Finite Group Databasebeta

← #16844812

Subgroups — #16844812

Order 3968.

250 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×D124C211yesmaximal, characteristic, complemented
2#14521395C211yesmaximal, characteristic, complemented
3#14521241C211yesmaximal, characteristic
4#14520796C211yesmaximal, characteristic, complemented
5Q8⋊(C31⋊C8)C211yesmaximal, characteristic, complemented
6#14520776C211yesmaximal, characteristic, complemented
7C31×#460C211yesmaximal, characteristic, complemented
8C2×Q8×D31—21no—
9C2×SD16×C31—21no—
10C31×#127C2211yescharacteristic
11C4×Q8×C31C2211yescharacteristic
12C2×#92479—21no—
13#11758474—21no—
14#11758474—22no—
15C4×D124—21no—
16C4×D124C2211yescharacteristic
17#11758446—21no—
18#11758446C2211yescharacteristic
19C31×C4⋊C8C2211yescharacteristic
20C31×#102—21no—
21#11758420—22no—
22#11758420—21no—
23C4⋊(C31⋊C8)C2211yescharacteristic
24C4×C31⋊C8C2211yescharacteristic
25C2×Q8×C31D411yescharacteristic
26C2×D4×C31—21no—
27C2×D4×C31—21no—
28C2×D4×C31—21no—
29Q8×D31—42no—
30Q8×D31—42no—
31C2×D124D411yescharacteristic
32C2×C4×D31—22no—
33C2×C4×D31—21no—
34C2×Dic62—22no—
35C2×Dic62—21no—
36SD16×C31—42no—
37C2×C8×C31—21no—
38C4⋊C4×C31—21no—
39C4⋊C4×C31D411yescharacteristic
40C42×C31—21no—
41C42×C31C2311yescharacteristic
42#92479—42no—
43(C2×D31)⋊C4—22no—
44(C2×D31)⋊C4—21no—
45C4⋊Dic31—22no—
46C4⋊Dic31—21no—
47C4⋊Dic31D411yescharacteristic
48C4⋊Dic31D412yes—
49C2×C31⋊C8—21no—
50C2×C31⋊C8—21no—
51C23×C31—41no—
52C22×D31—21no—
53Q8×C31—21no—
54Q8×C31D812yescomplemented
55D4×C31—42no—
56D4×C31—41no—
57D4×C31—41no—
58D4×C31—41no—
59D4×C31—41no—
60C2×C4×C31—21no—
61C2×C4×C31—21no—
62C2×C4×C31C2×D411yescharacteristic
63C2×C4×C31C2×D411yescharacteristic
64C2×C4×C31C2×D411yescharacteristic
65C2×Dic31—21no—
66C2×Dic31—22no—
67D124—21no—
68D124SD1612yescomplemented
69C4×D31—44no—
70C4×D31—41no—
71C4×D31—41no—
72Dic62—44no—
73Dic62—41no—
74Dic62—41no—
75C8×C31—41no—
76C31⋊C8—41no—
77C31⋊C8—41no—
78#1627—311nomaximal, Sylow
79C22×C31—81no—
80C22×C31—81no—
81C22×C31—81no—
82C22×C31#12011yescharacteristic
83C2×D31—22no—
84C2×D31—41no—
85C2×D31—41no—
86C4×C31—22no—
87C4×C31—21no—
88C4×C31—41no—
89C4×C31C2×D811yescharacteristic
90C4×C31C2×SD1611yescharacteristic
91C4×C31AGL(1,ℤ/8)11yescharacteristic
92C4×C31#13711yescharacteristic
93Dic31—41no—
94Dic31—42no—
95D4×Q8—311no—
96#486—311no—
97#460D3111yescharacteristic, complemented
98#459—311no—
99#332—311no—
100Q8⋊C8—311no—
101D4⋊C8—311no—
102C2×C31—161no—
103C2×C31#44711yescharacteristic
104C2×C31#44811yescharacteristic
105C2×C31#45411yescharacteristic
106D31—42no—
107C22×Q8—621no—
108C2×SD16—21no—
109C2×D8—621no—
110C4⋊Q8—311no—
111C4⋊Q8—621no—
112#127C2×D3111yescharacteristic
113C22⋊Q8—622no—
114C22⋊Q8—621no—
115C4×Q8C2×D3111yescharacteristic
116C4×D4—311no—
117C4×D4—621no—
118C4⋊C8—311no—
119C4⋊C8C2×D3111yescharacteristic
120#102—21no—
121#102—621no—
122#102—622no—
123C4×C8—311no—
124C31#162711yescharacteristic, complemented, Sylow
125C2×Q8—622no—
126C2×Q8—621no—
127C2×Q8—1242no—
128C2×Q8—1242no—
129C2×Q8C22⋊D3111yescharacteristic
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—311no—
134C22×C4—622no—
135C22×C4—621no—
136SD16—42no—
137D8—1242no—
138C2×C8—21no—
139C2×C8—621no—
140C2×C8—621no—
141C4⋊C4—21no—
142C4⋊C4—311no—
143C4⋊C4—312no—
144C4⋊C4—622no—
145C4⋊C4—621no—
146C4⋊C4C22⋊D3111yescharacteristic
147C22⋊C4—622no—
148C22⋊C4—621no—
149C42—21no—
150C42C22×D3111yescharacteristic
151C23—41no—
152C23—621no—
153Q8—21no—
154Q8—1244no—
155Q8—1241no—
156Q8—1241no—
157Q8#9247912yescomplemented
158D4—42no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—312no—
164D4—621no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—622no—
168C2×C4—621no—
169C2×C4—1244no—
170C2×C4—1241no—
171C2×C4—1241no—
172C2×C4D4×D3111yescharacteristic
173C2×C4D4×D3111yescharacteristic
174C2×C4C2×C22⋊D3111yescharacteristic
175C8—41no—
176C8—1241no—
177C8—1241no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22—622no—
182C22—1241no—
183C22—1241no—
184C22#1175851411yescharacteristic
185C4—22no—
186C4—21no—
187C4—41no—
188C4—1242no—
189C4—1241no—
190C4SD16×D3111yescharacteristic
191C4#1175849211yescharacteristic
192C4C2×#9247911yescharacteristic
193C4#1175852611yescharacteristic
194C2—161no—
195C2—1242no—
196C2#1452118211yescharacteristic
197C2#1452118611yescharacteristic
198C2#1452122011yescharacteristic
subgroupG/H
centerC22#11758514
commutator, cosocleC42×C31C23
FrattiniC42C22×D31
FittingC31×#460C2
socleC22×C31#120
prime radical#14521241C2
hypercenter#460D31
hypocenterC31#1627
LCSG ▷ C42×C31 ▷ C22×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. Q8×D124C2semidirect
2. #14521395C2semidirect
4. #14520796C2semidirect
5. Q8⋊(C31⋊C8)C2semidirect
6. #14520776C2semidirect
7. C31×#460C2semidirect
54. Q8×C31D8semidirect
68. D124SD16semidirect
97. #460D31semidirect
124. C31#1627semidirect
157. Q8#92479semidirect

Maximal subgroups

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

subgroupindex
Q8×D1242
#145213952
#145212412
#145207962
Q8⋊(C31⋊C8)2
#145207762
C31×#4602
#162731