Finite Group Databasebeta

← #27140885

Subgroups — #27140885

Order 12416.

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

normal subgroups only

This group has 234 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#20261389C211yesmaximal, characteristic
2#20261085C211yesmaximal, characteristic, complemented
3#20261022C211yesmaximal, characteristic, complemented
4#20260482C211yesmaximal, characteristic, complemented
5D4⋊#11715895C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7#476×C97C211yesmaximal, characteristic, complemented
8C2×#13095510—21no—
9#15690810C2211yescharacteristic
10#15690794—22no—
11#15690790—21no—
12#15690784—21no—
13C4×D388C2211yescharacteristic
14C2×C4⋊Dic97—21no—
15#15690741—21no—
16C4⋊#11715895C2211yescharacteristic
17#15690734—22no—
18#15690730—21no—
19C4×#11715895C2211yescharacteristic
20C4⋊Q8×C97C2211yescharacteristic
21C4×D4×C97C2211yescharacteristic
22#106×C97—21no—
23C4⋊C8×C97C2211yescharacteristic
24#102×C97—21no—
25#13095510—42no—
26#13095510—42no—
27C2×Dic194—21no—
28C2×D388D411yescharacteristic
29C2×C22⋊D97—22no—
30C22×Dic97—22no—
31C2×C4×D97—21no—
32C2×#11715895—21no—
33C2×#11715895—21no—
34C4⋊Dic97—22no—
35C4⋊Dic97—21no—
36C4⋊Dic97D411yescharacteristic
37C4⋊Dic97D412yes—
38#13095488—22no—
39#13095488—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—
50C22×D97—21no—
51Q8×C97—42no—
52D4×C97—21no—
53D4×C97SD1612yes—
54C2×C4×C97—21no—
55C2×C4×C97—41no—
56C2×C4×C97—41no—
57C2×C4×C97—41no—
58C2×C4×C97C2×D411yescharacteristic
59C2×C4×C97C2×D411yescharacteristic
60C2×C4×C97C2×D411yescharacteristic
61C22⋊D97—44no—
62C2×Dic97—21no—
63C2×Dic97—22no—
64C2×Dic97—44no—
65D388—21no—
66D388—21no—
67C4×D97—41no—
68C4×D97—41no—
69Dic194—41no—
70Dic194—41no—
71C8×C97—41no—
72#11715895—41no—
73#11715895—41no—
74C22×C97—22no—
75C22×C97—41no—
76C22×C97—41no—
77C22×C97#12011yescharacteristic
78C2×D97—41no—
79C2×D97—41no—
80C2×D97—41no—
81C4×C97—21no—
82C4×C97—41no—
83C4×C97—81no—
84C4×C97C2×SD1611yescharacteristic
85C4×C97#13511yescharacteristic
86C4×C97AGL(1,ℤ/8)11yescharacteristic
87C4×C97#13711yescharacteristic
88Dic97—42no—
89Dic97—41no—
90C2×C97—42no—
91C2×C97#45011yescharacteristic
92C2×C97#45211yescharacteristic
93C2×C97#45411yescharacteristic
94D97—81no—
95#1635—971nomaximal, Sylow
96C97#163511yescharacteristic, complemented, Sylow
97#547—971no—
98#487—971no—
99#476—971no—
100#476D9711yescharacteristic, complemented
101#332—971no—
102D4⋊C8—971no—
103D4⋊C8—971no—
104C2×#41—1941no—
105C4⋊Q8—971no—
106C4⋊Q8C2×D9711yescharacteristic
107#123—1942no—
108C22⋊Q8—1941no—
109#121—1941no—
110C4×D4—971no—
111C4×D4C2×D9711yescharacteristic
112C2×C4⋊C4—1941no—
113#106—21no—
114#106—1941no—
115C4⋊C8—971no—
116C4⋊C8C2×D9711yescharacteristic
117#103—1942no—
118#102—21no—
119#102—1941no—
120C4×C8—971no—
121#41—3882no—
122#41—3882no—
123C2×Q8—21no—
124C2×Q8—1941no—
125C2×D4—971no—
126C2×D4—1942no—
127C2×D4C22⋊D9711yescharacteristic
128C22×C4—21no—
129C22×C4—1942no—
130C22×C4—1941no—
131C2×C8—21no—
132C2×C8—1941no—
133C2×C8—1941no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—972no—
137C4⋊C4—971no—
138C4⋊C4—1942no—
139C4⋊C4—1941no—
140C4⋊C4C22⋊D9711yescharacteristic
141C22⋊C4—21no—
142C22⋊C4—1942no—
143C22⋊C4—1941no—
144C42C22×D9711yescharacteristic
145C23—21no—
146C23—1941no—
147Q8—42no—
148Q8—3881no—
149Q8—3881no—
150D4—21no—
151D4—3884no—
152D4—1941no—
153D4—1941no—
154D4#1309549812yes—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—3884no—
160C2×C4—3881no—
161C2×C4—3881no—
162C2×C4—1941no—
163C2×C4—1942no—
164C2×C4D4×D9711yescharacteristic
165C2×C4D4×D9711yescharacteristic
166C2×C4C2×C22⋊D9711yescharacteristic
167C8—41no—
168C8—3881no—
169C8—3881no—
170C22—22no—
171C22—41no—
172C22—41no—
173C22—3881no—
174C22—3881no—
175C22—3881no—
176C22#1569077811yescharacteristic
177C4—21no—
178C4—41no—
179C4—81no—
180C4—3881no—
181C4—3882no—
182C4C2×#1309549811yescharacteristic
183C4#1569083111yescharacteristic
184C4#1569083511yescharacteristic
185C4#1569083911yescharacteristic
186C2—42no—
187C2—7761no—
188C2#2026088511yescharacteristic
189C2#2026089411yescharacteristic
190C2#2026090611yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#476×C97C2
socleC22×C97#120
prime radical#20261389C2
hypercenter#476D97
hypocenterC97#1635
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. #20261022C2semidirect
4. #20260482C2semidirect
5. D4⋊#11715895C2semidirect
6. #20260462C2semidirect
7. #476×C97C2semidirect
96. C97#1635semidirect
100. #476D97semidirect

Maximal subgroups

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

subgroupindex
#202613892
#202610852
#202610222
#202604822
D4⋊#117158952
#202604622
#476×C972
#163597