Finite Group Databasebeta

← #30639632

Subgroups — #30639632

Order 16256.

296 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C22×#14523235C211yesmaximal, characteristic
2C2×#16861144C211yesmaximal, characteristic, complemented
3#22549880C211yesmaximal, characteristic
4C2×#16861128C211yesmaximal, characteristic, complemented
5C2×#16861125C211yesmaximal, characteristic
6#22549832C211yesmaximal, characteristic, complemented
7#419×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C2×#14523235—21no—
10C2×#14523235—21no—
11C2×#14523235—22no—
12C2×#14523235C414yescomplemented
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15C22×C127⋊C8C2211yescharacteristic
16#16861144—22no—
17#16861129—22no—
18#16861128—22no—
19#16861125—22no—
20#16861120—21no—
21C4×C127⋊C8—21no—
22C22×Q8×C127C2211yescharacteristic
23C2×M16×C127C2211yescharacteristic
24#117×C127C2211yescharacteristic
25#103×C127—22no—
26C23×D127—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508—21no—
32C2×D508C2×C412yes—
33#14523235—24no—
34#14523235—24no—
35#14523232—21no—
36#14523232—21no—
37#14523232—21no—
38#14523232D412yes—
39C2×C127⋊C8—21no—
40C2×C127⋊C8—21no—
41C2×C127⋊C8C2×C412yes—
42C2×C127⋊C8D411yescharacteristic
43C2×C127⋊C8D411yescharacteristic
44C2×C127⋊C8D412yes—
45C2×Q8×C127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127—21no—
50C2×Q8×C127C2×C412yes—
51C22×C4×C127—21no—
52C22×C4×C127C2311yescharacteristic
53M16×C127—21no—
54M16×C127—21no—
55C2×C8×C127—21no—
56C4⋊C4×C127D412yes—
57C22⋊C4×C127—21no—
58C42×C127—21no—
59C22×D127—22no—
60C22×D127—41no—
61C22×D127—41no—
62C22×D127—82no—
63D508—22no—
64D508—22no—
65D508—41no—
66D508—41no—
67C127⋊C8—22no—
68C127⋊C8—41no—
69C127⋊C8C22⋊C412yes—
70C127⋊C8C22⋊C412yes—
71C23×C127#4111yescharacteristic
72Q8×C127—22no—
73Q8×C127—22no—
74Q8×C127—41no—
75Q8×C127—41no—
76C2×C4×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127C22×C411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C8×C127—41no—
87C22×C127—22no—
88C22×C127C4×D411yescharacteristic
89C22×C127#12111yescharacteristic
90C22×C127#12411yescharacteristic
91C2×D127—42no—
92C2×D127—82no—
93C2×D127—82no—
94C2×D127—82no—
95C4×C127—42no—
96C4×C127—42no—
97C4×C127C2×C22⋊C411yescharacteristic
98C4×C127C4×D411yescharacteristic
99C4×C127#12111yescharacteristic
100C4×C127#12711yescharacteristic
101C2×C127—21no—
102C2×C127—21no—
103C2×C127#39011yescharacteristic
104C2×C127#47112yes—
105D127—82no—
106#1977—1271nomaximal, Sylow
107C127#197711yescharacteristic, complemented, Sylow
108C22×SD16—1271no—
109C2×#108—1271no—
110#419D12711yescharacteristic, complemented
111#418—1271no—
112C2×#101—1271no—
113C2×#100—1271no—
114#405—1271no—
115C22×Q8C2×D12711yescharacteristic
116C22×D4—1271no—
117C2×SD16—1274no—
118C2×SD16—2541no—
119C2×SD16—2541no—
120C2×SD16—2542no—
121C2×M16—1271no—
122C2×M16—1271no—
123C2×M16C2×D12711yescharacteristic
124C22×C8—1271no—
125#117C2×D12711yescharacteristic
126#108—2542no—
127#103—22no—
128#102—2542no—
129#101—2542no—
130#100—2542no—
131#97—2541no—
132C4×C8—2541no—
133C24—2541no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8C4×D12712yes—
140C2×D4—1272no—
141C2×D4—2541no—
142C2×D4—2541no—
143C2×D4—2541no—
144C2×D4—2541no—
145C2×D4—2541no—
146C22×C4—21no—
147C22×C4C22×D12711yescharacteristic
148SD16—2544no—
149SD16—2544no—
150M16—21no—
151M16—21no—
152M16—1272no—
153M16—2541no—
154M16—2541no—
155M16—2541no—
156C2×C8—21no—
157C2×C8—1272no—
158C2×C8—1271no—
159C2×C8—1271no—
160C2×C8—1272no—
161C2×C8—2541no—
162C2×C8—2541no—
163C4⋊C4C22⋊D12712yes—
164C22⋊C4—21no—
165C42—21no—
166C23—10162no—
167C23—5081no—
168C23—5081no—
169C23—2542no—
170C23#1452324711yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—41no—
174Q8—41no—
175D4—5081no—
176D4—5081no—
177D4—2542no—
178D4—2542no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4C2×C4×D12711yescharacteristic
184C2×C4D4×D12711yescharacteristic
185C2×C4D4×D12711yescharacteristic
186C2×C4D4×D12711yescharacteristic
187C2×C4C2×C22⋊D12711yescharacteristic
188C2×C4#1452324811yescharacteristic
189C8—41no—
190C8—1272no—
191C8—1272no—
192C8—5081no—
193C8—2542no—
194C22—22no—
195C22—10162no—
196C22—10162no—
197C22—10162no—
198C22—5082no—
199C22(C22⋊D127)⋊C411yescharacteristic
200C22#1686118111yescharacteristic
201C22#1686119511yescharacteristic
202C4—42no—
203C4—42no—
204C4C22⋊C4×D12711yescharacteristic
205C4C4×C22⋊D12711yescharacteristic
206C4#1686118211yescharacteristic
207C4#1686120511yescharacteristic
208C2—21no—
209C2—21no—
210C2—10162no—
211C2#2254977611yescharacteristic
212C2#2255014112yes—
subgroupG/H
centerC22#16861181
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#419×C127C2
socleC22×C127#121
prime radicalC22×D508C22
cosocleC22×C4×C127C23
hypercenter#419D127
hypocenterC127#1977
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
2. C2×#16861144C2semidirect
4. C2×#16861128C2semidirect
6. #22549832C2semidirect
7. #419×C127C2semidirect
12. C2×#14523235C4semidirect
107. C127#1977semidirect
110. #419D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232352
C2×#168611442
#225498802
C2×#168611282
C2×#168611252
#225498322
#419×C1272
#1977127