Finite Group Databasebeta

← #30637783

Subgroups — #30637783

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#22550252C211yesmaximal, characteristic, complemented
3#22550068C211yesmaximal, characteristic, complemented
4#22549619C211yesmaximal, characteristic, complemented
5D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic, complemented
7#459×C127C211yesmaximal, characteristic, complemented
8C22×D508—21no—
9C2×D4×D127—21no—
10C2×#14523233—21no—
11C2×#14523233—22no—
12#16861206C2211yescharacteristic
13#16861184—21no—
14#16861183—21no—
15#16861179—22no—
16C4×D508C2211yescharacteristic
17C4⋊(C127⋊C8)C2211yescharacteristic, complemented
18#16861130—21no—
19C4×C127⋊C8C2211yescharacteristic, complemented
20C2×D8×C127—21no—
21#127×C127C2211yescharacteristic, complemented
22C4×D4×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic, complemented
24#102×C127—21no—
25C23×D127—22no—
26C2×D508—22no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—42no—
30C2×D508—42no—
31C2×D508D411yescharacteristic
32C2×D508D412yes—
33C2×C22⋊D127—22no—
34D4×D127—42no—
35D4×D127—42no—
36C2×C4×D127—21no—
37#14523233—44no—
38#14523233—42no—
39C2×C127⋊C8—21no—
40C2×C127⋊C8—21no—
41C4⋊Dic127D411yescharacteristic
42(C2×D127)⋊C4—22no—
43(C2×D127)⋊C4—21no—
44C2×D4×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127D411yescharacteristic
48C22×C4×C127—21no—
49D8×C127—42no—
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—82no—
59D508—21no—
60D508—42no—
61D508—44no—
62D508—41no—
63D508—41no—
64D508—41no—
65D508—41no—
66D508D812yescomplemented
67C22⋊D127—44no—
68C2×Dic127—21no—
69C4×D127—41no—
70C4×D127—41no—
71C127⋊C8—22no—
72C127⋊C8—41no—
73C23×C127—21no—
74C23×C127—41no—
75D4×C127—21no—
76D4×C127—42no—
77D4×C127—41no—
78D4×C127—41no—
79D4×C127—41no—
80D4×C127—41no—
81D4×C127D812yescomplemented
82C2×C4×C127—21no—
83C2×C4×C127—41no—
84C2×C4×C127—41no—
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C8×C127—41no—
89C22×C127—22no—
90C22×C127—41no—
91C22×C127—41no—
92C22×C127—81no—
93C22×C127—81no—
94C22×C127—81no—
95C22×C127#12011yescharacteristic
96C2×D127—22no—
97C2×D127—41no—
98C2×D127—41no—
99C2×D127—42no—
100C2×D127—42no—
101C2×D127—42no—
102C2×D127—88no—
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—161no—
112C2×C127#44711yescharacteristic
113C2×C127#44711yescharacteristic
114C2×C127#45311yescharacteristic
115D127—42no—
116D127—82no—
117#1619—1271nomaximal, Sylow
118C127#161911yescharacteristic, complemented, Sylow
119D42—1271no—
120#493—1271no—
121#459—1271no—
122#459D12711yescharacteristic, complemented
123#331—1271no—
124D4⋊C8—1271no—
125D4⋊C8—1271no—
126C22×D4—2541no—
127C22×D4—2541no—
128C2×D8—21no—
129C2×D8—2541no—
130C2×D8—2542no—
131#127—1271no—
132#127C2×D12711yescharacteristic, complemented
133#121—2541no—
134#121—2541no—
135#120—2542no—
136C4×D4—1271no—
137C4×D4C2×D12711yescharacteristic
138C4⋊C8—1271no—
139C4⋊C8C2×D12711yescharacteristic, complemented
140#102—21no—
141#102—2541no—
142C4×C8—1271no—
143C24—2542no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—1271no—
148C2×D4—1272no—
149C2×D4—5082no—
150C2×D4—5082no—
151C2×D4—5082no—
152C2×D4—5082no—
153C2×D4—2542no—
154C2×D4—2542no—
155C2×D4—2541no—
156C2×D4—2541no—
157C2×D4C22⋊D12711yescharacteristic
158C22×C4—21no—
159C22×C4—2541no—
160D8—42no—
161D8—5082no—
162D8—5084no—
163C2×C8—21no—
164C2×C8—2541no—
165C2×C8—2541no—
166C4⋊C4—1271no—
167C4⋊C4C22⋊D12711yescharacteristic
168C22⋊C4—21no—
169C22⋊C4—2542no—
170C22⋊C4—2541no—
171C42C22×D12711yescharacteristic
172C23—21no—
173C23—41no—
174C23—10162no—
175C23—5084no—
176C23—5084no—
177C23—2542no—
178C23—2541no—
179D4—21no—
180D4—42no—
181D4—41no—
182D4—41no—
183D4—41no—
184D4—41no—
185D4—1272no—
186D4—5082no—
187D4—5084no—
188D4—5084no—
189D4—5081no—
190D4—5081no—
191D4—5081no—
192D4—5081no—
193D4—2541no—
194D4#1452323312yescomplemented
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×C22⋊D12711yescharacteristic
204C8—41no—
205C8—5081no—
206C8—2542no—
207C22—22no—
208C22—41no—
209C22—41no—
210C22—81no—
211C22—81no—
212C22—81no—
213C22—10168no—
214C22—5082no—
215C22—5082no—
216C22—5082no—
217C22—5081no—
218C22—5081no—
219C22—2542no—
220C22#1686117711yescharacteristic
221C4—21no—
222C4—41no—
223C4—5081no—
224C4D8×D12711yescharacteristic
225C4C2×#1452323311yescharacteristic
226C4#1686123311yescharacteristic
227C4#1686123411yescharacteristic
228C2—42no—
229C2—161no—
230C2—10162no—
231C2—5082no—
232C2#2255000811yescharacteristic
233C2#2255000911yescharacteristic
234C2#2255004211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#459×C127C2
socleC22×C127#120
hypercenter#459D127
hypocenterC127#1619
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. D4×D508C2semidirect
2. #22550252C2semidirect
3. #22550068C2semidirect
4. #22549619C2semidirect
5. D4⋊(C127⋊C8)C2semidirect
6. #22549603C2semidirect
7. #459×C127C2semidirect
17. C4⋊(C127⋊C8)C22semidirect
19. C4×C127⋊C8C22semidirect
21. #127×C127C22semidirect
23. C4⋊C8×C127C22semidirect
66. D508D8semidirect
81. D4×C127D8semidirect
118. C127#1619semidirect
122. #459D127semidirect
132. #127C2×D127semidirect
139. C4⋊C8C2×D127semidirect
194. D4#14523233semidirect

Maximal subgroups

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

subgroupindex
D4×D5082
#225502522
#225500682
#225496192
D4⋊(C127⋊C8)2
#225496032
#459×C1272
#1619127