Finite Group Databasebeta

← #30638529

Subgroups — #30638529

Order 16256.

284 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16861217C211yesmaximal, characteristic
2#22549858C211yesmaximal, characteristic
3#22549849C211yesmaximal, characteristic
4#22549834C211yesmaximal, characteristic, complemented
5#22549832C211yesmaximal, characteristic, complemented
6#22549647C211yesmaximal, characteristic, complemented
7#344×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861217—21no—
10#16861217—21no—
11#16861217—21no—
12#16861217—21no—
13C2×#14523232C2211yescharacteristic
14C2×#14523231C412yescomplemented
15C22×C127⋊C8C2211yescharacteristic
16C2×C8×D127C412yescomplemented
17#16861168C2211yescharacteristic
18#16861127C412yescomplemented
19#16861125C412yescomplemented
20#16861123C414yes—
21#16861120—21no—
22#16861119—21no—
23C4×C127⋊C8—21no—
24C8×Dic127—21no—
25C2×M16×C127C2211yescharacteristic
26C2×M16×C127C2211yescharacteristic
27#117×C127C2211yescharacteristic
28#14523248—21no—
29#14523248—21no—
30#14523248—21no—
31#14523248—21no—
32C2×Dic254C2×C411yescharacteristic
33C2×D508C2×C411yescharacteristic
34C2×C22⋊D127C2×C412yes—
35C2×C4×D127C2×C412yes—
36#14523232—21no—
37#14523232C2×C412yes—
38#14523231—22no—
39#14523231—22no—
40C2×C127⋊C8—21no—
41C2×C127⋊C8C2×C411yescharacteristic
42C2×C127⋊C8C2×C411yescharacteristic
43C2×C127⋊C8D411yescharacteristic
44C2×C127⋊C8D411yescharacteristic
45C2×C127⋊C8D411yescharacteristic
46C2×C127⋊C8D411yescharacteristic
47C8×D127—22no—
48C8×D127—22no—
49C4⋊Dic127—21no—
50C22⋊Dic127C2×C412yes—
51C4×Dic127—21no—
52C22×C4×C127C2311yescharacteristic
53M16×C127—21no—
54M16×C127—21no—
55M16×C127—21no—
56M16×C127C2×C412yes—
57C2×C8×C127—21no—
58C2×C8×C127C2×C412yes—
59C4⋊C4×C127—21no—
60C22⋊C4×C127C2×C412yes—
61C42×C127—21no—
62C22×D127C4212yes—
63Dic254—21no—
64Dic254—21no—
65D508—21no—
66D508—21no—
67C22⋊D127—22no—
68C22⋊D127—22no—
69C2×Dic127—22no—
70C2×Dic127C4212yes—
71C4×D127—22no—
72C4×D127—22no—
73C127⋊C8—22no—
74C127⋊C8C22⋊C411yescharacteristic
75C127⋊C8C22⋊C411yescharacteristic
76C127⋊C8C22⋊C411yescharacteristic
77C127⋊C8C22⋊C411yescharacteristic
78C23×C127C22×C411yescharacteristic
79C2×C4×C127—22no—
80C2×C4×C127C22×C411yescharacteristic
81C2×C4×C127C22×C411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127#4111yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C8×C127—22no—
87C8×C127—22no—
88C22×C127—41no—
89C22×C127C2×C4211yescharacteristic
90C22×C127C4×D411yescharacteristic
91C22×C127C4×D411yescharacteristic
92C2×D127—22no—
93C2×D127—42no—
94C4×C127—42no—
95C4×C127C2×C22⋊C411yescharacteristic
96C4×C127#11711yescharacteristic
97C4×C127C4×D411yescharacteristic
98C4×C127C4×D411yescharacteristic
99Dic127—22no—
100Dic127—42no—
101C2×C127—21no—
102C2×C127—21no—
103C2×C127—21no—
104C2×C127C4×C22⋊C411yescharacteristic
105D127—42no—
106#1757—1271nomaximal, Sylow
107C127#175711yescharacteristic, complemented, Sylow
108C2×#131—1271no—
109#413—1271no—
110#410—1271no—
111#405—1271no—
112#405—1271no—
113#344—1271no—
114#344D12711yescharacteristic, complemented
115C2×#41—1271no—
116#131—2541no—
117#131—2541no—
118#131—2541no—
119#131—2541no—
120C2×M16—1272no—
121C2×M16—1271no—
122C2×M16C2×D12711yescharacteristic
123C2×M16C2×D12711yescharacteristic
124C22×C8—1271no—
125C22×C8—1272no—
126#117—1271no—
127#117C2×D12711yescharacteristic
128#101—1272no—
129#100—1272no—
130#99—1274no—
131#97—2541no—
132#97—2541no—
133C4×C8—2541no—
134C4×C8—2541no—
135#41—2541no—
136#41—2541no—
137#41—2541no—
138#41—2541no—
139C2×Q8—1271no—
140C2×D4—1272no—
141C2×D4—1271no—
142C22×C4—1272no—
143C22×C4C22×D12711yescharacteristic
144M16—21no—
145M16—21no—
146M16—21no—
147M16—1272no—
148M16—2542no—
149M16—2542no—
150M16—2541no—
151M16C4×D12712yes—
152C2×C8—21no—
153C2×C8—1271no—
154C2×C8—1271no—
155C2×C8—1271no—
156C2×C8—1271no—
157C2×C8—1271no—
158C2×C8—1271no—
159C2×C8—2542no—
160C2×C8—2542no—
161C2×C8—2541no—
162C2×C8C4×D12712yes—
163C4⋊C4—21no—
164C4⋊C4—2541no—
165C22⋊C4—1272no—
166C22⋊C4C4×D12712yes—
167C42—21no—
168C42—2541no—
169C23—1272no—
170C23C2×C4×D12711yescharacteristic
171Q8—2541no—
172Q8—2541no—
173D4—2542no—
174D4—2542no—
175D4—2541no—
176D4—2541no—
177C2×C4—22no—
178C2×C4—1272no—
179C2×C4—2542no—
180C2×C4—2542no—
181C2×C4—2542no—
182C2×C4C2×C4×D12711yescharacteristic
183C2×C4C2×C4×D12711yescharacteristic
184C2×C4D4×D12711yescharacteristic
185C2×C4D4×D12711yescharacteristic
186C2×C4#1452324711yescharacteristic
187C2×C4#1452324911yescharacteristic
188C8—22no—
189C8—22no—
190C8—1271no—
191C8—1271no—
192C8—1271no—
193C8—1271no—
194C8—2542no—
195C22—41no—
196C22—5082no—
197C22—2542no—
198C22C42×D12711yescharacteristic
199C22(C22⋊D127)⋊C411yescharacteristic
200C22#1686117111yescharacteristic
201C4—42no—
202C4—5082no—
203C4—2542no—
204C4C22⋊C4×D12711yescharacteristic
205C4#1686116711yescharacteristic
206C4(C22⋊D127)⋊C411yescharacteristic
207C4#1686117111yescharacteristic
208C2—21no—
209C2—21no—
210C2—21no—
211C2—5082no—
212C2#2254972311yescharacteristic
subgroupG/H
centerC4C22⋊C4×D127
commutatorC22×C127C2×C42
FrattiniC22×C4C22×D127
Fitting#344×C127C2
socleC2×C127C4×C22⋊C4
prime radicalC2×#14523248C22
cosocleC22×C4×C127C23
hypercenter#344D127
hypocenterC127#1757
LCSG ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
4. #22549834C2semidirect
5. #22549832C2semidirect
6. #22549647C2semidirect
7. #344×C127C2semidirect
14. C2×#14523231C4semidirect
16. C2×C8×D127C4semidirect
18. #16861127C4semidirect
19. #16861125C4semidirect
107. C127#1757semidirect
114. #344D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612172
#225498582
#225498492
#225498342
#225498322
#225496472
#344×C1272
#1757127