Finite Group Databasebeta

← #27140997

Subgroups — #27140997

Order 12416.

234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×Dic194C211yesmaximal, characteristic
2#20261131C211yesmaximal, characteristic, complemented
3#20261017C211yesmaximal, characteristic, complemented
4#20260480C211yesmaximal, characteristic, complemented
5#20260465C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic
7#461×C97C211yesmaximal, characteristic
8C22×Dic194—21no—
9#15690810C2211yescharacteristic
10#15690809—21no—
11#15690789—21no—
12#15690787—22no—
13C4×Dic194C2211yescharacteristic
14D4×Dic97—21no—
15#15690741—22no—
16#15690741—21no—
17C4⋊#11715895C2211yescharacteristic
18#15690733—21no—
19C4×#11715895C2211yescharacteristic
20C2×SD16×C97—21no—
21C4⋊Q8×C97C2211yescharacteristic
22C4×D4×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24#103×C97—21no—
25C2×Dic194—22no—
26C2×Dic194—21no—
27C2×Dic194—42no—
28C2×Dic194—42no—
29C2×Dic194D411yescharacteristic
30C22×Dic97—22no—
31C2×#11715895—21no—
32C2×#11715895—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97D411yescharacteristic
35C4⋊Dic97D412yes—
36#13095490—22no—
37#13095490—21no—
38C22⋊Dic97—22no—
39C4×Dic97—21no—
40C2×Q8×C97—21no—
41C2×D4×C97D411yescharacteristic
42C22×C4×C97—21no—
43SD16×C97—42no—
44C2×C8×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97—21no—
47C4⋊C4×C97D411yescharacteristic
48C22⋊C4×C97—21no—
49C42×C97C2311yescharacteristic
50C23×C97—21no—
51Q8×C97—41no—
52Q8×C97—41no—
53D4×C97—21no—
54D4×C97SD1612yes—
55C2×C4×C97—21no—
56C2×C4×C97—41no—
57C2×C4×C97—41no—
58C2×C4×C97—41no—
59C2×C4×C97C2×D411yescharacteristic
60C2×C4×C97C2×D411yescharacteristic
61C2×C4×C97C2×D411yescharacteristic
62C2×Dic97—21no—
63C2×Dic97—21no—
64C2×Dic97—22no—
65C2×Dic97—44no—
66Dic194—21no—
67Dic194—44no—
68Dic194—41no—
69Dic194—41no—
70Dic194SD1612yescomplemented
71C8×C97—41no—
72#11715895—22no—
73#11715895—41no—
74C22×C97—22no—
75C22×C97—41no—
76C22×C97—41no—
77C22×C97#12011yescharacteristic
78C4×C97—21no—
79C4×C97—41no—
80C4×C97—81no—
81C4×C97C2×SD1611yescharacteristic
82C4×C97C2×SD1611yescharacteristic
83C4×C97AGL(1,ℤ/8)11yescharacteristic
84C4×C97#13711yescharacteristic
85Dic97—22no—
86Dic97—42no—
87Dic97—41no—
88C2×C97—42no—
89C2×C97#44811yescharacteristic
90C2×C97#45011yescharacteristic
91C2×C97#45611yescharacteristic
92#1651—971nomaximal, Sylow
93C97#165111yescharacteristic, complemented, Sylow
94D4×Q8—971no—
95#498—971no—
96#475—971no—
97#461D9711yescharacteristic
98#332—971no—
99Q8⋊C8—971no—
100D4⋊C8—971no—
101C22×Q8—1941no—
102C2×SD16—21no—
103C4⋊Q8—971no—
104C4⋊Q8—1941no—
105C4⋊Q8C2×D9711yescharacteristic
106C22⋊Q8—1942no—
107C22⋊Q8—1941no—
108C4×Q8—971no—
109C4×D4—1941no—
110C4×D4C2×D9711yescharacteristic
111#106—1941no—
112#106—1942no—
113C4⋊C8—971no—
114C4⋊C8C2×D9711yescharacteristic
115#103—21no—
116#103—1941no—
117C4×C8—971no—
118C2×Q8—21no—
119C2×Q8—971no—
120C2×Q8—3882no—
121C2×Q8—3882no—
122C2×Q8—1942no—
123C2×Q8—1941no—
124C2×D4C22⋊D9711yescharacteristic
125C22×C4—21no—
126C22×C4—1942no—
127SD16—42no—
128C2×C8—21no—
129C2×C8—1941no—
130C2×C8—1941no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—971no—
134C4⋊C4—972no—
135C4⋊C4—1942no—
136C4⋊C4—1941no—
137C4⋊C4—1941no—
138C4⋊C4C22⋊D9711yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—1942no—
141C42—1941no—
142C42C22×D9711yescharacteristic
143C23—21no—
144Q8—41no—
145Q8—41no—
146Q8—972no—
147Q8—3884no—
148Q8—3881no—
149Q8—3881no—
150Q8—1941no—
151D4—21no—
152D4#1309549912yes—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—3884no—
158C2×C4—1941no—
159C2×C4—1942no—
160C2×C4—1941no—
161C2×C4D4×D9711yescharacteristic
162C2×C4D4×D9711yescharacteristic
163C2×C4C2×C22⋊D9711yescharacteristic
164C8—41no—
165C8—3881no—
166C8—1942no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#1569077811yescharacteristic
171C4—21no—
172C4—41no—
173C4—81no—
174C4—3882no—
175C4—3881no—
176C4—1942no—
177C4SD16×D9711yescharacteristic
178C4C2×#1309549911yescharacteristic
179C4#1569083711yescharacteristic
180C4#1569084011yescharacteristic
181C2—42no—
182C2#2026087311yescharacteristic
183C2#2026088411yescharacteristic
184C2#2026091611yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#461×C97C2
socleC22×C97#120
prime radicalC2×D4×C97D4
hypercenter#461D97
hypocenterC97#1651
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #20261131C2semidirect
3. #20261017C2semidirect
4. #20260480C2semidirect
5. #20260465C2semidirect
70. Dic194SD16semidirect
93. C97#1651semidirect

Maximal subgroups

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

subgroupindex
D4×Dic1942
#202611312
#202610172
#202604802
#202604652
D4⋊#117158952
#461×C972
#165197