Finite Group Databasebeta

← #72259367

Subgroups — #72259367

Order 8112.

256 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 49 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#72259216C311yesmaximal, characteristic, complemented, Hall
2#72229261C211yesmaximal, characteristic, complemented
3#72229237C211yesmaximal, characteristic, complemented
4#72229237C211yesmaximal, characteristic, complemented
5#72229203C211yesmaximal, characteristic, complemented
6C2×#14523128C211yesmaximal, characteristic, complemented
7#72214085C211yesmaximal, characteristic, complemented
8#72214085C211yesmaximal, characteristic, complemented
9C22×#92788C2211yescharacteristic
10C2×#11759817C2211yescharacteristic
11C2×#11759817C2211yescharacteristic
12#14523130C2211yescharacteristic
13#14523130C2211yescharacteristic
14#14523128C2211yescharacteristic
15#14523128C2211yescharacteristic
16C2×#10621725C2×C311yescharacteristic, complemented
17C13×C22⋊D13C2×C311yescharacteristic, complemented
18C13×C22⋊D13C2×C311yescharacteristic, complemented
19#13062580C2×C311yescharacteristic, complemented
20#13062578C2×C311yescharacteristic, complemented
21D13×Dic13C2×C311yescharacteristic, complemented
22D13×Dic13C2×C311yescharacteristic, complemented
23C2×#92788—21no—
24C2×#92788C2311yescharacteristic
25#11759817—21no—
26#11759817—21no—
27C22×C132C22×C311yescharacteristic
28C2×C13×D13C22×C311yescharacteristic
29C2×C13×D13C22×C311yescharacteristic
30#10621725C22×C311yescharacteristic
31#10621725C22×C311yescharacteristic
32C13×Dic13C22×C311yescharacteristic
33C13×Dic13C22×C311yescharacteristic
34#6129⋊C3—131nomaximal
35#6129⋊C3—131nomaximal
36#92788#4111yescharacteristic, complemented, Hall
37C2×C132—21no—
38C2×C132C23×C311yescharacteristic
39C13×D13—21no—
40C13×D13—21no—
41D4×C13⋊C3—131no—
42D4×C13⋊C3—131no—
43(C22⋊D13)⋊C3—131no—
44(C22⋊D13)⋊C3—131no—
45(C22⋊D13)⋊C3—261no—
46(C22⋊D13)⋊C3—261no—
47C2×Dic13⋊C3—131no—
48C2×Dic13⋊C3—131no—
49C2×Dic13⋊C3—131no—
50C2×Dic13⋊C3—131no—
51C2×Dic13⋊C3—261no—
52C2×Dic13⋊C3—265no—
53C4×D13⋊C3—131no—
54C4×D13⋊C3—131no—
55(C13⋊C3)⋊Q8—131no—
56(C13⋊C3)⋊Q8—131no—
57#6129—131no—
58#6129—131no—
59C132C3×#4111yescharacteristic, complemented, Sylow
60C22×C13⋊C3—131no—
61C22×C13⋊C3—131no—
62C22×C13⋊C3—261no—
63C22×C13⋊C3—261no—
64C22×C13⋊C3—261no—
65C22×C13⋊C3—525no—
66C2×D13⋊C3—261no—
67C2×D13⋊C3—261no—
68C4×C13⋊C3—131no—
69C4×C13⋊C3—131no—
70Dic13⋊C3—131no—
71Dic13⋊C3—131no—
72Dic13⋊C3—131no—
73Dic13⋊C3—131no—
74Dic13⋊C3—261no—
75Dic13⋊C3—261no—
76Dic13⋊C3—265no—
77Dic13⋊C3—265no—
78Dic13⋊C3—261no—
79Dic13⋊C3—261no—
80D4×C13—131no—
81D4×C13—131no—
82C22⋊D13—131no—
83C22⋊D13—131no—
84C22⋊D13D13⋊C311yescharacteristic, complemented
85C22⋊D13D13⋊C311yescharacteristic, complemented
86C2×Dic13—131no—
87C2×Dic13—131no—
88C2×Dic13—131no—
89C2×Dic13—131no—
90C2×Dic13—261no—
91C2×Dic13—265no—
92C4×D13—131no—
93C4×D13—131no—
94Dic26—131no—
95Dic26—131no—
96C2×C13⋊C3—261no—
97C2×C13⋊C3—261no—
98C2×C13⋊C3—261no—
99C2×C13⋊C3—261no—
100C2×C13⋊C3—261no—
101C2×C13⋊C3—261no—
102C2×C13⋊C3—521no—
103C2×C13⋊C3—521no—
104C2×C13⋊C3—521no—
105C2×C13⋊C3—525no—
106C2×C13⋊C3—5210no—
107D13⋊C3—521no—
108D13⋊C3—521no—
109C22×C13—21no—
110C22×C13—25no—
111C22×C13—131no—
112C22×C13—131no—
113C22×C13C2×D13⋊C311yescharacteristic
114C22×C13C2×D13⋊C311yescharacteristic
115C2×D13C2×D13⋊C311yescharacteristic
116C2×D13C2×D13⋊C311yescharacteristic
117C4×C13—131no—
118C4×C13—131no—
119Dic13—131no—
120Dic13—131no—
121Dic13—131no—
122Dic13—131no—
123Dic13—261no—
124Dic13—265no—
125Dic13—261no—
126Dic13—265no—
127Dic13C2×D13⋊C311yescharacteristic
128Dic13C2×D13⋊C311yescharacteristic
129C3×#41—1691noHall
130C13⋊C3—261no—
131C13⋊C3—261no—
132C13⋊C3—261no—
133C13⋊C3—525no—
134C2×C13—21no—
135C2×C13—21no—
136C2×C13—210no—
137C2×C13—21no—
138C2×C13—21no—
139C2×C13—21no—
140C2×C13—25no—
141C2×C13—261no—
142C2×C13—261no—
143C2×C13C22×D13⋊C311yescharacteristic
144C2×C13C22×D13⋊C311yescharacteristic
145D13—21no—
146D13—21no—
147C3×Q8—1691no—
148C3×D4—3381no—
149C3×D4—1691no—
150C3×D4—1691no—
151C2×C3×C4—1691no—
152C2×C3×C4—1691no—
153C2×C3×C4—1691no—
154#41—1691noSylow
155C13—21no—
156C13—25no—
157C13#6129⋊C311yescharacteristic, complemented
158C13#6129⋊C311yescharacteristic, complemented
159C22×C3—3381no—
160C22×C3—6761no—
161C22×C3—1691no—
162C3×C4—3381no—
163C3×C4—1691no—
164C3×C4—1691no—
165C3×C4—1691no—
166Q8—1691no—
167D4—131no—
168D4—131no—
169D4—1691no—
170C2×C4—1691no—
171C2×C4—1691no—
172C2×C4—1691no—
173C2×C3—3381no—
174C2×C3—6761no—
175C2×C3—6761no—
176C2×C3—6761no—
177C22—131no—
178C22—131no—
179C22#11759815⋊C211yescharacteristic
180C4—131no—
181C4—131no—
182C4—1691no—
183C4—1691no—
184C3—6761noSylow
185C2—21no—
186C2—261no—
187C2—261no—
188C2C2×#11759815⋊C211yescharacteristic
subgroupG/H
center, FrattiniC2C2×#11759815⋊C2
commutator, socle, cosocleC2×C132C23×C3
FittingC22×C132C22×C3
hypercenterC22#11759815⋊C2
hypocenterC132C3×#41
LCSG ▷ C2×C132 ▷ C132
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #72259216C3semidirect
2. #72229261C2semidirect
3. #72229237C2semidirect
4. #72229237C2semidirect
5. #72229203C2semidirect
6. C2×#14523128C2semidirect
7. #72214085C2semidirect
8. #72214085C2semidirect
16. C2×#10621725C2×C3semidirect
17. C13×C22⋊D13C2×C3semidirect
18. C13×C22⋊D13C2×C3semidirect
19. #13062580C2×C3semidirect
20. #13062578C2×C3semidirect
21. D13×Dic13C2×C3semidirect
22. D13×Dic13C2×C3semidirect
36. #92788#41semidirect
59. C132C3×#41semidirect
84. C22⋊D13D13⋊C3semidirect
85. C22⋊D13D13⋊C3semidirect
157. C13#6129⋊C3semidirect
158. C13#6129⋊C3semidirect

Maximal subgroups

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

subgroupindex
#722592163
#722292612
#722292372
#722292032
C2×#145231282
#722140852
#6129⋊C313