Finite Group Databasebeta

← #16847380

Subgroups — #16847380

Order 3968.

286 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 73 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×#11758422C212yesmaximal, complemented
2#14520844C212yesmaximal, complemented
3#14520842C212yesmaximal, complemented
4C31×#535C211yesmaximal, characteristic
5C2×#41×C31C411yescharacteristic, complemented
6C22×D4×C31C2211yescharacteristic
7C31×#127C411yescharacteristic, complemented
8C31×#127C2211yescharacteristic
9C31×#124C411yescharacteristic, complemented
10C31×#124C2211yescharacteristic
11C31×#121—22no—
12C31×#120—22no—
13C31×#117C411yescharacteristic, complemented
14C2×C22⋊Dic31—22no—
15#11758422—22no—
16#11758422C2214yes—
17C24×C31D412yes—
18#41×C31—42no—
19C2×Q8×C31C2×C411yescharacteristic, complemented
20C2×D4×C31—21no—
21C2×D4×C31—22no—
22C2×D4×C31—22no—
23C2×D4×C31—42no—
24C2×D4×C31C2×C411yescharacteristic
25C2×D4×C31C2×C411yescharacteristic, complemented
26C2×D4×C31D412yes—
27C2×D4×C31C2311yescharacteristic
28C22×C4×C31—21no—
29C22×C4×C31C2×C411yescharacteristic
30C22×Dic31—42no—
31C4⋊C4×C31—21no—
32C22⋊C4×C31—21no—
33C22⋊C4×C31—22no—
34C22⋊C4×C31—22no—
35C42×C31C2×C411yescharacteristic, complemented
36C42×C31C2×C411yescharacteristic, complemented
37C22⋊Dic31—24no—
38C22⋊Dic31—42no—
39C23×C31—22no—
40C23×C31—22no—
41C23×C31—21no—
42C23×C31—82no—
43C23×C31C22⋊C412yes—
44C23×C31C22⋊C411yescharacteristic
45C23×C31C2×D412yes—
46Q8×C31—41no—
47D4×C31—41no—
48D4×C31—42no—
49D4×C31—41no—
50D4×C31—41no—
51D4×C31—42no—
52D4×C31—82no—
53D4×C31—82no—
54C2×C4×C31—22no—
55C2×C4×C31—21no—
56C2×C4×C31—21no—
57C2×C4×C31—21no—
58C2×C4×C31—42no—
59C2×C4×C31C22⋊C411yescharacteristic
60C2×C4×C31C22×C411yescharacteristic
61C2×Dic31—44no—
62C2×Dic31—84no—
63#2127—311nomaximal, Sylow
64C22×C31—22no—
65C22×C31—22no—
66C22×C31—41no—
67C22×C31—42no—
68C22×C31—42no—
69C22×C31—44no—
70C22×C31—42no—
71C22×C31—41no—
72C22×C31—82no—
73C22×C31#9912yes—
74C22×C31C2×C22⋊C411yescharacteristic
75C4×C31—21no—
76C4×C31—21no—
77C4×C31—41no—
78C4×C31—41no—
79C4×C31—41no—
80Dic31—84no—
81#535D3111yescharacteristic
82C2×#99—312no—
83#354—312no—
84#353—312no—
85C2×C31—21no—
86C2×C31—22no—
87C2×C31—42no—
88C2×C31—42no—
89C2×C31—81no—
90C2×C31C2×#9911yescharacteristic
91C2×#41Dic3111yescharacteristic, complemented
92C22×D4C2×D3111yescharacteristic
93#127Dic3111yescharacteristic, complemented
94#127C2×D3111yescharacteristic
95#124Dic3111yescharacteristic, complemented
96#124C2×D3111yescharacteristic
97#121—22no—
98#120—22no—
99#117Dic3111yescharacteristic, complemented
100C2×C22⋊C4—622no—
101#99—314no—
102#99—622no—
103C31#212711yescharacteristic, complemented, Sylow
104C24C22⋊D3112yes—
105#41—42no—
106C2×Q8C2×Dic3111yescharacteristic, complemented
107C2×D4—21no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—42no—
111C2×D4C2×Dic3111yescharacteristic
112C2×D4C2×Dic3111yescharacteristic, complemented
113C2×D4C22⋊D3112yes—
114C2×D4C22×D3111yescharacteristic
115C22×C4—21no—
116C22×C4—1242no—
117C22×C4C2×Dic3111yescharacteristic
118C4⋊C4—21no—
119C22⋊C4—22no—
120C22⋊C4—21no—
121C22⋊C4—22no—
122C22⋊C4—624no—
123C22⋊C4—1242no—
124C42C2×Dic3111yescharacteristic, complemented
125C42C2×Dic3111yescharacteristic, complemented
126C23—22no—
127C23—22no—
128C23—21no—
129C23—82no—
130C23C22⋊Dic3112yes—
131C23C22⋊Dic3111yescharacteristic
132C23C2×C22⋊D3112yes—
133Q8—41no—
134D4—41no—
135D4—42no—
136D4—41no—
137D4—41no—
138D4—42no—
139D4—82no—
140D4—82no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—42no—
146C2×C4—1244no—
147C2×C4—2484no—
148C2×C4C22⋊Dic3111yescharacteristic
149C2×C4C22×Dic3111yescharacteristic
150C22—22no—
151C22—22no—
152C22—41no—
153C22—42no—
154C22—42no—
155C22—44no—
156C22—42no—
157C22—41no—
158C22—82no—
159C22#1175842212yes—
160C22C2×C22⋊Dic3111yescharacteristic
161C4—21no—
162C4—21no—
163C4—41no—
164C4—41no—
165C4—41no—
166C4—2484no—
167C2—21no—
168C2—22no—
169C2—42no—
170C2—42no—
171C2—81no—
172C2C2×#1175842211yescharacteristic
subgroupG/H
centerC2C2×#11758422
commutatorC2×C4×C31C22×C4
FrattiniC2×D4C22×D31
Fitting, prime radicalC31×#535C2
socleC2×C31C2×#99
cosocleC2×D4×C31C23
hypercenter#535D31
hypocenterC31#2127
LCSG ▷ C2×C4×C31 ▷ C22×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#11758422C2semidirect
2. #14520844C2semidirect
3. #14520842C2semidirect
5. C2×#41×C31C4semidirect
7. C31×#127C4semidirect
9. C31×#124C4semidirect
13. C31×#117C4semidirect
19. C2×Q8×C31C2×C4semidirect
25. C2×D4×C31C2×C4semidirect
35. C42×C31C2×C4semidirect
36. C42×C31C2×C4semidirect
91. C2×#41Dic31semidirect
93. #127Dic31semidirect
95. #124Dic31semidirect
99. #117Dic31semidirect
103. C31#2127semidirect
106. C2×Q8C2×Dic31semidirect
112. C2×D4C2×Dic31semidirect
124. C42C2×Dic31semidirect
125. C42C2×Dic31semidirect

Maximal subgroups

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

subgroupindex
C2×#117584222
#145208442
#145208422
C31×#5352
#212731