Finite Group Databasebeta

← #30637978

Subgroups — #30637978

Order 16256.

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

normal subgroups only

This group has 234 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
1D4×Dic254C211yesmaximal, characteristic
2#22550279C211yesmaximal, characteristic, complemented
3#22550083C211yesmaximal, characteristic, complemented
4#22549621C211yesmaximal, characteristic, complemented
5#22549606C211yesmaximal, characteristic, complemented
6D4⋊(C127⋊C8)C211yesmaximal, characteristic
7#474×C127C211yesmaximal, characteristic
8C22×Dic254—21no—
9C2×#14523238—21no—
10#16861209C2211yescharacteristic
11#16861208—21no—
12#16861188—21no—
13#16861186—22no—
14C4×Dic254C2211yescharacteristic
15D4×Dic127—21no—
16#16861142—22no—
17C4⋊(C127⋊C8)C2211yescharacteristic
18#16861133—21no—
19C4×C127⋊C8C2211yescharacteristic
20C4⋊Q8×C127C2211yescharacteristic
21C4×D4×C127C2211yescharacteristic
22#107×C127—21no—
23C4⋊C8×C127C2211yescharacteristic
24#102×C127—21no—
25C2×Dic254—22no—
26C2×Dic254—21no—
27C2×Dic254—42no—
28C2×Dic254—42no—
29C2×Dic254D411yescharacteristic
30C22×Dic127—22no—
31#14523238—42no—
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127D412yes—
36C4⋊Dic127D411yescharacteristic
37Dic127⋊C4—22no—
38Dic127⋊C4—21no—
39C22⋊Dic127—22no—
40C4×Dic127—21no—
41C2×Q8×C127—21no—
42C2×D4×C127D411yescharacteristic
43C22×C4×C127—21no—
44C2×C8×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127D411yescharacteristic
48C22⋊C4×C127—21no—
49C42×C127C2311yescharacteristic
50Dic254—21no—
51Dic254—44no—
52Dic254—41no—
53Dic254—41no—
54Dic254D812yes—
55C2×Dic127—21no—
56C2×Dic127—21no—
57C2×Dic127—22no—
58C2×Dic127—44no—
59C127⋊C8—22no—
60C127⋊C8—41no—
61C23×C127—21no—
62Q8×C127—41no—
63Q8×C127—41no—
64D4×C127—21no—
65D4×C127Q1612yescomplemented
66C2×C4×C127—21no—
67C2×C4×C127—41no—
68C2×C4×C127—41no—
69C2×C4×C127—41no—
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C8×C127—41no—
74C22×C127—22no—
75C22×C127—41no—
76C22×C127—41no—
77C22×C127#12011yescharacteristic
78C4×C127—21no—
79C4×C127—41no—
80C4×C127—81no—
81C4×C127C2×D811yescharacteristic
82C4×C127C2×Q1611yescharacteristic
83C4×C127AGL(1,ℤ/8)11yescharacteristic
84C4×C127#13711yescharacteristic
85Dic127—22no—
86Dic127—42no—
87Dic127—41no—
88C2×C127—42no—
89C2×C127#44711yescharacteristic
90C2×C127#45111yescharacteristic
91C2×C127#45611yescharacteristic
92#1649—1271nomaximal, Sylow
93C127#164911yescharacteristic, complemented, Sylow
94D4×Q8—1271no—
95#500—1271no—
96#474D12711yescharacteristic
97#462—1271no—
98#332—1271no—
99Q8⋊C8—1271no—
100D4⋊C8—1271no—
101C22×Q8—2541no—
102C2×Q16—2541no—
103C4⋊Q8—1271no—
104C4⋊Q8—2541no—
105C4⋊Q8C2×D12711yescharacteristic
106C22⋊Q8—2542no—
107C22⋊Q8—2541no—
108C4×Q8—1271no—
109C4×D4—2541no—
110C4×D4C2×D12711yescharacteristic
111#107—21no—
112#107—2542no—
113C4⋊C8—1271no—
114C4⋊C8C2×D12711yescharacteristic
115#103—2541no—
116#102—21no—
117C4×C8—1271no—
118C2×Q8—21no—
119C2×Q8—1271no—
120C2×Q8—5082no—
121C2×Q8—5082no—
122C2×Q8—2542no—
123C2×Q8—2541no—
124C2×D4C22⋊D12711yescharacteristic
125C22×C4—21no—
126C22×C4—2542no—
127Q16—5082no—
128C2×C8—21no—
129C2×C8—2541no—
130C2×C8—2541no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—1271no—
134C4⋊C4—1272no—
135C4⋊C4—2542no—
136C4⋊C4—2541no—
137C4⋊C4—2541no—
138C4⋊C4C22⋊D12711yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—2542no—
141C42—2541no—
142C42C22×D12711yescharacteristic
143C23—21no—
144Q8—41no—
145Q8—41no—
146Q8—1272no—
147Q8—5084no—
148Q8—5081no—
149Q8—5081no—
150Q8—2541no—
151D4—21no—
152D4#1452323812yescomplemented
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—5084no—
158C2×C4—2541no—
159C2×C4—2542no—
160C2×C4—2541no—
161C2×C4D4×D12711yescharacteristic
162C2×C4D4×D12711yescharacteristic
163C2×C4C2×C22⋊D12711yescharacteristic
164C8—41no—
165C8—5081no—
166C8—2542no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#1686117711yescharacteristic
171C4—21no—
172C4—41no—
173C4—81no—
174C4—5081no—
175C4—5082no—
176C4—2542no—
177C4D8×D12711yescharacteristic
178C4C2×#1452323811yescharacteristic
179C4#1686123311yescharacteristic
180C4#1686123911yescharacteristic
181C2—42no—
182C2#2255000811yescharacteristic
183C2#2255003111yescharacteristic
184C2#2255005711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#474×C127C2
socleC22×C127#120
prime radicalC2×D4×C127D4
hypercenter#474D127
hypocenterC127#1649
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #22550279C2semidirect
3. #22550083C2semidirect
4. #22549621C2semidirect
5. #22549606C2semidirect
65. D4×C127Q16semidirect
93. C127#1649semidirect
152. D4#14523238semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2542
#225502792
#225500832
#225496212
#225496062
D4⋊(C127⋊C8)2
#474×C1272
#1649127