Finite Group Databasebeta

← #30650179

Subgroups — #30650179

Order 16256.

322 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 109 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#22550486C212yesmaximal
2#22550185C212yesmaximal, complemented
3#22550182C212yesmaximal, complemented
4#22550132C212yesmaximal
5#22550117C211yesmaximal, characteristic
6#22550085C212yesmaximal, complemented
7#22550084C212yesmaximal, complemented
8#22549902C211yesmaximal, characteristic
9#467×C127C211yesmaximal, characteristic
10C2×Dic508C2211yescharacteristic
11C2×#14523238C2212yes—
12C2×#14523232C2211yescharacteristic
13#16861207C2212yes—
14#16861202—22no—
15#16861202—22no—
16#16861200—22no—
17#16861200C2212yes—
18#16861199—22no—
19#16861188C2212yes—
20Q8×Dic127C2212yes—
21#16861174C2212yes—
22#16861168C2211yescharacteristic
23#16861143C2211yescharacteristic
24#16861140C2212yes—
25Dic127⋊C8C2212yes—
26Dic127⋊C8C2212yes—
27#16861134C2212yes—
28#16861133C2212yes—
29#16861132C2212yes—
30C2×Q16×C127C2211yescharacteristic
31C22×C8×C127C2211yescharacteristic
32C22⋊Q8×C127C2212yes—
33#107×C127C2211yescharacteristic
34#103×C127C2212yes—
35C2×Dic254C2312yes—
36Dic508—41no—
37#14523238—42no—
38#14523232—41no—
39C2×C127⋊C8C2312yes—
40C4⋊Dic127—22no—
41C4⋊Dic127—22no—
42C4⋊Dic127D411yescharacteristic
43C4⋊Dic127D411yescharacteristic
44C4⋊Dic127C2312yes—
45Dic127⋊C4—22no—
46Dic127⋊C4—22no—
47Dic127⋊C4—22no—
48Dic127⋊C4—22no—
49Dic127⋊C4—22no—
50C22⋊Dic127—22no—
51C22⋊Dic127D412yes—
52C4×Dic127—22no—
53C4×Dic127—22no—
54C4×Dic127C2312yes—
55C2×Q8×C127C2312yes—
56C22×C4×C127C2311yescharacteristic
57Q16×C127—41no—
58C2×C8×C127—41no—
59C2×C8×C127C2311yescharacteristic
60C2×C8×C127C2311yescharacteristic
61C4⋊C4×C127—22no—
62C4⋊C4×C127C2312yes—
63C22⋊C4×C127—22no—
64Dic254—22no—
65Dic254—22no—
66C2×Dic127—22no—
67C2×Dic127—22no—
68C2×Dic127C2×D412yes—
69C2×Dic127C2×D412yes—
70C127⋊C8—22no—
71C23×C127C2×D411yescharacteristic
72Q8×C127—22no—
73Q8×C127—22no—
74C2×C4×C127—21no—
75C2×C4×C127—21no—
76C2×C4×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127C2×D411yescharacteristic
79C2×C4×C127C2411yescharacteristic
80C8×C127—21no—
81C8×C127—21no—
82C22×C127—21no—
83C22×C127—21no—
84C22×C127—21no—
85C22×C127C22×D411yescharacteristic
86C4×C127—21no—
87C4×C127—42no—
88C4×C127—42no—
89C4×C127H(5,𝔽2)11yescharacteristic
90C4×C1272−1+411yescharacteristic
91Dic127—22no—
92Dic127—22no—
93Dic127—42no—
94Dic127—42no—
95C2×C127—41no—
96C2×C127#53711yescharacteristic
97C2×C127#57811yescharacteristic
98C2×C127#57811yescharacteristic
99#3261—1271nomaximal, Sylow
100C127#326111yescharacteristic, complemented, Sylow
101#541—1272no—
102#479—1272no—
103#479—1272no—
104#470—1272no—
105#467—1271no—
106#467D12711yescharacteristic
107#462—1272no—
108#462—1272no—
109#424—1271no—
110C2×Q16—1272no—
111C2×Q16—1271no—
112C2×Q16C2×D12711yescharacteristic
113C2×M16—1271no—
114C22×C8C2×D12711yescharacteristic
115C4⋊Q8—1272no—
116#126—2542no—
117#126—2542no—
118#125—1272no—
119#125—2542no—
120#125—2542no—
121C22⋊Q8—1272no—
122C22⋊Q8C2×D12712yes—
123C4×Q8—1272no—
124C4×Q8—1272no—
125#117—1271no—
126#107—1271no—
127#107C2×D12711yescharacteristic
128#106—1272no—
129C4⋊C8—1272no—
130C4⋊C8—1272no—
131#103—1272no—
132#103—1272no—
133#103—1272no—
134#103C2×D12712yes—
135C2×Q8—1272no—
136C2×Q8C22×D12712yes—
137C22×C4C22×D12711yescharacteristic
138Q16—41no—
139Q16—5081no—
140Q16—5082no—
141M16—5081no—
142C2×C8—41no—
143C2×C8—1272no—
144C2×C8C22×D12711yescharacteristic
145C2×C8C22×D12711yescharacteristic
146C4⋊C4—22no—
147C4⋊C4—1271no—
148C4⋊C4—1271no—
149C4⋊C4—1272no—
150C4⋊C4—2542no—
151C4⋊C4—2542no—
152C4⋊C4—2542no—
153C4⋊C4—2542no—
154C4⋊C4—2542no—
155C4⋊C4—2542no—
156C4⋊C4—2542no—
157C4⋊C4C22×D12712yes—
158C22⋊C4—22no—
159C22⋊C4—1272no—
160C22⋊C4—2542no—
161C42—1272no—
162C42—2542no—
163C42—2542no—
164C23D4×D12711yescharacteristic
165Q8—22no—
166Q8—22no—
167Q8—2542no—
168Q8—2542no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—1272no—
174C2×C4—1272no—
175C2×C4—2542no—
176C2×C4—2542no—
177C2×C4D4×D12711yescharacteristic
178C2×C4C23×D12711yescharacteristic
179C8—21no—
180C8—21no—
181C8—2542no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22C2×D4×D12711yescharacteristic
186C4—21no—
187C4—42no—
188C4—42no—
189C4—5082no—
190C4—5082no—
191C4—2542no—
192C4—2542no—
193C4#1686125411yescharacteristic
194C4#1686125611yescharacteristic
195C2—41no—
196C2#2255045311yescharacteristic
197C2#2255068911yescharacteristic
198C2#2255069011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#467×C127C2
socleC22×C127C22×D4
prime radicalC23×C127C2×D4
hypercenter#467D127
hypocenterC127#3261
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. #22550185C2semidirect
3. #22550182C2semidirect
6. #22550085C2semidirect
7. #22550084C2semidirect
100. C127#3261semidirect

Maximal subgroups

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

subgroupindex
#225504862
#225501852
#225501822
#225501322
#225501172
#225500852
#225500842
#225499022
#467×C1272
#3261127