C8—2022no—
190C22—1012no—
191C22—4041no—
192C22—4041no—
193C22—4041no—
194C22—2021no—
195C22—2021no—
196C22C22⋊C4×D10111yescharacteristic
197C4—21no—
198C4—41no—
199C4—41no—
200C4—4041no—
201C4—2021no—
202C4—2021no—
203C4—2021no—
204C4#1569226111yescharacteristic
205C4#1569226311yescharacteristic
206C4#1569228211yescharacteristic
207C4#1569228711yescharacteristic
208C2—8081no—
209C2—2022no—
210C2#2026514211yescharacteristic
211C2#2026517711yescharacteristic
212C2#104×D10111yescharacteristic
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#1483
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. C4⋊C8×D101C2semidirect
3. C4×#13108057C2semidirect
4. Q8⋊#11717579C2semidirect
5. #20264906C2semidirect
6. #20264904C2semidirect
7. Q8⋊C8×C101C2semidirect
107. C101#1483semidirect
112. Q8⋊C8D101semidirect

Maximal subgroups

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

subgroupindex
#202657142
C4⋊C8×D1012
C4×#131080572
Q8⋊#117175792
#202649062
#202649042
Q8⋊C8×C1012
#1483101