Finite Group Databasebeta

← #27153183

Subgroups — #27153183

Order 12416.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#20261347C211yesmaximal, characteristic, complemented
2#20261340C211yesmaximal, characteristic
3#20261043C212yesmaximal, complemented
4#20261036C212yesmaximal, complemented
5#20260981C212yesmaximal, complemented
6#20260964C212yesmaximal, complemented
7#20260941C212yesmaximal, complemented
8#20260936C212yesmaximal, complemented
9#424×C97C211yesmaximal, characteristic, complemented
10C2×#13095500C2212yes—
11C2×#13095500C2212yes—
12#15690810C2211yescharacteristic
13#15690807C2211yescharacteristic
14#15690803—22no—
15#15690802C2211yescharacteristic
16#15690802C2211yescharacteristic
17#15690800—22no—
18#15690795—22no—
19#15690789C2212yes—
20#15690784C2212yes—
21#15690783—22no—
22C4×Dic194C2212yes—
23C4×D388C2212yes—
24#15690744C2212yes—
25#15690742C2212yes—
26#15690735C2212yes—
27#15690735C2212yes—
28#15690732C2212yes—
29#15690732C2212yes—
30C2×M16×C97C2211yescharacteristic
31C22×C8×C97C2211yescharacteristic
32#117×C97C2211yescharacteristic
33C4⋊C8×C97C2212yes—
34C4⋊C8×C97C2212yes—
35C2×Dic194C2312yes—
36C2×D388—22no—
37C2×D388C2312yes—
38C2×C22⋊D97—22no—
39C2×C4×D97—22no—
40#13095500—42no—
41#13095500—42no—
42C4⋊Dic97—22no—
43C4⋊Dic97C2312yes—
44C4⋊Dic97C2312yes—
45#13095490—22no—
46#13095490—22no—
47#13095490—22no—
48#13095490—22no—
49C22⋊Dic97—22no—
50#13095488—22no—
51#13095488—22no—
52C4×Dic97—22no—
53C22×C4×C97C2311yescharacteristic
54M16×C97—41no—
55C2×C8×C97—41no—
56C2×C8×C97C2312yes—
57C2×C8×C97C2312yes—
58C4⋊C4×C97D412yes—
59C22⋊C4×C97D412yes—
60C42×C97C2311yescharacteristic
61C42×C97C2311yescharacteristic
62C23×C97C2×D411yescharacteristic
63C22×D97—22no—
64C2×C4×C97—21no—
65C2×C4×C97—21no—
66C2×C4×C97C2×D412yes—
67C2×C4×C97C2×D412yes—
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97C2411yescharacteristic
70C22⋊D97—82no—
71C2×Dic97—22no—
72C2×Dic97—22no—
73C2×Dic97—22no—
74D388—22no—
75D388—22no—
76D388—82no—
77C4×D97—82no—
78Dic194—22no—
79Dic194—22no—
80C8×C97—22no—
81C8×C97—22no—
82C22×C97—21no—
83C22×C97—22no—
84C22×C97C22×D411yescharacteristic
85C2×D97—42no—
86C2×D97—42no—
87C2×D97—42no—
88C4×C97—21no—
89C4×C97—22no—
90C4×C97—22no—
91C4×C97H(5,𝔽2)11yescharacteristic
92C4×C972−1+411yescharacteristic
93Dic97—42no—
94Dic97—42no—
95Dic97—42no—
96C2×C97—41no—
97C2×C97#53711yescharacteristic
98C2×C97#57712yes—
99D97—82no—
100#3260—971nomaximal, Sylow
101C97#326011yescharacteristic, complemented, Sylow
102#541—971no—
103#540—971no—
104#479—972no—
105#478—972no—
106#468—972no—
107#465—972no—
108#461—972no—
109#460—972no—
110#424D9711yescharacteristic, complemented
111C2×SD16—972no—
112C2×SD16—972no—
113C2×M16C2×D9711yescharacteristic
114C22×C8C2×D9711yescharacteristic
115C4⋊Q8—971no—
116#127—971no—
117#126—1942no—
118#125—971no—
119#125—971no—
120#125—1942no—
121#123—1942no—
122C22⋊Q8—972no—
123#121—972no—
124#121—1942no—
125C4×Q8—972no—
126C4×D4—972no—
127#117C2×D9711yescharacteristic
128#107—972no—
129#106—972no—
130C4⋊C8C2×D9712yes—
131C4⋊C8C2×D9712yes—
132#103—972no—
133#103—972no—
134#102—972no—
135#102—972no—
136C2×Q8—972no—
137C2×D4—972no—
138C2×D4—1942no—
139C2×D4—1942no—
140C22×C4—1942no—
141C22×C4C22×D9711yescharacteristic
142SD16—3882no—
143SD16—3882no—
144M16—41no—
145C2×C8—41no—
146C2×C8C22×D9712yes—
147C2×C8C22×D9712yes—
148C4⋊C4—972no—
149C4⋊C4—972no—
150C4⋊C4—1942no—
151C4⋊C4—1942no—
152C4⋊C4—1942no—
153C4⋊C4—1942no—
154C4⋊C4—1942no—
155C4⋊C4D38812yes—
156C22⋊C4—1942no—
157C22⋊C4—1942no—
158C22⋊C4—1942no—
159C22⋊C4D38812yes—
160C42—1942no—
161C42C22×D9711yescharacteristic
162C42C22×D9711yescharacteristic
163C23—1942no—
164C23C2×D38811yescharacteristic
165Q8—1942no—
166Q8—1942no—
167D4—7762no—
168D4—7762no—
169D4—1942no—
170D4—1942no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—7762no—
174C2×C4—1942no—
175C2×C4—1942no—
176C2×C4—1942no—
177C2×C4C2×D38812yes—
178C2×C4C2×D38812yes—
179C2×C4C2×D38811yescharacteristic
180C2×C4C23×D9711yescharacteristic
181C8—22no—
182C8—22no—
183C22—22no—
184C22—21no—
185C22—3882no—
186C22—3882no—
187C22—3882no—
188C22C22×D38811yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—3882no—
193C4—3882no—
194C4—3882no—
195C4#1569085511yescharacteristic
196C4#1569085811yescharacteristic
197C2—41no—
198C2—7762no—
199C2#2026131611yescharacteristic
200C2#2026154712yes—
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#424×C97C2
socleC22×C97C22×D4
prime radical#20261340C2
hypercenter#424D97
hypocenterC97#3260
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #20261347C2semidirect
3. #20261043C2semidirect
4. #20261036C2semidirect
5. #20260981C2semidirect
6. #20260964C2semidirect
7. #20260941C2semidirect
8. #20260936C2semidirect
9. #424×C97C2semidirect
101. C97#3260semidirect
110. #424D97semidirect

Maximal subgroups

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

subgroupindex
#202613472
#202613402
#202610432
#202610362
#202609812
#202609642
#202609412
#202609362
#424×C972
#326097