Finite Group Databasebeta

← #27153207

Subgroups — #27153207

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#20261356C212yesmaximal, complemented
2#20261026C212yesmaximal, complemented
3#20261021C212yesmaximal, complemented
4#20260993C211yesmaximal, characteristic, complemented
5#20260988C211yesmaximal, characteristic
6#20260964C212yesmaximal, complemented
7#20260958C212yesmaximal, complemented
8#20260951C212yesmaximal, complemented
9#424×C97C211yesmaximal, characteristic, complemented
10C2×Dic388C2211yescharacteristic
11C2×#13095500C2212yes—
12C2×D776C2211yescharacteristic
13#15690810C2212yes—
14#15690805—22no—
15#15690798C2212yes—
16#15690796—22no—
17#15690792—22no—
18#15690789C2212yes—
19#15690788—22no—
20#15690784C2212yes—
21C4×Dic194C2212yes—
22C4×D388C2212yes—
23#15690742C2212yes—
24#15690742C2211yescharacteristic
25#15690742C2211yescharacteristic
26#15690735C2212yes—
27#15690735C2212yes—
28#15690732C2212yes—
29#15690732C2212yes—
30C2×M16×C97C2211yescharacteristic
31C22×C8×C97C2211yescharacteristic
32#117×C97C2211yescharacteristic
33C4⋊C8×C97C2212yes—
34C4⋊C8×C97C2212yes—
35C2×Dic194—22no—
36C2×Dic194C2312yes—
37C2×D388C2312yes—
38C2×C22⋊D97—22no—
39C2×C4×D97—22no—
40Dic388—41no—
41#13095500—42no—
42D776—41no—
43C4⋊Dic97—22no—
44C4⋊Dic97C2312yes—
45C4⋊Dic97C2312yes—
46#13095490—22no—
47#13095490—22no—
48C22⋊Dic97—22no—
49#13095488—22no—
50#13095488—22no—
51#13095488—22no—
52#13095488—22no—
53C4×Dic97—22no—
54C22×C4×C97C2311yescharacteristic
55M16×C97—41no—
56C2×C8×C97—41no—
57C2×C8×C97C2311yescharacteristic
58C2×C8×C97C2312yes—
59C2×C8×C97C2311yescharacteristic
60C4⋊C4×C97D412yes—
61C22⋊C4×C97D411yescharacteristic
62C22⋊C4×C97D411yescharacteristic
63C42×C97C2312yes—
64C23×C97C2×D411yescharacteristic
65C22×D97—22no—
66C2×C4×C97—21no—
67C2×C4×C97—21no—
68C2×C4×C97C2×D412yes—
69C2×C4×C97C2×D412yes—
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97C2411yescharacteristic
72C22⋊D97—82no—
73C2×Dic97—22no—
74C2×Dic97—22no—
75C2×Dic97—22no—
76D388—22no—
77D388—22no—
78C4×D97—82no—
79Dic194—22no—
80Dic194—22no—
81Dic194—82no—
82C8×C97—22no—
83C8×C97—21no—
84C8×C97—21no—
85C22×C97—21no—
86C22×C97—22no—
87C22×C97C22×D411yescharacteristic
88C2×D97—42no—
89C2×D97—42no—
90C2×D97—42no—
91C4×C97—21no—
92C4×C97—22no—
93C4×C97—22no—
94C4×C97H(5,𝔽2)11yescharacteristic
95C4×C972−1+411yescharacteristic
96Dic97—42no—
97Dic97—42no—
98Dic97—42no—
99C2×C97—41no—
100C2×C97#53711yescharacteristic
101C2×C97#57712yes—
102D97—82no—
103#3263—971nomaximal, Sylow
104C97#326311yescharacteristic, complemented, Sylow
105#542—972no—
106#476—972no—
107#475—972no—
108#470—971no—
109#469—971no—
110#465—972no—
111#464—972no—
112#463—972no—
113#424D9711yescharacteristic, complemented
114C2×Q16—971no—
115C2×SD16—972no—
116C2×D8—971no—
117C2×M16C2×D9711yescharacteristic
118C22×C8C2×D9711yescharacteristic
119C4⋊Q8—972no—
120#126—1942no—
121#124—972no—
122#124—1942no—
123#123—1942no—
124C22⋊Q8—972no—
125C22⋊Q8—1942no—
126#121—972no—
127C4×Q8—972no—
128C4×D4—972no—
129#117C2×D9711yescharacteristic
130#106—972no—
131#106—971no—
132#106—971no—
133C4⋊C8C2×D9712yes—
134C4⋊C8C2×D9712yes—
135#103—972no—
136#103—972no—
137#102—972no—
138#102—972no—
139C2×Q8—972no—
140C2×Q8—1942no—
141C2×D4—972no—
142C2×D4—1942no—
143C22×C4—1942no—
144C22×C4C22×D9711yescharacteristic
145Q16—3881no—
146SD16—3882no—
147D8—3881no—
148M16—41no—
149C2×C8—41no—
150C2×C8C22×D9711yescharacteristic
151C2×C8C22×D9712yes—
152C2×C8C22×D9711yescharacteristic
153C4⋊C4—972no—
154C4⋊C4—972no—
155C4⋊C4—1942no—
156C4⋊C4—1942no—
157C4⋊C4—1942no—
158C4⋊C4D38812yes—
159C22⋊C4—1942no—
160C22⋊C4—1942no—
161C22⋊C4—1942no—
162C22⋊C4—1942no—
163C22⋊C4—1942no—
164C22⋊C4D38811yescharacteristic
165C22⋊C4D38811yescharacteristic
166C42—1942no—
167C42C22×D9712yes—
168C23—1942no—
169C23C2×D38811yescharacteristic
170Q8—7762no—
171Q8—1942no—
172Q8—1942no—
173D4—7762no—
174D4—1942no—
175D4—1942no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—7762no—
179C2×C4—1942no—
180C2×C4—1942no—
181C2×C4—1942no—
182C2×C4C2×D38812yes—
183C2×C4C2×D38812yes—
184C2×C4C2×D38811yescharacteristic
185C2×C4C23×D9711yescharacteristic
186C8—22no—
187C8—21no—
188C8—21no—
189C22—22no—
190C22—21no—
191C22—3882no—
192C22—3882no—
193C22—3882no—
194C22C22×D38811yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—3882no—
199C4—3882no—
200C4—3882no—
201C4#1569085511yescharacteristic
202C4#1569085811yescharacteristic
203C2—41no—
204C2—7762no—
205C2#2026131611yescharacteristic
206C2#2026154712yes—
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#424×C97C2
socleC22×C97C22×D4
prime radical#20260988C2
hypercenter#424D97
hypocenterC97#3263
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. #20261356C2semidirect
2. #20261026C2semidirect
3. #20261021C2semidirect
4. #20260993C2semidirect
6. #20260964C2semidirect
7. #20260958C2semidirect
8. #20260951C2semidirect
9. #424×C97C2semidirect
104. C97#3263semidirect
113. #424D97semidirect

Maximal subgroups

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

subgroupindex
#202613562
#202610262
#202610212
#202609932
#202609882
#202609642
#202609582
#202609512
#424×C972
#326397