Finite Group Databasebeta

← #30639226

Subgroups — #30639226

Order 16256.

310 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 93 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#22550354C211yesmaximal, characteristic, complemented
2#22549893C211yesmaximal, characteristic
3#22549885C211yesmaximal, characteristic, complemented
4#22549850C211yesmaximal, characteristic
5#22549647C211yesmaximal, characteristic, complemented
6#22549633C211yesmaximal, characteristic, complemented
7C2×#137×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9Q8×Dic127—22no—
10Q8×Dic127—22no—
11#16861168—22no—
12#16861168C2211yescharacteristic
13#16861168C2211yescharacteristic
14#16861168C2211yescharacteristic
15#16861136—22no—
16#16861133—22no—
17#16861124—22no—
18C2×#41×C127C2211yescharacteristic
19C22×Q8×C127C2211yescharacteristic
20#137×C127—22no—
21#137×C127—22no—
22C2×Q16×C127C412yescomplemented
23C2×SD16×C127C412yescomplemented
24C2×M16×C127C2211yescharacteristic
25#14523232—21no—
26#14523232—21no—
27C2×C127⋊C8—21no—
28C4⋊Dic127—24no—
29C4⋊Dic127—21no—
30C4⋊Dic127—21no—
31C4⋊Dic127—21no—
32C22⋊Dic127—21no—
33C22⋊Dic127—21no—
34C22⋊Dic127D412yes—
35C4×Dic127—24no—
36C4×Dic127—21no—
37C4×Dic127—21no—
38C4×Dic127D412yes—
39#41×C127—21no—
40#41×C127—21no—
41#41×C127—21no—
42#41×C127—21no—
43C2×Q8×C127—21no—
44C2×Q8×C127—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127C2×C411yescharacteristic
47C2×Q8×C127C2×C412yes—
48C2×Q8×C127D412yes—
49C2×Q8×C127D412yes—
50C2×D4×C127—21no—
51C2×D4×C127C2×C411yescharacteristic
52C22×C4×C127—21no—
53C22×C4×C127—21no—
54C22×C4×C127C2311yescharacteristic
55Q16×C127—24no—
56SD16×C127—24no—
57M16×C127—21no—
58M16×C127—21no—
59C2×C8×C127C2×C412yes—
60C2×Dic127—22no—
61C2×Dic127—22no—
62C2×Dic127—41no—
63C127⋊C8—41no—
64C23×C127—21no—
65C23×C127C2×D411yescharacteristic
66Q8×C127—22no—
67Q8×C127—22no—
68Q8×C127—22no—
69Q8×C127—21no—
70Q8×C127—21no—
71Q8×C127C22⋊C414yes—
72D4×C127—21no—
73D4×C127—21no—
74D4×C127—41no—
75D4×C127—41no—
76C2×C4×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127—21no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127C22×C411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C2×C4×C127#4111yescharacteristic
88C8×C127—22no—
89C22×C127—41no—
90C22×C127—41no—
91C22×C127—81no—
92C22×C127C4×D411yescharacteristic
93C22×C127#12011yescharacteristic
94C22×C127#12111yescharacteristic
95C4×C127—24no—
96C4×C127—41no—
97C4×C127C2×C22⋊C411yescharacteristic
98C4×C127C4×D411yescharacteristic
99C4×C127#12111yescharacteristic
100C4×C127#12311yescharacteristic
101Dic127—42no—
102Dic127—42no—
103Dic127—81no—
104C2×C127—21no—
105C2×C127—21no—
106C2×C127—21no—
107C2×C127—81no—
108C2×C127#38611yescharacteristic
109#1905—1271nomaximal, Sylow
110C127#190511yescharacteristic, complemented, Sylow
111C2×#137D12711yescharacteristic
112#519—1271no—
113#421—1271no—
114#419—1271no—
115#410—1271no—
116#344—1271no—
117H(3,ℤ/4)—1271no—
118C2×#41C2×D12711yescharacteristic
119C22×Q8C2×D12711yescharacteristic
120#137—22no—
121#137—22no—
122C2×Q16Dic12712yescomplemented
123C2×SD16Dic12712yescomplemented
124C2×M16—1271no—
125C2×M16C2×D12711yescharacteristic
126C4×Q8—2542no—
127C4×Q8—2542no—
128#117—1271no—
129#117—1271no—
130#117—1271no—
131#117—2542no—
132#104—2542no—
133#103—2542no—
134#99—2542no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8C2×Dic12711yescharacteristic
143C2×Q8C2×Dic12712yes—
144C2×Q8C22⋊D12712yes—
145C2×Q8C22⋊D12712yes—
146C2×D4—21no—
147C2×D4C2×Dic12711yescharacteristic
148C22×C4—21no—
149C22×C4—21no—
150C22×C4C22×D12711yescharacteristic
151Q16—24no—
152SD16—24no—
153M16—21no—
154M16—21no—
155M16—2541no—
156M16—2541no—
157C2×C8—2541no—
158C2×C8C2×Dic12712yes—
159C4⋊C4—2544no—
160C4⋊C4—2541no—
161C4⋊C4—2541no—
162C4⋊C4—2541no—
163C22⋊C4—1272no—
164C22⋊C4—2541no—
165C22⋊C4—2541no—
166C42—1272no—
167C42—2544no—
168C42—2541no—
169C42—2541no—
170C23—21no—
171C23D4×D12711yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—22no—
175Q8—21no—
176Q8—21no—
177Q8C22⋊Dic12714yes—
178D4—21no—
179D4—21no—
180D4—41no—
181D4—41no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—5081no—
189C2×C4—2542no—
190C2×C4—2542no—
191C2×C4C22×Dic12711yescharacteristic
192C2×C4D4×D12711yescharacteristic
193C2×C4C2×C22⋊D12711yescharacteristic
194C2×C4C2×C22⋊D12711yescharacteristic
195C2×C4#1452324711yescharacteristic
196C2×C4#1452324711yescharacteristic
197C8—22no—
198C8—5081no—
199C22—41no—
200C22—41no—
201C22—81no—
202C22D4×Dic12711yescharacteristic
203C22#1686117711yescharacteristic
204C22#1686118111yescharacteristic
205C4—24no—
206C4—41no—
207C4—10161no—
208C4—5082no—
209C4—5082no—
210C4C2×C22⋊Dic12711yescharacteristic
211C4D4×Dic12711yescharacteristic
212C4#1686118111yescharacteristic
213C4#1686119011yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—81no—
218C2#2254975611yescharacteristic
subgroupG/H
centerC2#22549756
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#137×C127C2
socleC2×C127#386
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#137D127
hypocenterC127#1905
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. #22550354C2semidirect
3. #22549885C2semidirect
5. #22549647C2semidirect
6. #22549633C2semidirect
22. C2×Q16×C127C4semidirect
23. C2×SD16×C127C4semidirect
110. C127#1905semidirect
122. C2×Q16Dic127semidirect
123. C2×SD16Dic127semidirect

Maximal subgroups

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

subgroupindex
#225503542
#225498932
#225498852
#225498502
#225496472
#225496332
C2×#137×C1272
#1905127