Finite Group Databasebeta

← #46809349

Subgroups — #46809349

Order 34432.

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#31771142C211yesmaximal, characteristic
2#31770656C211yesmaximal, characteristic, complemented
3#31770627C211yesmaximal, characteristic, complemented
4Q8⋊#14524130C211yesmaximal, characteristic, complemented
5#31770334C211yesmaximal, characteristic, complemented
6#31770332C211yesmaximal, characteristic, complemented
7Q8⋊C8×C269C211yesmaximal, characteristic, complemented
8C2×#16864385—21no—
9#22575448—21no—
10#22575444C411yescharacteristic
11#22575442C411yescharacteristic
12#22575438—21no—
13#22575429C411yescharacteristic
14#22575428C411yescharacteristic
15C4×D1076—21no—
16C4×D1076C2211yescharacteristic
17#22575410—21no—
18C42×D269C2211yescharacteristic
19C4⋊#14524130C2211yescharacteristic
20#22575382—21no—
21#22575382—21no—
22#22575365—21no—
23C4×#14524130C2211yescharacteristic
24C4×Q8×C269C2211yescharacteristic
25C4⋊C8×C269C2211yescharacteristic
26C4×C8×C269C2211yescharacteristic
27C2×D1076—21no—
28C2×D1076C2×C411yescharacteristic
29C2×C4×D269—21no—
30C2×C4×D269C2×C411yescharacteristic
31C2×C4×D269D411yescharacteristic
32C2×C4×D269D411yescharacteristic
33#16864385—42no—
34C2×#14524130—21no—
35C2×#14524130—21no—
36C4⋊Dic269—21no—
37C4⋊Dic269C2×C411yescharacteristic
38#16864381—21no—
39#16864381—21no—
40#16864379—21no—
41#16864379—21no—
42#16864379—21no—
43#16864379—21no—
44#16864379—21no—
45C4×Dic269C2×C411yescharacteristic
46C4×Dic269D411yescharacteristic
47C4×Dic269D411yescharacteristic
48C2×Q8×C269C2×C411yescharacteristic
49C2×C8×C269—21no—
50C2×C8×C269—21no—
51C4⋊C4×C269—21no—
52C4⋊C4×C269C2×C411yescharacteristic
53C42×C269—21no—
54C42×C269C2311yescharacteristic
55C22×D269—21no—
56C22×D269C22⋊C411yescharacteristic
57D1076—21no—
58D1076—21no—
59D1076—42no—
60C2×Dic269—21no—
61C2×Dic269C22⋊C411yescharacteristic
62C2×Dic269C22⋊C411yescharacteristic
63C2×Dic269C22⋊C411yescharacteristic
64C4×D269—21no—
65C4×D269—21no—
66C4×D269—21no—
67C4×D269—21no—
68C4×D269—41no—
69C4×D269—41no—
70C4×D269—41no—
71#14524130—41no—
72#14524130—41no—
73Q8×C269—21no—
74Q8×C269—21no—
75C2×C4×C269—21no—
76C2×C4×C269—21no—
77C2×C4×C269C22×C411yescharacteristic
78C2×C4×C269C2×D411yescharacteristic
79C2×C4×C269C2×D411yescharacteristic
80C8×C269—41no—
81C8×C269—41no—
82C22×C269C2×C22⋊C411yescharacteristic
83C2×D269—21no—
84C2×D269—21no—
85C2×D269—21no—
86C2×D269—41no—
87C2×D269—41no—
88C2×D269—41no—
89C4×C269—21no—
90C4×C269—41no—
91C4×C269—41no—
92C4×C269#13111yescharacteristic
93C4×C269#13111yescharacteristic
94C4×C269AGL(1,ℤ/8)11yescharacteristic
95C4×C269#13711yescharacteristic
96Dic269—21no—
97Dic269—21no—
98Dic269—21no—
99Dic269—41no—
100C2×C269#40811yescharacteristic
101C2×C269#41711yescharacteristic
102C2×C269#42111yescharacteristic
103D269—41no—
104D269—81no—
105C269#149511yescharacteristic, complemented, Sylow
106#1495—2691nomaximal, Sylow
107#529—2691no—
108#431—2691no—
109#423—2691no—
110Q8⋊C8—2691no—
111Q8⋊C8D26911yescharacteristic, complemented
112D4⋊C8—2691no—
113D4⋊C8—2691no—
114C2×M16—5381no—
115#126—5381no—
116#125—2691no—
117#124—2691no—
118#123—5381no—
119C22⋊Q8—2691no—
120#121—2691no—
121C4×Q8C2×D26911yescharacteristic
122C4×D4—2691no—
123C4×D4—5381no—
124#117—5381no—
125C2×C42—2691no—
126C4⋊C8—2691no—
127C4⋊C8—5381no—
128C4⋊C8—5381no—
129C4⋊C8C2×D26911yescharacteristic
130C22⋊C8—5381no—
131C4×C8—2691no—
132C4×C8C2×D26911yescharacteristic
133C2×Q8C4×D26911yescharacteristic
134C2×D4—2691no—
135C2×D4—5381no—
136C22×C4—2691no—
137C22×C4—2691no—
138C22×C4—2691no—
139C22×C4—5381no—
140M16—10762no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—5381no—
144C2×C8—5381no—
145C4⋊C4—21no—
146C4⋊C4—2691no—
147C4⋊C4—5381no—
148C4⋊C4—5381no—
149C4⋊C4—5381no—
150C4⋊C4C4×D26911yescharacteristic
151C22⋊C4—5381no—
152C22⋊C4—5381no—
153C22⋊C4—5381no—
154C22⋊C4—5381no—
155C22⋊C4—5381no—
156C42—21no—
157C42—2691no—
158C42—2691no—
159C42—2691no—
160C42C22×D26911yescharacteristic
161C23—2691no—
162C23—5381no—
163Q8—21no—
164Q8—21no—
165D4—5381no—
166D4—5381no—
167D4—10762no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—2691no—
171C2×C4—2691no—
172C2×C4—2691no—
173C2×C4—5381no—
174C2×C4—5381no—
175C2×C4—5381no—
176C2×C4—5381no—
177C2×C4—5381no—
178C2×C4—10761no—
179C2×C4—10761no—
180C2×C4—10761no—
181C2×C4C2×C4×D26911yescharacteristic
182C2×C4D4×D26911yescharacteristic
183C2×C4D4×D26911yescharacteristic
184C8—41no—
185C8—41no—
186C8—10761no—
187C8—10761no—
188C22—5381no—
189C22—5381no—
190C22—5381no—
191C22—10761no—
192C22—10761no—
193C22—10761no—
194C22C22⋊C4×D26911yescharacteristic
195C4—21no—
196C4—41no—
197C4—41no—
198C4—5381no—
199C4—5381no—
200C4—5381no—
201C4—10761no—
202C4#2257545911yescharacteristic
203C4#2257546111yescharacteristic
204C4#2257548011yescharacteristic
205C4#2257548511yescharacteristic
206C2—10761no—
207C2—21521no—
208C2#3177057011yescharacteristic
209C2#3177060511yescharacteristic
210C2#3177062011yescharacteristic
subgroupG/H
centerC2×C4D4×D269
commutatorC2×C4×C269C22×C4
FrattiniC42C22×D269
FittingQ8⋊C8×C269C2
socleC22×C269C2×C22⋊C4
prime radical#22575428C4
cosocleC42×C269C23
hypercenterQ8⋊C8D269
hypocenterC269#1495
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #31770656C2semidirect
3. #31770627C2semidirect
4. Q8⋊#14524130C2semidirect
5. #31770334C2semidirect
6. #31770332C2semidirect
7. Q8⋊C8×C269C2semidirect
105. C269#1495semidirect
111. Q8⋊C8D269semidirect

Maximal subgroups

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

subgroupindex
#317711422
#317706562
#317706272
Q8⋊#145241302
#317703342
#317703322
Q8⋊C8×C2692
#1495269