Finite Group Databasebeta

← #30639205

Subgroups — #30639205

Order 16256.

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

normal subgroups only

This group has 304 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#22550643C211yesmaximal, characteristic
2#22549893C211yesmaximal, characteristic
3#22549885C211yesmaximal, characteristic, complemented
4C2×#16861128C211yesmaximal, characteristic, complemented
5#22549647C211yesmaximal, characteristic, complemented
6C127⋊#338C211yesmaximal, characteristic, complemented
7C2×#137×C127C211yesmaximal, characteristic
8#16861216—22no—
9#16861216—22no—
10C2×#14523232—22no—
11C2×#14523232C2211yescharacteristic
12C2×#14523232C2211yescharacteristic
13C2×#14523232C2211yescharacteristic
14#16861168C2211yescharacteristic
15#16861136—22no—
16#16861133—22no—
17#16861128—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—24no—
26#14523232—21no—
27#14523232—21no—
28#14523232—21no—
29#14523232—21no—
30#14523232—21no—
31#14523232D412yes—
32C2×C127⋊C8—24no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C2×C127⋊C8D412yes—
36C4⋊Dic127—21no—
37C22⋊Dic127—21no—
38C4×Dic127—21no—
39#41×C127—21no—
40#41×C127—21no—
41#41×C127D412yes—
42#41×C127D412yes—
43C2×Q8×C127—21no—
44C2×Q8×C127—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127C2×C412yes—
49C2×Q8×C127C2×C411yescharacteristic
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—41no—
61C127⋊C8—22no—
62C127⋊C8—22no—
63C127⋊C8—41no—
64C23×C127—21no—
65C23×C127#4111yescharacteristic
66Q8×C127—21no—
67Q8×C127—22no—
68Q8×C127—22no—
69Q8×C127—41no—
70Q8×C127—41no—
71Q8×C127C22⋊C412yes—
72D4×C127—22no—
73D4×C127—22no—
74D4×C127—21no—
75D4×C127C22⋊C412yes—
76C2×C4×C127—21no—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
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×C127C2×D411yescharacteristic
87C2×C4×C127#4111yescharacteristic
88C8×C127—22no—
89C22×C127—22no—
90C22×C127—41no—
91C22×C127—81no—
92C22×C127C4×D411yescharacteristic
93C22×C127#12111yescharacteristic
94C22×C127#12311yescharacteristic
95C4×C127—22no—
96C4×C127—42no—
97C4×C127C2×C22⋊C411yescharacteristic
98C4×C127C4×D411yescharacteristic
99C4×C127#12011yescharacteristic
100C4×C127#12111yescharacteristic
101Dic127—81no—
102C2×C127—21no—
103C2×C127—21no—
104C2×C127—21no—
105C2×C127—42no—
106C2×C127#38611yescharacteristic
107#1902—1271nomaximal, Sylow
108C127#190211yescharacteristic, complemented, Sylow
109C2×#137D12711yescharacteristic
110#568—1271no—
111#421—1271no—
112#419—1271no—
113C2×#101—1271no—
114#344—1271no—
115#338—1271no—
116C2×#41C2×D12711yescharacteristic
117C22×Q8C2×D12711yescharacteristic
118#137—22no—
119#137—22no—
120C2×Q16Dic12712yescomplemented
121C2×SD16Dic12712yescomplemented
122#131—2542no—
123#131—2542no—
124C2×M16—1271no—
125C2×M16—1271no—
126C2×M16—1271no—
127C2×M16—2542no—
128C2×M16C2×D12711yescharacteristic
129#117—1271no—
130#104—2542no—
131#103—2542no—
132#101—2542no—
133#41—21no—
134#41—21no—
135#41C22⋊D12712yes—
136#41C22⋊D12712yes—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8C2×Dic12712yes—
143C2×Q8C2×Dic12711yescharacteristic
144C2×D4—21no—
145C2×D4C2×Dic12711yescharacteristic
146C22×C4—21no—
147C22×C4—21no—
148C22×C4C22×D12711yescharacteristic
149Q16—24no—
150SD16—24no—
151M16—21no—
152M16—21no—
153M16—1272no—
154M16—2541no—
155M16—2541no—
156M16—2544no—
157M16—2541no—
158M16—2541no—
159M16—2541no—
160C2×C8—1272no—
161C2×C8—2541no—
162C2×C8—2544no—
163C2×C8—2541no—
164C2×C8C2×Dic12712yes—
165C4⋊C4—2541no—
166C22⋊C4—2541no—
167C42—2541no—
168C23—21no—
169C23#1452324711yescharacteristic
170Q8—21no—
171Q8—22no—
172Q8—22no—
173Q8—41no—
174Q8—41no—
175Q8C22⋊Dic12712yes—
176D4—22no—
177D4—22no—
178D4—21no—
179D4C22⋊Dic12712yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—5081no—
187C2×C4C22×Dic12711yescharacteristic
188C2×C4D4×D12711yescharacteristic
189C2×C4D4×D12711yescharacteristic
190C2×C4C2×C22⋊D12711yescharacteristic
191C2×C4C2×C22⋊D12711yescharacteristic
192C2×C4#1452324711yescharacteristic
193C8—22no—
194C8—5081no—
195C8—2542no—
196C8—2542no—
197C22—22no—
198C22—41no—
199C22—81no—
200C22D4×Dic12711yescharacteristic
201C22#1686118111yescharacteristic
202C22#1686119011yescharacteristic
203C4—22no—
204C4—42no—
205C4—10161no—
206C4C2×C22⋊Dic12711yescharacteristic
207C4D4×Dic12711yescharacteristic
208C4#1686117711yescharacteristic
209C4#1686118111yescharacteristic
210C2—21no—
211C2—21no—
212C2—21no—
213C2—42no—
214C2#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#1902
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
3. #22549885C2semidirect
4. C2×#16861128C2semidirect
5. #22549647C2semidirect
6. C127⋊#338C2semidirect
22. C2×Q16×C127C4semidirect
23. C2×SD16×C127C4semidirect
108. C127#1902semidirect
120. C2×Q16Dic127semidirect
121. C2×SD16Dic127semidirect

Maximal subgroups

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

subgroupindex
#225506432
#225498932
#225498852
C2×#168611282
#225496472
C127⋊#3382
C2×#137×C1272
#1902127