Finite Group Databasebeta

← #30637786

Subgroups — #30637786

Order 16256.

336 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×D508C211yesmaximal, characteristic, complemented
2#22550253C211yesmaximal, characteristic, complemented
3#22550072C211yesmaximal, characteristic, complemented
4#22550071C211yesmaximal, characteristic, complemented
5#22549619C211yesmaximal, characteristic, complemented
6#22549605C211yesmaximal, characteristic, complemented
7D4⋊C8×C127C211yesmaximal, characteristic, complemented
8C22×D508—21no—
9C2×D4×D127—21no—
10C2×D1016—21no—
11C2×D1016—22no—
12C2×#14523233—21no—
13#16861206C2211yescharacteristic
14#16861206C2211yescharacteristic, complemented
15#16861184—21no—
16#16861183—21no—
17#16861179—22no—
18C4×D508C2211yescharacteristic
19C4⋊(C127⋊C8)C2211yescharacteristic, complemented
20#16861131—21no—
21#16861129—21no—
22C4×D4×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic, complemented
24C4×C8×C127C2211yescharacteristic, complemented
25C23×D127—22no—
26C2×D508—22no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508—21no—
32C2×D508—42no—
33C2×D508—42no—
34C2×D508D411yescharacteristic
35C2×D508D412yes—
36C2×C22⋊D127—22no—
37D4×D127—42no—
38D4×D127—42no—
39C2×C4×D127—21no—
40D1016—44no—
41D1016—42no—
42#14523233—42no—
43C2×C127⋊C8—21no—
44C4⋊Dic127D411yescharacteristic
45(C2×D127)⋊C4—22no—
46(C2×D127)⋊C4—21no—
47C2×D4×C127D411yescharacteristic
48C22×C4×C127—21no—
49C2×C8×C127—21no—
50C2×C8×C127—21no—
51C4⋊C4×C127D411yescharacteristic
52C22⋊C4×C127—21no—
53C42×C127C2311yescharacteristic
54C22×D127—21no—
55C22×D127—22no—
56C22×D127—44no—
57C22×D127—44no—
58C22×D127—41no—
59C22×D127—82no—
60D508—21no—
61D508—42no—
62D508—42no—
63D508—44no—
64D508—41no—
65D508—41no—
66D508—41no—
67D508—41no—
68D508—41no—
69D508—41no—
70D508—41no—
71D508—41no—
72D508D812yescomplemented
73C22⋊D127—44no—
74C2×Dic127—21no—
75C4×D127—41no—
76C4×D127—41no—
77C127⋊C8—41no—
78C23×C127—21no—
79D4×C127—21no—
80D4×C127D812yescomplemented
81C2×C4×C127—21no—
82C2×C4×C127—41no—
83C2×C4×C127—41no—
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C8×C127—22no—
88C8×C127—41no—
89C22×C127—22no—
90C22×C127—41no—
91C22×C127—41no—
92C22×C127#12011yescharacteristic
93C2×D127—22no—
94C2×D127—41no—
95C2×D127—41no—
96C2×D127—42no—
97C2×D127—42no—
98C2×D127—42no—
99C2×D127—88no—
100C2×D127—81no—
101C2×D127—81no—
102C2×D127—81no—
103C4×C127—21no—
104C4×C127—41no—
105C4×C127C2×D811yescharacteristic
106C4×C127C2×D811yescharacteristic
107C4×C127AGL(1,ℤ/8)11yescharacteristic
108C4×C127AGL(1,ℤ/8)11yescharacteristic
109Dic127—41no—
110C2×C127—42no—
111C2×C127#44711yescharacteristic
112C2×C127#44711yescharacteristic
113C2×C127#45311yescharacteristic
114D127—42no—
115D127—82no—
116D127—161no—
117#1619—1271nomaximal, Sylow
118C127#161911yescharacteristic, complemented, Sylow
119D42—1271no—
120#493—1271no—
121#459—1271no—
122#459—1271no—
123#331—1271no—
124D4⋊C8—1271no—
125D4⋊C8D12711yescharacteristic, complemented
126C22×D4—2541no—
127C22×D4—2541no—
128C2×D8—2541no—
129C2×D8—2542no—
130C2×D8—2541no—
131#127—1271no—
132#127—1271no—
133#121—2541no—
134#121—2541no—
135#120—2542no—
136C4×D4—1271no—
137C4×D4C2×D12711yescharacteristic
138C4⋊C8—1271no—
139C4⋊C8C2×D12711yescharacteristic, complemented
140#102—2541no—
141#102—2541no—
142C4×C8C2×D12711yescharacteristic, complemented
143C24—2542no—
144C2×D4—1271no—
145C2×D4—1272no—
146C2×D4—5082no—
147C2×D4—5082no—
148C2×D4—5082no—
149C2×D4—5082no—
150C2×D4—2542no—
151C2×D4—2542no—
152C2×D4—2541no—
153C2×D4—2541no—
154C2×D4—2541no—
155C2×D4—2541no—
156C2×D4—2541no—
157C2×D4D50811yescharacteristic
158C22×C4—21no—
159C22×C4—2541no—
160D8—5084no—
161D8—5082no—
162D8—5082no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—2541no—
166C4⋊C4—1271no—
167C4⋊C4D50811yescharacteristic
168C22⋊C4—21no—
169C22⋊C4—2542no—
170C22⋊C4—2541no—
171C42C22×D12711yescharacteristic
172C23—21no—
173C23—10162no—
174C23—5084no—
175C23—5084no—
176C23—5081no—
177C23—2542no—
178C23—2541no—
179D4—21no—
180D4—1272no—
181D4—5082no—
182D4—5082no—
183D4—5084no—
184D4—5084no—
185D4—5081no—
186D4—5081no—
187D4—5081no—
188D4—5081no—
189D4—5081no—
190D4—5081no—
191D4—5081no—
192D4—5081no—
193D4—2541no—
194D4D101612yescomplemented
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—5081no—
199C2×C4—5081no—
200C2×C4—2541no—
201C2×C4D4×D12711yescharacteristic
202C2×C4D4×D12711yescharacteristic
203C2×C4C2×D50811yescharacteristic
204C8—22no—
205C8—41no—
206C8—5081no—
207C22—22no—
208C22—41no—
209C22—41no—
210C22—10168no—
211C22—10161no—
212C22—10161no—
213C22—10161no—
214C22—5082no—
215C22—5082no—
216C22—5082no—
217C22—5081no—
218C22—5081no—
219C22—2542no—
220C22#1686117911yescharacteristic
221C4—21no—
222C4—41no—
223C4—5081no—
224C4D8×D12711yescharacteristic
225C4C2×D101611yescharacteristic
226C4#1686123611yescharacteristic
227C4#1686123711yescharacteristic
228C2—42no—
229C2—20321no—
230C2—10162no—
231C2—5082no—
232C2#2255001111yescharacteristic
233C2#2255001211yescharacteristic
234C2#2255004511yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingD4⋊C8×C127C2
socleC22×C127#120
hypercenterD4⋊C8D127
hypocenterC127#1619
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
1. D4×D508C2semidirect
2. #22550253C2semidirect
3. #22550072C2semidirect
4. #22550071C2semidirect
5. #22549619C2semidirect
6. #22549605C2semidirect
7. D4⋊C8×C127C2semidirect
14. #16861206C22semidirect
19. C4⋊(C127⋊C8)C22semidirect
23. C4⋊C8×C127C22semidirect
24. C4×C8×C127C22semidirect
72. D508D8semidirect
80. D4×C127D8semidirect
118. C127#1619semidirect
125. D4⋊C8D127semidirect
139. C4⋊C8C2×D127semidirect
142. C4×C8C2×D127semidirect
194. D4D1016semidirect

Maximal subgroups

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

subgroupindex
D4×D5082
#225502532
#225500722
#225500712
#225496192
#225496052
D4⋊C8×C1272
#1619127