Finite Group Databasebeta

← #30639206

Subgroups — #30639206

Order 16256.

304 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 79 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×#16861240C211yesmaximal, characteristic
2#22550642C211yesmaximal, characteristic
3#22549892C211yesmaximal, characteristic
4#22549885C211yesmaximal, characteristic, complemented
5#22549648C211yesmaximal, characteristic, complemented
6C127⋊#338C211yesmaximal, characteristic, complemented
7C2×#101×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—22no—
10#16861240—22no—
11C2×#14523238C412yescomplemented
12C2×#14523235C412yescomplemented
13#16861217—22no—
14#16861215—22no—
15C2×#14523232C2211yescharacteristic
16C2×#14523232C2211yescharacteristic
17M16×D127—22no—
18#16861168C2211yescharacteristic
19#16861135—22no—
20#16861133—22no—
21C22×Q8×C127C2211yescharacteristic
22C2×M16×C127C2211yescharacteristic
23C2×M16×C127C2211yescharacteristic
24#101×C127—22no—
25#14523248—21no—
26#14523248—21no—
27#14523248D412yes—
28#14523248D412yes—
29C2×Dic254C2×C411yescharacteristic
30C2×D508C2×C411yescharacteristic
31C2×C22⋊D127—21no—
32C2×C4×D127—21no—
33#14523238—24no—
34#14523235—24no—
35#14523232—21no—
36#14523232—21no—
37#14523232—21no—
38#14523232—21no—
39#14523231—24no—
40C2×C127⋊C8C2×C412yes—
41C2×C127⋊C8D412yes—
42C8×D127—24no—
43C4⋊Dic127—21no—
44C22⋊Dic127—21no—
45C4×Dic127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127—21no—
50C2×Q8×C127—21no—
51C2×Q8×C127C2×C412yes—
52C22×C4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54M16×C127—21no—
55M16×C127—21no—
56M16×C127—21no—
57M16×C127D412yes—
58C2×C8×C127—21no—
59C2×C8×C127—21no—
60C22×D127—21no—
61Dic254—21no—
62Dic254C22⋊C412yes—
63D508—21no—
64D508C22⋊C412yes—
65C22⋊D127—22no—
66C22⋊D127—22no—
67C2×Dic127—21no—
68C2×Dic127—41no—
69C4×D127—22no—
70C4×D127—22no—
71C127⋊C8—22no—
72C127⋊C8—22no—
73C23×C127#4111yescharacteristic
74Q8×C127—22no—
75Q8×C127—22no—
76Q8×C127—41no—
77Q8×C127—41no—
78C2×C4×C127—22no—
79C2×C4×C127—41no—
80C2×C4×C127—41no—
81C2×C4×C127C22×C411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C8×C127—22no—
88C8×C127—41no—
89C22×C127—41no—
90C22×C127C4×D411yescharacteristic
91C22×C127#12111yescharacteristic
92C22×C127#12311yescharacteristic
93C2×D127—22no—
94C2×D127—81no—
95C4×C127—42no—
96C4×C127C2×C22⋊C411yescharacteristic
97C4×C127C4×D411yescharacteristic
98C4×C127#12011yescharacteristic
99C4×C127#12111yescharacteristic
100Dic127—22no—
101Dic127—81no—
102C2×C127—21no—
103C2×C127—21no—
104C2×C127—21no—
105C2×C127#38611yescharacteristic
106D127—42no—
107#1902—1271nomaximal, Sylow
108C127#190211yescharacteristic, complemented, Sylow
109C2×#137—1271no—
110#568—1271no—
111#421—1271no—
112#419—1271no—
113C2×#101D12711yescharacteristic, complemented
114#344—1271no—
115#338—1271no—
116C2×#41—1271no—
117C22×Q8C2×D12711yescharacteristic
118#137—2542no—
119#137—2542no—
120C2×Q16—1272no—
121C2×SD16—1272no—
122#131—2542no—
123#131—2542no—
124C2×M16—1271no—
125C2×M16—1271no—
126C2×M16—2542no—
127C2×M16C2×D12711yescharacteristic
128C2×M16C2×D12711yescharacteristic
129#117—1271no—
130#104—2542no—
131#103—2542no—
132#101—22no—
133#41—1272no—
134#41—1272no—
135#41—2541no—
136#41—2541no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—1271no—
143C2×Q8C4×D12712yes—
144C2×D4—1271no—
145C2×D4—2541no—
146C22×C4—21no—
147C22×C4—2541no—
148C22×C4C22×D12711yescharacteristic
149Q16—2544no—
150SD16—2544no—
151M16—21no—
152M16—21no—
153M16—21no—
154M16—2544no—
155M16—2541no—
156M16—2541no—
157M16—2541no—
158M16—2541no—
159M16C22⋊D12712yes—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—1272no—
163C2×C8—1272no—
164C2×C8—2544no—
165C4⋊C4—2541no—
166C22⋊C4—2541no—
167C42—2541no—
168C23—2541no—
169C23#1452324811yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—41no—
173Q8—41no—
174Q8—1272no—
175Q8—2541no—
176D4—1272no—
177D4—2542no—
178D4—2542no—
179D4—2541no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—5081no—
184C2×C4—2541no—
185C2×C4—2542no—
186C2×C4—2542no—
187C2×C4C2×C4×D12711yescharacteristic
188C2×C4D4×D12711yescharacteristic
189C2×C4D4×D12711yescharacteristic
190C2×C4D4×D12711yescharacteristic
191C2×C4C2×C22⋊D12711yescharacteristic
192C2×C4#1452324711yescharacteristic
193C8—22no—
194C8—41no—
195C8—2542no—
196C8—2542no—
197C22—41no—
198C22—10161no—
199C22—2542no—
200C22C4×C22⋊D12711yescharacteristic
201C22#1686118211yescharacteristic
202C22#1686119111yescharacteristic
203C4—42no—
204C4—10161no—
205C4—2542no—
206C4C22⋊C4×D12711yescharacteristic
207C4(C22⋊D127)⋊C411yescharacteristic
208C4#1686117711yescharacteristic
209C4#1686118011yescharacteristic
210C2—21no—
211C2—21no—
212C2—21no—
213C2—5082no—
214C2#2254975211yescharacteristic
subgroupG/H
centerC2#22549752
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#101×C127C2
socleC2×C127#386
prime radicalC2×#14523248C22
cosocleC22×C4×C127C23
hypercenterC2×#101D127
hypocenterC127#1902
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
4. #22549885C2semidirect
5. #22549648C2semidirect
6. C127⋊#338C2semidirect
7. C2×#101×C127C2semidirect
11. C2×#14523238C4semidirect
12. C2×#14523235C4semidirect
108. C127#1902semidirect
113. C2×#101D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
#225506422
#225498922
#225498852
#225496482
C127⋊#3382
C2×#101×C1272
#1902127