Finite Group Databasebeta

← #27143348

Subgroups — #27143348

Order 12416.

228 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×#15690816C211yesmaximal, characteristic
2C2×#15690749C211yesmaximal, characteristic, complemented
3C2×C16×D97C211yesmaximal, characteristic, complemented
4C22⋊#13095486C211yesmaximal, characteristic, complemented
5#20260514C211yesmaximal, characteristic, complemented
6#20260514C211yesmaximal, characteristic, complemented
7C22⋊C16×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C411yescharacteristic
9#15690816—22no—
10#15690816—22no—
11C2×#13095495C411yescharacteristic
12C2×#13095494C411yescharacteristic
13C2×#13095494C411yescharacteristic
14C2×C8×D97C2211yescharacteristic
15C2×C8×D97C2211yescharacteristic
16#15690749—22no—
17C2×#13095486C2211yescharacteristic
18C2×#13095486C2211yescharacteristic
19C16×D97—22no—
20C22×C8×C97C2211yescharacteristic
21C2×C16×C97C2211yescharacteristic
22C2×C16×C97C2211yescharacteristic
23#13095511—22no—
24#13095511—22no—
25C2×Dic194C811yescharacteristic
26C2×D388C811yescharacteristic
27C2×C22⋊D97C811yescharacteristic
28C2×C22⋊D97C811yescharacteristic
29C2×C4×D97C2×C411yescharacteristic
30C2×C4×D97C2×C411yescharacteristic
31#13095495—22no—
32#13095494—22no—
33#13095494—22no—
34C2×#11715895C2×C411yescharacteristic
35C2×#11715895C2×C411yescharacteristic
36C8×D97—21no—
37C8×D97—21no—
38C8×D97D412yes—
39C8×D97D412yes—
40#13095486—21no—
41#13095486—21no—
42C22×C4×C97C2×C411yescharacteristic
43C2×C8×C97—21no—
44C2×C8×C97—21no—
45C2×C8×C97C2×C411yescharacteristic
46C2×C8×C97C2311yescharacteristic
47C16×C97—21no—
48C16×C97—21no—
49C23×C97C2×C811yescharacteristic
50C22×D97C2×C811yescharacteristic
51C22×D97C2×C811yescharacteristic
52C2×C4×C97—21no—
53C2×C4×C97—21no—
54C2×C4×C97C2×C811yescharacteristic
55C2×C4×C97C22×C411yescharacteristic
56C22⋊D97—22no—
57C22⋊D97—22no—
58C2×Dic97C2×C811yescharacteristic
59C2×Dic97C2×C811yescharacteristic
60D388—22no—
61C4×D97—21no—
62C4×D97—21no—
63C4×D97C22⋊C412yes—
64C4×D97M1612yes—
65Dic194—22no—
66C8×C97—21no—
67C8×C97C2×D411yescharacteristic
68C8×C97C2×D411yescharacteristic
69#11715895—21no—
70#11715895C22⋊C412yes—
71C22×C97—21no—
72C22×C97—21no—
73C22×C97—21no—
74C22×C97C22×C811yescharacteristic
75C2×D97—21no—
76C2×D97—21no—
77C2×D97—21no—
78C2×D97—21no—
79C2×D97—21no—
80C2×D97C22⋊C812yes—
81C4×C97—21no—
82C4×C97C2×C22⋊C411yescharacteristic
83C4×C97C2×M1611yescharacteristic
84Dic97—21no—
85Dic97C22⋊C812yes—
86C2×C97—41no—
87C2×C97C2×C22⋊C811yescharacteristic
88C2×C97#50411yescharacteristic
89C2×C97#50411yescharacteristic
90D97—22no—
91D97—41no—
92#2113—971nomaximal, Sylow
93C97#211311yescharacteristic, complemented, Sylow
94C2×#131—971no—
95C2×M32—971no—
96C22×C16—971no—
97C22⋊C16—971no—
98C22⋊C16—971no—
99C22⋊C16—971no—
100C22⋊C16D9711yescharacteristic, complemented
101C2×#41—971no—
102#131—1942no—
103#131—1942no—
104C2×M16—971no—
105C2×M16—971no—
106C2×M16—971no—
107C22×C8—971no—
108C22×C8—971no—
109C22×C8C2×D9711yescharacteristic
110M32—1942no—
111C2×C16—971no—
112C2×C16—971no—
113C2×C16—1942no—
114C2×C16C2×D9711yescharacteristic
115C2×C16C2×D9711yescharacteristic
116#41—1942no—
117#41—1942no—
118C2×Q8—971no—
119C2×D4—971no—
120C2×D4—971no—
121C2×D4—971no—
122C22×C4—971no—
123C22×C4—971no—
124C22×C4C4×D9711yescharacteristic
125M16—1942no—
126M16—1942no—
127M16—1942no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—971no—
131C2×C8—972no—
132C2×C8—972no—
133C2×C8—971no—
134C2×C8—1941no—
135C2×C8—1941no—
136C2×C8C4×D9711yescharacteristic
137C2×C8C22×D9711yescharacteristic
138C16—21no—
139C16—21no—
140C16—1941no—
141C16—1941no—
142C23—971no—
143C23—971no—
144C23C8×D9711yescharacteristic
145Q8—1942no—
146D4—1942no—
147D4—1942no—
148D4—1942no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—971no—
152C2×C4—972no—
153C2×C4—972no—
154C2×C4—971no—
155C2×C4—1941no—
156C2×C4—1941no—
157C2×C4C8×D9711yescharacteristic
158C2×C4C2×C4×D9711yescharacteristic
159C8—21no—
160C8—972no—
161C8—1941no—
162C8D4×D9711yescharacteristic
163C8D4×D9711yescharacteristic
164C22—21no—
165C22—21no—
166C22—21no—
167C22—972no—
168C22—1941no—
169C22—1941no—
170C22—1941no—
171C22—1941no—
172C22—1941no—
173C22C2×C8×D9711yescharacteristic
174C4—21no—
175C4—972no—
176C4—1941no—
177C4C22⋊C4×D9711yescharacteristic
178C4M16×D9711yescharacteristic
179C2—41no—
180C2—3881no—
181C2—1942no—
182C2C22⋊C8×D9711yescharacteristic
183C2#2026115111yescharacteristic
184C2#2026115311yescharacteristic
subgroupG/H
center, FrattiniC2×C8C22×D97
commutator, socleC22×C97C22×C8
FittingC22⋊C16×C97C2
prime radicalC2×#13095511C4
cosocleC2×C8×C97C23
hypercenterC22⋊C16D97
hypocenterC97#2113
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C8 ◁ C22⋊C16

Complements

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

NH ≅ G/Ntype
2. C2×#15690749C2semidirect
3. C2×C16×D97C2semidirect
4. C22⋊#13095486C2semidirect
5. #20260514C2semidirect
6. #20260514C2semidirect
7. C22⋊C16×C97C2semidirect
93. C97#2113semidirect
100. C22⋊C16D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908162
C2×#156907492
C2×C16×D972
C22⋊#130954862
#202605142
C22⋊C16×C972
#211397