Finite Group Databasebeta

← #27154818

Subgroups — #27154818

Order 12416.

356 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 113 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#20261454C212yesmaximal, complemented
2#20261126C212yesmaximal, complemented
3#20260954C212yesmaximal, complemented
4#20260948C212yesmaximal
5#20260943C212yesmaximal, complemented
6#20260865C211yesmaximal, characteristic, complemented
7#20260836C212yesmaximal, complemented
8C4×Dic388C211yesmaximal, characteristic, complemented
9#495×C97C211yesmaximal, characteristic
10C2×Dic388C2211yescharacteristic
11C2×#13095501C2212yes—
12C2×#13095499C2212yes—
13#15690808C2212yes—
14#15690803—22no—
15#15690803—22no—
16#15690797C2212yes—
17#15690787—22no—
18Q8×Dic97C2212yes—
19C4×Dic194C2212yes—
20C4×Dic194C2211yescharacteristic
21D4×Dic97C2212yes—
22#15690766—22no—
23#15690744C2211yescharacteristic
24C4⋊#11715895C2211yescharacteristic
25#15690739C2212yes—
26#15690735C2212yes—
27#15690734C2212yes—
28#15690731C2212yes—
29#15690720C2212yes—
30C2×Q16×C97C2211yescharacteristic
31C2×SD16×C97C2212yes—
32C2×D8×C97C2211yescharacteristic
33#124×C97C2212yes—
34C4×C8×C97C2211yescharacteristic
35C2×Dic194D411yescharacteristic
36C2×Dic194C2312yes—
37C22×Dic97—22no—
38Dic388—22no—
39#13095501—42no—
40#13095499—42no—
41C2×#11715895C2312yes—
42C4⋊Dic97—22no—
43C4⋊Dic97D411yescharacteristic
44C4⋊Dic97C2312yes—
45#13095490—22no—
46#13095490—22no—
47#13095490—22no—
48#13095490D412yes—
49C22⋊Dic97—22no—
50C22⋊Dic97—22no—
51C22⋊Dic97—22no—
52C4×Dic97—22no—
53C4×Dic97—22no—
54C4×Dic97C2312yes—
55C2×Q8×C97C2312yes—
56C2×D4×C97C2312yes—
57Q16×C97—41no—
58SD16×C97—24no—
59D8×C97—41no—
60C2×C8×C97C2311yescharacteristic
61C2×C8×C97C2311yescharacteristic
62C22⋊C4×C97—22no—
63C22⋊C4×C97—22no—
64C42×C97C2311yescharacteristic
65C23×C97—22no—
66Q8×C97—22no—
67Q8×C97—22no—
68D4×C97—22no—
69D4×C97—22no—
70C2×C4×C97—22no—
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97C2×D411yescharacteristic
73C2×C4×C97C2411yescharacteristic
74C2×Dic97—22no—
75C2×Dic97—22no—
76C2×Dic97—44no—
77C2×Dic97C2×D412yes—
78C2×Dic97C2×D412yes—
79Dic194—22no—
80Dic194—22no—
81Dic194—41no—
82C8×C97—21no—
83C8×C97#4112yes—
84#11715895—22no—
85C22×C97—42no—
86C22×C97—42no—
87C22×C97—42no—
88C22×C97C22×D411yescharacteristic
89C4×C97—21no—
90C4×C97—21no—
91C4×C97—42no—
92C4×C97C2×#4111yescharacteristic
93C4×C97H(5,𝔽2)11yescharacteristic
94Dic97—22no—
95Dic97—22no—
96Dic97—42no—
97Dic97—42no—
98C2×C97—82no—
99C2×C97#55011yescharacteristic
100C2×C97#57511yescharacteristic
101C2×C97#57811yescharacteristic
102#3402—971nomaximal, Sylow
103C97#340211yescharacteristic, complemented, Sylow
104#556—972no—
105#497—972no—
106#495D9711yescharacteristic
107#464—972no—
108#463—972no—
109#462—972no—
110#446—971no—
111#440—972no—
112C4×Q16—971no—
113C2×Q16—972no—
114C2×Q16—971no—
115C2×Q16C2×D9711yescharacteristic
116C2×SD16—972no—
117C2×SD16C2×D9712yes—
118C2×D8C2×D9711yescharacteristic
119C4⋊Q8—972no—
120#126—1942no—
121#126—1942no—
122#124—972no—
123#124C2×D9712yes—
124C22⋊Q8—1942no—
125C4×Q8—971no—
126C4×Q8—972no—
127C4×Q8—972no—
128C4×D4—972no—
129#117—1942no—
130#107—971no—
131C4⋊C8—972no—
132C4⋊C8—971no—
133#103—972no—
134#103—972no—
135#102—972no—
136#97—972no—
137C4×C8C2×D9711yescharacteristic
138C2×Q8—971no—
139C2×Q8—972no—
140C2×Q8C22×D9712yes—
141C2×D4C22×D9712yes—
142C22×C4—1942no—
143Q16—41no—
144Q16—3882no—
145Q16—1942no—
146SD16—24no—
147SD16—3882no—
148D8—41no—
149C2×C8—972no—
150C2×C8C22×D9711yescharacteristic
151C2×C8C22×D9711yescharacteristic
152C4⋊C4—972no—
153C4⋊C4—971no—
154C4⋊C4—972no—
155C4⋊C4—1942no—
156C4⋊C4—1942no—
157C4⋊C4—1942no—
158C4⋊C4—1942no—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—1942no—
162C22⋊C4—1942no—
163C22⋊C4—1942no—
164C42—972no—
165C42—1942no—
166C42—1942no—
167C42C22×D9711yescharacteristic
168C23—22no—
169Q8—22no—
170Q8—22no—
171Q8—3881no—
172Q8—1942no—
173Q8—1942no—
174D4—22no—
175D4—22no—
176C2×C4—22no—
177C2×C4—972no—
178C2×C4—972no—
179C2×C4—3884no—
180C2×C4—1942no—
181C2×C4—1942no—
182C2×C4D4×D9711yescharacteristic
183C2×C4D4×D9711yescharacteristic
184C2×C4C23×D9711yescharacteristic
185C8—21no—
186C8—1942no—
187C8#1309551012yes—
188C22—42no—
189C22—42no—
190C22—42no—
191C22C2×D4×D9711yescharacteristic
192C4—21no—
193C4—21no—
194C4—42no—
195C4—3882no—
196C4—3882no—
197C4—1942no—
198C4—1942no—
199C4C2×#1309551011yescharacteristic
200C4#1569085511yescharacteristic
201C2—82no—
202C2#2026140511yescharacteristic
203C2#2026153211yescharacteristic
204C2#2026154911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#495×C97C2
socleC22×C97C22×D4
prime radicalC2×D8×C97C22
hypercenter#495D97
hypocenterC97#3402
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #20261454C2semidirect
2. #20261126C2semidirect
3. #20260954C2semidirect
5. #20260943C2semidirect
6. #20260865C2semidirect
7. #20260836C2semidirect
8. C4×Dic388C2semidirect
103. C97#3402semidirect

Maximal subgroups

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

subgroupindex
#202614542
#202611262
#202609542
#202609482
#202609432
#202608652
#202608362
C4×Dic3882
#495×C972
#340297