Finite Group Databasebeta

← #27140922

Subgroups — #27140922

Order 12416.

212 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 57 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#20261081C211yesmaximal, characteristic
2#20261020C211yesmaximal, characteristic, complemented
3#20260945C211yesmaximal, characteristic, complemented
4#20260479C211yesmaximal, characteristic
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6Q8⋊#11715895C211yesmaximal, characteristic, complemented
7#559×C97C211yesmaximal, characteristic
8C2×#13095501—21no—
9#15690810C2211yescharacteristic
10#15690741—21no—
11C4⋊#11715895C2211yescharacteristic
12C4⋊#11715895C2211yescharacteristic
13#15690734—21no—
14#15690733—21no—
15#15690731—22no—
16C4×#11715895C2211yescharacteristic
17#127×C97C2211yescharacteristic
18#124×C97—21no—
19#124×C97—21no—
20#123×C97—22no—
21C4×Q8×C97C2211yescharacteristic
22C4×Q8×C97C2211yescharacteristic
23C4×D4×C97—21no—
24C4×D4×C97—21no—
25C2×Dic194—21no—
26#13095501—42no—
27C2×#11715895—21no—
28C2×#11715895—21no—
29C2×#11715895—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97—21no—
32C2×Q8×C97D411yescharacteristic
33C2×Q8×C97D411yescharacteristic
34C2×D4×C97—21no—
35C2×D4×C97—21no—
36C2×D4×C97D412yes—
37C22×C4×C97—22no—
38C4⋊C4×C97—21no—
39C4⋊C4×C97—21no—
40C4⋊C4×C97D411yescharacteristic
41C4⋊C4×C97D411yescharacteristic
42C22⋊C4×C97—22no—
43C22⋊C4×C97—22no—
44C22⋊C4×C97—22no—
45C42×C97—21no—
46C42×C97—21no—
47C42×C97C2311yescharacteristic
48C23×C97—22no—
49Q8×C97—21no—
50Q8×C97—21no—
51Q8×C97—21no—
52Q8×C97—21no—
53D4×C97—42no—
54D4×C97—41no—
55D4×C97—41no—
56D4×C97—41no—
57D4×C97—41no—
58C2×C4×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97—21no—
62C2×C4×C97—82no—
63C2×C4×C97C2×D411yescharacteristic
64C2×C4×C97C2×D411yescharacteristic
65C2×C4×C97C2×D411yescharacteristic
66C2×Dic97—41no—
67Dic194—41no—
68Dic194—41no—
69#11715895—41no—
70#11715895—41no—
71#11715895—41no—
72C22×C97—42no—
73C22×C97—42no—
74C22×C97—42no—
75C22×C97#12011yescharacteristic
76C4×C97—21no—
77C4×C97—41no—
78C4×C97—41no—
79C4×C97—41no—
80C4×C97—41no—
81C4×C97#13511yescharacteristic
82C4×C97#13511yescharacteristic
83C4×C97AGL(1,ℤ/8)11yescharacteristic
84C4×C97#13711yescharacteristic
85Dic97—81no—
86C2×C97—82no—
87C2×C97#44911yescharacteristic
88C2×C97#45211yescharacteristic
89C2×C97#45511yescharacteristic
90#1640—971nomaximal, Sylow
91C97#164011yescharacteristic, complemented, Sylow
92#559D9711yescharacteristic
93#486—971no—
94#475—971no—
95#462—971no—
96#331—971no—
97Q8⋊C8—971no—
98Q8⋊C8—971no—
99C2×Q16—1941no—
100C4⋊Q8—971no—
101#127C2×D9711yescharacteristic
102#124—21no—
103#124—21no—
104#123—22no—
105C4×Q8C2×D9711yescharacteristic
106C4×Q8C2×D9711yescharacteristic
107C4×D4—21no—
108C4×D4—21no—
109#106—1941no—
110C4⋊C8—971no—
111C4⋊C8—971no—
112#103—1941no—
113#103—1941no—
114#102—1942no—
115C4×C8—971no—
116C2×Q8—1941no—
117C2×Q8C22⋊D9711yescharacteristic
118C2×Q8C22⋊D9711yescharacteristic
119C2×D4—21no—
120C2×D4—21no—
121C2×D4C22⋊D9712yes—
122C22×C4—22no—
123Q16—3882no—
124C2×C8—1941no—
125C2×C8—1941no—
126C2×C8—1941no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—1941no—
130C4⋊C4—1941no—
131C4⋊C4C22⋊D9711yescharacteristic
132C4⋊C4C22⋊D9711yescharacteristic
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C42—21no—
137C42—21no—
138C42C22×D9711yescharacteristic
139C23—22no—
140Q8—21no—
141Q8—21no—
142Q8—21no—
143Q8—21no—
144Q8—3881no—
145Q8—3881no—
146D4—42no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—82no—
156C2×C4—3881no—
157C2×C4C2×C22⋊D9711yescharacteristic
158C2×C4C2×C22⋊D9711yescharacteristic
159C2×C4C2×C22⋊D9711yescharacteristic
160C8—3881no—
161C8—3881no—
162C8—3881no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22#1569077911yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—7761no—
173C4#1569083011yescharacteristic
174C4#1569083011yescharacteristic
175C4#1569083511yescharacteristic
176C4#1569084011yescharacteristic
177C2—82no—
178C2#2026088011yescharacteristic
179C2#2026089711yescharacteristic
180C2#2026091311yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#559×C97C2
socleC22×C97#120
prime radical#127×C97C22
hypercenter#559D97
hypocenterC97#1640
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #559

Complements

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

NH ≅ G/Ntype
2. #20261020C2semidirect
3. #20260945C2semidirect
5. Q8⋊#11715895C2semidirect
6. Q8⋊#11715895C2semidirect
91. C97#1640semidirect

Maximal subgroups

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

subgroupindex
#202610812
#202610202
#202609452
#202604792
Q8⋊#117158952
#559×C972
#164097