Finite Group Databasebeta

← #30639630

Subgroups — #30639630

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×#14523236C211yesmaximal, characteristic
2C2×#16861144C211yesmaximal, characteristic, complemented
3#22549886C211yesmaximal, characteristic, complemented
4#22549881C211yesmaximal, characteristic
5C2×#16861127C211yesmaximal, characteristic, complemented
6#22549834C211yesmaximal, characteristic, complemented
7C2×#100×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523236—21no—
10C2×#14523236—21no—
11C2×#14523236—22no—
12C2×#14523236C414yescomplemented
13C2×#14523232C2211yescharacteristic
14C22×C127⋊C8C2211yescharacteristic
15#16861168C2211yescharacteristic
16#16861144—22no—
17#16861134—22no—
18#16861130—22no—
19#16861127—22no—
20#16861119—21no—
21C8×Dic127—21no—
22C22×D4×C127C2211yescharacteristic
23C2×M16×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
25#100×C127—22no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254C2×C412yes—
32C22×Dic127—21no—
33#14523236—24no—
34#14523236—24no—
35#14523232—21no—
36#14523232—21no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8C2×C412yes—
39C2×C127⋊C8D411yescharacteristic
40C2×C127⋊C8D411yescharacteristic
41C2×C127⋊C8D412yes—
42C4⋊Dic127D412yes—
43C22⋊Dic127—21no—
44C4×Dic127—21no—
45C24×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127C2×C412yes—
52C22×C4×C127C2311yescharacteristic
53M16×C127—21no—
54M16×C127—21no—
55M16×C127—21no—
56M16×C127D412yes—
57C2×C8×C127—21no—
58C2×C8×C127—21no—
59Dic254—22no—
60Dic254—22no—
61Dic254—41no—
62Dic254—41no—
63C2×Dic127—22no—
64C2×Dic127—22no—
65C2×Dic127—41no—
66C2×Dic127—41no—
67C127⋊C8—41no—
68C127⋊C8C22⋊C412yes—
69C127⋊C8C22⋊C412yes—
70C23×C127—22no—
71C23×C127—41no—
72C23×C127—41no—
73C23×C127—82no—
74C23×C127#4111yescharacteristic
75D4×C127—22no—
76D4×C127—22no—
77D4×C127—41no—
78D4×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×C127—22no—
86C8×C127—41no—
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×C127C2×C22⋊C411yescharacteristic
96C4×C127C4×D411yescharacteristic
97C4×C127#12111yescharacteristic
98C4×C127#12711yescharacteristic
99Dic127—42no—
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×SD16—1271no—
109C2×#108—1271no—
110#419—1271no—
111#418—1271no—
112C2×#101—1271no—
113C2×#100D12711yescharacteristic
114#405—1271no—
115C22×Q8—1271no—
116C22×D4C2×D12711yescharacteristic
117C2×SD16—1274no—
118C2×SD16—2541no—
119C2×SD16—2541no—
120C2×SD16—2542no—
121C2×M16—1271no—
122C2×M16C2×D12711yescharacteristic
123C2×M16C2×D12711yescharacteristic
124C22×C8—1271no—
125#117—1271no—
126#108—2542no—
127#103—2542no—
128#102—2542no—
129#101—2542no—
130#100—22no—
131#97—2541no—
132C4×C8—2541no—
133C24—21no—
134C2×Q8—1272no—
135C2×Q8—2541no—
136C2×Q8—2541no—
137C2×Q8—2541no—
138C2×Q8—2541no—
139C2×Q8—2541no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4C4×D12712yes—
146C22×C4—2541no—
147C22×C4C22×D12711yescharacteristic
148SD16—2544no—
149SD16—2544no—
150M16—21no—
151M16—21no—
152M16—21no—
153M16—2541no—
154M16—2541no—
155M16C22⋊D12712yes—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—1272no—
159C2×C8—1271no—
160C2×C8—1271no—
161C2×C8—1272no—
162C2×C8—2541no—
163C4⋊C4—1272no—
164C22⋊C4—2541no—
165C42—2541no—
166C23—22no—
167C23—41no—
168C23—41no—
169C23—82no—
170C23#1452324811yescharacteristic
171Q8—5081no—
172Q8—5081no—
173Q8—2542no—
174Q8—2542no—
175D4—22no—
176D4—22no—
177D4—41no—
178D4—41no—
179C2×C4—5081no—
180C2×C4—5081no—
181C2×C4—2542no—
182C2×C4—2542no—
183C2×C4C2×C4×D12711yescharacteristic
184C2×C4D4×D12711yescharacteristic
185C2×C4D4×D12711yescharacteristic
186C2×C4D4×D12711yescharacteristic
187C2×C4C2×C22⋊D12711yescharacteristic
188C2×C4#1452324711yescharacteristic
189C8—22no—
190C8—41no—
191C8—1272no—
192C8—1272no—
193C8—5081no—
194C22—22no—
195C22—42no—
196C22—82no—
197C22—82no—
198C22—82no—
199C22C4×C22⋊D12711yescharacteristic
200C22#1686118211yescharacteristic
201C22#1686119511yescharacteristic
202C4—5082no—
203C4—5082no—
204C4C22⋊C4×D12711yescharacteristic
205C4(C22⋊D127)⋊C411yescharacteristic
206C4#1686118111yescharacteristic
207C4#1686120511yescharacteristic
208C2—21no—
209C2—21no—
210C2—82no—
211C2#2254977611yescharacteristic
212C2#2255013812yes—
subgroupG/H
centerC22#16861182
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#100×C127C2
socleC22×C127#121
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#100D127
hypocenterC127#1977
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. C2×#16861144C2semidirect
3. #22549886C2semidirect
5. C2×#16861127C2semidirect
6. #22549834C2semidirect
12. C2×#14523236C4semidirect
107. C127#1977semidirect

Maximal subgroups

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

subgroupindex
C22×#145232362
C2×#168611442
#225498862
#225498812
C2×#168611272
#225498342
C2×#100×C1272
#1977127