Finite Group Databasebeta

← #27140900

Subgroups — #27140900

Order 12416.

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

normal subgroups only

This group has 228 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#20261085C211yesmaximal, characteristic, complemented
3#20261027C211yesmaximal, characteristic, complemented
4#20260480C211yesmaximal, characteristic, complemented
5#20260465C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic
7#476×C97C211yesmaximal, characteristic
8C22×Dic194—21no—
9#15690810C2211yescharacteristic
10#15690809—21no—
11#15690789—21no—
12#15690787—22no—
13C4×Dic194C2211yescharacteristic
14D4×Dic97—21no—
15#15690743—21no—
16C4⋊#11715895C2211yescharacteristic
17#15690734—22no—
18#15690733—21no—
19C4×#11715895C2211yescharacteristic
20C4⋊Q8×C97C2211yescharacteristic
21C4×D4×C97C2211yescharacteristic
22#106×C97—21no—
23C4⋊C8×C97C2211yescharacteristic
24#102×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—
43C2×C8×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97D411yescharacteristic
47C22⋊C4×C97—21no—
48C42×C97C2311yescharacteristic
49C23×C97—21no—
50Q8×C97—42no—
51D4×C97—21no—
52D4×C97Q1612yes—
53C2×C4×C97—21no—
54C2×C4×C97—41no—
55C2×C4×C97—41no—
56C2×C4×C97—41no—
57C2×C4×C97C2×D411yescharacteristic
58C2×C4×C97C2×D411yescharacteristic
59C2×C4×C97C2×D411yescharacteristic
60C2×Dic97—21no—
61C2×Dic97—21no—
62C2×Dic97—22no—
63C2×Dic97—44no—
64Dic194—21no—
65Dic194—44no—
66Dic194—41no—
67Dic194—41no—
68Dic194SD1612yes—
69C8×C97—41no—
70#11715895—41no—
71#11715895—41no—
72C22×C97—22no—
73C22×C97—41no—
74C22×C97—41no—
75C22×C97#12011yescharacteristic
76C4×C97—21no—
77C4×C97—41no—
78C4×C97—81no—
79C4×C97C2×SD1611yescharacteristic
80C4×C97C2×Q1611yescharacteristic
81C4×C97AGL(1,ℤ/8)11yescharacteristic
82C4×C97#13711yescharacteristic
83Dic97—22no—
84Dic97—42no—
85Dic97—41no—
86C2×C97—42no—
87C2×C97#45011yescharacteristic
88C2×C97#45111yescharacteristic
89C2×C97#45411yescharacteristic
90#1637—971nomaximal, Sylow
91C97#163711yescharacteristic, complemented, Sylow
92D4×Q8—971no—
93#487—971no—
94#477—971no—
95#476D9711yescharacteristic
96#332—971no—
97Q8⋊C8—971no—
98D4⋊C8—971no—
99C22×Q8—1941no—
100C4⋊Q8—971no—
101C4⋊Q8—1941no—
102C4⋊Q8C2×D9711yescharacteristic
103C22⋊Q8—1942no—
104C22⋊Q8—1941no—
105C4×Q8—971no—
106C4×D4—1941no—
107C4×D4C2×D9711yescharacteristic
108#107—1941no—
109#106—21no—
110C4⋊C8—971no—
111C4⋊C8C2×D9711yescharacteristic
112#103—1941no—
113#103—1942no—
114#102—21no—
115C4×C8—971no—
116C2×Q8—21no—
117C2×Q8—971no—
118C2×Q8—3882no—
119C2×Q8—3882no—
120C2×Q8—1942no—
121C2×Q8—1941no—
122C2×D4C22⋊D9711yescharacteristic
123C22×C4—21no—
124C22×C4—1942no—
125C2×C8—21no—
126C2×C8—1941no—
127C2×C8—1941no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—972no—
131C4⋊C4—971no—
132C4⋊C4—1942no—
133C4⋊C4—1941no—
134C4⋊C4—1941no—
135C4⋊C4C22⋊D9711yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—1942no—
138C42—1941no—
139C42C22×D9711yescharacteristic
140C23—21no—
141Q8—42no—
142Q8—972no—
143Q8—3884no—
144Q8—3881no—
145Q8—3881no—
146Q8—1941no—
147D4—21no—
148D4#1309550112yes—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—3884no—
154C2×C4—1941no—
155C2×C4—1942no—
156C2×C4—1941no—
157C2×C4D4×D9711yescharacteristic
158C2×C4D4×D9711yescharacteristic
159C2×C4C2×C22⋊D9711yescharacteristic
160C8—41no—
161C8—3881no—
162C8—3881no—
163C22—22no—
164C22—41no—
165C22—41no—
166C22#1569077811yescharacteristic
167C4—21no—
168C4—41no—
169C4—81no—
170C4—3882no—
171C4—3881no—
172C4—1942no—
173C4SD16×D9711yescharacteristic
174C4C2×#1309550111yescharacteristic
175C4#1569083711yescharacteristic
176C4#1569084011yescharacteristic
177C2—42no—
178C2#2026088411yescharacteristic
179C2#2026089011yescharacteristic
180C2#2026090611yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#476×C97C2
socleC22×C97#120
prime radicalC2×D4×C97D4
hypercenter#476D97
hypocenterC97#1637
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #20261085C2semidirect
3. #20261027C2semidirect
4. #20260480C2semidirect
5. #20260465C2semidirect
91. C97#1637semidirect

Maximal subgroups

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

subgroupindex
D4×Dic1942
#202610852
#202610272
#202604802
#202604652
D4⋊#117158952
#476×C972
#163797