Finite Group Databasebeta

← #30649108

Subgroups — #30649108

Order 16256.

380 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 129 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#22550429C211yesmaximal, characteristic
2C2×#16861207C211yesmaximal, characteristic
3#22550243C214yesmaximal, complemented
4#22550225C214yesmaximal, complemented
5#22549886C211yesmaximal, characteristic
6#22549885C211yesmaximal, characteristic
7#22549884C211yesmaximal, characteristic
8M16×Dic127C211yesmaximal, characteristic
9#419×C127C211yesmaximal, characteristic
10C22×Dic254C2211yescharacteristic
11C2×#14523232C2211yescharacteristic
12#16861208C2212yes—
13#16861207—22no—
14#16861207C2212yes—
15#16861207C2212yes—
16#16861200C2212yes—
17#16861186—22no—
18C4×Dic254—22no—
19#16861168C2211yescharacteristic
20C2×Dic127⋊C4—22no—
21C2×C4×Dic127C2211yescharacteristic
22#16861134C2214yes—
23#16861133C2214yes—
24#16861132C2214yes—
25#16861119C2212yes—
26C8×Dic127C2212yes—
27C22×Q8×C127C2211yescharacteristic
28C2×M16×C127C2211yescharacteristic
29#117×C127C2211yescharacteristic
30#103×C127C2214yes—
31C2×Dic254—22no—
32C2×Dic254—21no—
33C2×Dic254—22no—
34C2×Dic254—22no—
35C2×Dic254C2312yes—
36C22×Dic127—21no—
37C22×Dic127D412yes—
38#14523232—22no—
39C2×C127⋊C8C2312yes—
40C4⋊Dic127—22no—
41C4⋊Dic127C2312yes—
42Dic127⋊C4—22no—
43Dic127⋊C4—22no—
44Dic127⋊C4—24no—
45Dic127⋊C4—24no—
46C22⋊Dic127—21no—
47C4×Dic127—21no—
48C4×Dic127D412yes—
49C4×Dic127C2311yescharacteristic
50C4×Dic127C2311yescharacteristic
51C2×Q8×C127—21no—
52C2×Q8×C127—22no—
53C2×Q8×C127—22no—
54C2×Q8×C127C2312yes—
55C22×C4×C127—21no—
56C22×C4×C127C2311yescharacteristic
57M16×C127—22no—
58C2×C8×C127C2312yes—
59C4⋊C4×C127C2312yes—
60C22⋊C4×C127—21no—
61C42×C127—21no—
62Dic254—22no—
63Dic254—22no—
64Dic254—42no—
65Dic254—44no—
66C2×Dic127—24no—
67C2×Dic127—22no—
68C2×Dic127—22no—
69C2×Dic127—42no—
70C2×Dic127C2×D412yes—
71C2×Dic127C2×D412yes—
72C127⋊C8—22no—
73C23×C127C2×D411yescharacteristic
74Q8×C127—22no—
75Q8×C127—22no—
76Q8×C127—42no—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127—42no—
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127#4112yes—
82C2×C4×C127#4112yes—
83C2×C4×C127C2411yescharacteristic
84C8×C127—22no—
85C22×C127—22no—
86C22×C127#12412yes—
87C22×C127C22×D411yescharacteristic
88C4×C127—42no—
89C4×C127—42no—
90C4×C127#12412yes—
91C4×C127C2×#4111yescharacteristic
92C4×C127C2×#4111yescharacteristic
93Dic127—22no—
94Dic127—42no—
95Dic127—42no—
96Dic127#13714yes—
97C2×C127—22no—
98C2×C127C2×#12411yescharacteristic
99C2×C127C2×#13712yes—
100#3139—1271nomaximal, Sylow
101C127#313911yescharacteristic, complemented, Sylow
102#533—1271no—
103C2×C4⋊Q8—1271no—
104#491—1274no—
105#487—1274no—
106#419—1271no—
107#419—1271no—
108#419—1271no—
109#419D12711yescharacteristic
110C4×M16—1271no—
111C22×Q8—1271no—
112C22×Q8C2×D12711yescharacteristic
113C2×M16—1271no—
114C2×M16C2×D12711yescharacteristic
115C4⋊Q8—1272no—
116C4⋊Q8—1272no—
117C4⋊Q8—1272no—
118C4⋊Q8—2542no—
119#125—1272no—
120C22⋊Q8—2542no—
121C4×Q8—2542no—
122#117—1271no—
123#117C2×D12711yescharacteristic
124C2×C4⋊C4—2542no—
125C2×C42—1271no—
126#103—1274no—
127#103—1274no—
128#103—1274no—
129#103C2×D12714yes—
130#97—1272no—
131C4×C8—1272no—
132C2×Q8—21no—
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—1272no—
136C2×Q8—2542no—
137C2×Q8—2542no—
138C2×Q8—2542no—
139C2×Q8—2541no—
140C2×Q8C22×D12712yes—
141C22×C4—21no—
142C22×C4—1272no—
143C22×C4—2541no—
144C22×C4C22×D12711yescharacteristic
145M16—22no—
146M16—2542no—
147C2×C8—1272no—
148C2×C8C22×D12712yes—
149C4⋊C4—1272no—
150C4⋊C4—2544no—
151C4⋊C4—2544no—
152C4⋊C4—2542no—
153C4⋊C4—2542no—
154C4⋊C4—2542no—
155C4⋊C4C22×D12712yes—
156C22⋊C4—21no—
157C22⋊C4—2541no—
158C42—21no—
159C42—1272no—
160C42—1271no—
161C42—1271no—
162C42—2541no—
163C23D4×D12711yescharacteristic
164Q8—22no—
165Q8—22no—
166Q8—42no—
167Q8—5084no—
168Q8—5082no—
169Q8—2542no—
170Q8—2542no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—42no—
174C2×C4—1272no—
175C2×C4—1272no—
176C2×C4—5082no—
177C2×C4—2544no—
178C2×C4—2542no—
179C2×C4—2542no—
180C2×C4D4×D12711yescharacteristic
181C2×C4#1452324712yes—
182C2×C4#1452324812yes—
183C2×C4C23×D12711yescharacteristic
184C8—22no—
185C8—2542no—
186C22—22no—
187C22#1686119512yes—
188C22C2×D4×D12711yescharacteristic
189C4—42no—
190C4—42no—
191C4—1274no—
192C4—5082no—
193C4—5082no—
194C4—2542no—
195C4#1686119512yes—
196C4C2×#1452324711yescharacteristic
197C4C2×#1452324811yescharacteristic
198C2—22no—
199C2C2×#1686119511yescharacteristic
200C2#137×D12712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#419×C127C2
socleC22×C127C22×D4
prime radicalC23×C127C2×D4
hypercenter#419D127
hypocenterC127#3139
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #22550243C2semidirect
4. #22550225C2semidirect
101. C127#3139semidirect

Maximal subgroups

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

subgroupindex
#225504292
C2×#168612072
#225502432
#225502252
#225498862
#225498852
#225498842
M16×Dic1272
#419×C1272
#3139127