Finite Group Databasebeta

← #30639635

Subgroups — #30639635

Order 16256.

296 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16861145C211yesmaximal, characteristic, complemented
2#22549885C211yesmaximal, characteristic, complemented
3#22549881C211yesmaximal, characteristic
4C2×#16861128C211yesmaximal, characteristic, complemented
5C2×#16861126C211yesmaximal, characteristic
6#22549831C211yesmaximal, characteristic, complemented
7C22×SD16×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11#16861168C2211yescharacteristic
12#16861145—22no—
13#16861133—22no—
14#16861130—22no—
15#16861128—22no—
16#16861126—22no—
17#16861119—21no—
18C8×Dic127—21no—
19C22×Q8×C127C2211yescharacteristic
20C22×D4×C127C2211yescharacteristic
21C2×SD16×C127—21no—
22C2×SD16×C127—21no—
23C2×SD16×C127—22no—
24C2×SD16×C127C414yescomplemented
25C22×C8×C127C2211yescharacteristic
26#14523232—21no—
27#14523232—21no—
28#14523232—21no—
29#14523232—21no—
30#14523232—21no—
31#14523232D412yes—
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127D412yes—
36C22⋊Dic127—21no—
37C4×Dic127—21no—
38C24×C127—21no—
39C2×Q8×C127—21no—
40C2×Q8×C127—21no—
41C2×Q8×C127—21no—
42C2×Q8×C127—21no—
43C2×Q8×C127—21no—
44C2×Q8×C127C2×C412yes—
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127C2×C412yes—
51C22×C4×C127—21no—
52C22×C4×C127C2311yescharacteristic
53SD16×C127—24no—
54SD16×C127—24no—
55C2×C8×C127C2×C412yes—
56C2×C8×C127D411yescharacteristic
57C2×C8×C127D411yescharacteristic
58C2×C8×C127D412yes—
59C2×Dic127—22no—
60C127⋊C8—22no—
61C127⋊C8—41no—
62C127⋊C8—41no—
63C23×C127—22no—
64C23×C127—41no—
65C23×C127—41no—
66C23×C127—82no—
67C23×C127#4111yescharacteristic
68Q8×C127—22no—
69Q8×C127—22no—
70Q8×C127—41no—
71Q8×C127—41no—
72D4×C127—22no—
73D4×C127—22no—
74D4×C127—41no—
75D4×C127—41no—
76C2×C4×C127—22no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127C22×C411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127#4111yescharacteristic
85C8×C127C22⋊C412yes—
86C8×C127C22⋊C412yes—
87C22×C127—22no—
88C22×C127—42no—
89C22×C127—82no—
90C22×C127—82no—
91C22×C127—82no—
92C22×C127C4×D411yescharacteristic
93C22×C127#12111yescharacteristic
94C22×C127#12411yescharacteristic
95C4×C127—42no—
96C4×C127C2×C22⋊C411yescharacteristic
97C4×C127C4×D411yescharacteristic
98C4×C127#12111yescharacteristic
99C4×C127#12711yescharacteristic
100Dic127—42no—
101C2×C127—21no—
102C2×C127—21no—
103C2×C127—82no—
104C2×C127#39011yescharacteristic
105C2×C127#47112yes—
106#1977—1271nomaximal, Sylow
107C127#197711yescharacteristic, complemented, Sylow
108C22×SD16D12711yescharacteristic
109C2×#108—1271no—
110#419—1271no—
111#418—1271no—
112C2×#101—1271no—
113C2×#100—1271no—
114#405—1271no—
115C22×Q8C2×D12711yescharacteristic
116C22×D4C2×D12711yescharacteristic
117C2×SD16—21no—
118C2×SD16—21no—
119C2×SD16—22no—
120C2×SD16Dic12714yescomplemented
121C2×M16—1271no—
122C2×M16—1271no—
123C2×M16—1271no—
124C22×C8C2×D12711yescharacteristic
125#117—1271no—
126#108—2542no—
127#103—2542no—
128#102—2542no—
129#101—2542no—
130#100—2542no—
131#97—2541no—
132C4×C8—2541no—
133C24—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8C2×Dic12712yes—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4C2×Dic12712yes—
146C22×C4—21no—
147C22×C4C22×D12711yescharacteristic
148SD16—24no—
149SD16—24no—
150M16—1272no—
151M16—2541no—
152M16—2541no—
153M16—2541no—
154M16—2541no—
155M16—2541no—
156C2×C8—2541no—
157C2×C8—2541no—
158C2×C8—2541no—
159C2×C8C2×Dic12712yes—
160C2×C8C22⋊D12711yescharacteristic
161C2×C8C22⋊D12711yescharacteristic
162C2×C8C22⋊D12712yes—
163C4⋊C4—1272no—
164C22⋊C4—2541no—
165C42—2541no—
166C23—22no—
167C23—41no—
168C23—41no—
169C23—82no—
170C23#1452324711yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—41no—
174Q8—41no—
175D4—22no—
176D4—22no—
177D4—41no—
178D4—41no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—2542no—
183C2×C4C22×Dic12711yescharacteristic
184C2×C4D4×D12711yescharacteristic
185C2×C4D4×D12711yescharacteristic
186C2×C4C2×C22⋊D12711yescharacteristic
187C2×C4C2×C22⋊D12711yescharacteristic
188C2×C4#1452324711yescharacteristic
189C8—5081no—
190C8—5081no—
191C8—2542no—
192C8C22⋊Dic12712yes—
193C8C22⋊Dic12712yes—
194C22—22no—
195C22—42no—
196C22—82no—
197C22—82no—
198C22—82no—
199C22D4×Dic12711yescharacteristic
200C22#1686118011yescharacteristic
201C22#1686119611yescharacteristic
202C4—42no—
203C4—5082no—
204C4C2×C22⋊Dic12711yescharacteristic
205C4D4×Dic12711yescharacteristic
206C4#1686118011yescharacteristic
207C4#1686120511yescharacteristic
208C2—21no—
209C2—21no—
210C2—82no—
211C2#2254977911yescharacteristic
212C2#2255013612yes—
subgroupG/H
centerC22#16861180
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC22×SD16×C127C2
socleC22×C127#121
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenterC22×SD16D127
hypocenterC127#1977
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16861145C2semidirect
2. #22549885C2semidirect
4. C2×#16861128C2semidirect
6. #22549831C2semidirect
24. C2×SD16×C127C4semidirect
107. C127#1977semidirect
120. C2×SD16Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611452
#225498852
#225498812
C2×#168611282
C2×#168611262
#225498312
C22×SD16×C1272
#1977127