Finite Group Databasebeta

← #30639904

Subgroups — #30639904

Order 16256.

244 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 65 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#22550429C211yesmaximal, characteristic, complemented
2#22549893C211yesmaximal, characteristic
3C2×#16861136C211yesmaximal, characteristic
4#22549646C211yesmaximal, characteristic, complemented
5#22549635C211yesmaximal, characteristic, complemented
6#22549632C211yesmaximal, characteristic, complemented
7#410×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861208—21no—
11#16861200—21no—
12#16861186—22no—
13C4×Dic254—22no—
14#16861168C2211yescharacteristic
15C2×C4×Dic127C2211yescharacteristic
16#16861136—21no—
17#16861136—21no—
18#16861136—22no—
19C2×#41×C127C2211yescharacteristic
20#117×C127C2211yescharacteristic
21#117×C127C2211yescharacteristic
22#99×C127—22no—
23C2×Dic254—22no—
24C22×Dic127—21no—
25#14523232—21no—
26#14523232—21no—
27#14523232—21no—
28#14523232—21no—
29C2×C127⋊C8—21no—
30C2×C127⋊C8—21no—
31C4⋊Dic127—22no—
32C4⋊Dic127—21no—
33Dic127⋊C4—22no—
34Dic127⋊C4—22no—
35C22⋊Dic127—21no—
36C4×Dic127—21no—
37C4×Dic127D411yescharacteristic
38C4×Dic127D411yescharacteristic
39C4×Dic127D412yes—
40#41×C127—21no—
41#41×C127—21no—
42#41×C127—21no—
43#41×C127—21no—
44C2×Q8×C127Q811yescharacteristic, complemented
45C2×D4×C127—21no—
46C2×D4×C127Q811yescharacteristic, complemented
47C22×C4×C127—21no—
48C22×C4×C127C2311yescharacteristic
49C4⋊C4×C127—21no—
50C4⋊C4×C127—21no—
51C22⋊C4×C127—21no—
52C22⋊C4×C127D412yes—
53C42×C127—21no—
54C42×C127—21no—
55Dic254—44no—
56C2×Dic127—21no—
57C2×Dic127—21no—
58C2×Dic127—22no—
59C2×Dic127—42no—
60C127⋊C8—41no—
61C127⋊C8—41no—
62C23×C127—21no—
63C23×C127C2×D411yescharacteristic
64Q8×C127—21no—
65Q8×C127—21no—
66D4×C127—21no—
67D4×C127—21no—
68D4×C127—41no—
69D4×C127—41no—
70C2×C4×C127—22no—
71C2×C4×C127—21no—
72C2×C4×C127—41no—
73C2×C4×C127—41no—
74C2×C4×C127—41no—
75C2×C4×C127C2×D411yescharacteristic
76C2×C4×C127C2×D411yescharacteristic
77C2×C4×C127C2×Q811yescharacteristic
78C2×C4×C127#4111yescharacteristic
79C2×C4×C127#4111yescharacteristic
80C2×C4×C127#4111yescharacteristic
81C22×C127—41no—
82C22×C127—41no—
83C22×C127—81no—
84C22×C127#12011yescharacteristic
85C22×C127C22⋊Q811yescharacteristic
86C22×C127#12411yescharacteristic
87C4×C127—42no—
88C4×C127—41no—
89C4×C127—81no—
90C4×C127C22⋊Q811yescharacteristic
91C4×C127C22⋊Q811yescharacteristic
92C4×C127#12411yescharacteristic
93C4×C127#12411yescharacteristic
94Dic127—22no—
95Dic127—41no—
96Dic127—42no—
97C2×C127—21no—
98C2×C127—21no—
99C2×C127—21no—
100C2×C127—81no—
101C2×C127#39311yescharacteristic
102#2031—1271nomaximal, Sylow
103C127#203111yescharacteristic, complemented, Sylow
104#533—1271no—
105#421—1271no—
106C2×#104—1271no—
107#410D12711yescharacteristic
108#344—1271no—
109#339—1271no—
110H(3,ℤ/4)—1271no—
111C2×#41C2×D12711yescharacteristic
112C2×M16—1271no—
113C2×M16—1271no—
114C4⋊Q8—2541no—
115#125—2541no—
116C22⋊Q8—2542no—
117C4×Q8—2542no—
118#117—1271no—
119#117C2×D12711yescharacteristic
120#117C2×D12711yescharacteristic
121C2×C42—1271no—
122#104—2541no—
123#104—2541no—
124#104—2542no—
125#99—22no—
126#41—21no—
127#41—21no—
128#41—21no—
129#41—21no—
130C2×Q8—2542no—
131C2×Q8Dic25411yescharacteristic, complemented
132C2×D4—21no—
133C2×D4Dic25411yescharacteristic, complemented
134C22×C4—21no—
135C22×C4—2541no—
136C22×C4C22×D12711yescharacteristic
137M16—2541no—
138M16—2541no—
139M16—2541no—
140M16—2541no—
141C2×C8—2541no—
142C2×C8—2541no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—2542no—
146C4⋊C4—2542no—
147C4⋊C4—2542no—
148C4⋊C4—2541no—
149C22⋊C4—21no—
150C22⋊C4—2541no—
151C22⋊C4C22⋊D12712yes—
152C42—21no—
153C42—21no—
154C42—1271no—
155C42—1271no—
156C42—1272no—
157C42—2541no—
158C23—21no—
159C23C2×C22⋊D12711yescharacteristic
160Q8—21no—
161Q8—21no—
162Q8—5084no—
163D4—21no—
164D4—21no—
165D4—41no—
166D4—41no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—5082no—
173C2×C4—2541no—
174C2×C4—2542no—
175C2×C4—2541no—
176C2×C4D4×D12711yescharacteristic
177C2×C4D4×D12711yescharacteristic
178C2×C4C2×Dic25411yescharacteristic
179C2×C4#1452324711yescharacteristic
180C2×C4#1452324711yescharacteristic
181C2×C4#1452324811yescharacteristic
182C8—5081no—
183C8—5081no—
184C22—41no—
185C22—41no—
186C22—81no—
187C22#1686117711yescharacteristic
188C22#1686118811yescharacteristic
189C22#1686119611yescharacteristic
190C4—42no—
191C4—41no—
192C4—81no—
193C4—5081no—
194C4—5082no—
195C4—2542no—
196C4#1686118611yescharacteristic
197C4#1686118611yescharacteristic
198C4#1686119511yescharacteristic
199C4#1686119511yescharacteristic
200C2—21no—
201C2—21no—
202C2—21no—
203C2—81no—
204C2#2254978811yescharacteristic
subgroupG/H
centerC4#16861195
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#410×C127C2
socleC2×C127#393
prime radicalC2×#41×C127C22
hypercenter#410D127
hypocenterC127#2031
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #22550429C2semidirect
4. #22549646C2semidirect
5. #22549635C2semidirect
6. #22549632C2semidirect
44. C2×Q8×C127Q8semidirect
46. C2×D4×C127Q8semidirect
103. C127#2031semidirect
131. C2×Q8Dic254semidirect
133. C2×D4Dic254semidirect

Maximal subgroups

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

subgroupindex
#225504292
#225498932
C2×#168611362
#225496462
#225496352
#225496322
#410×C1272
#2031127