Finite Group Databasebeta

← #30650669

Subgroups — #30650669

Order 16256.

396 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550596C211yesmaximal, characteristic, complemented
2#22550222C211yesmaximal, characteristic, complemented
3#22550197C212yesmaximal, complemented
4#22550161C211yesmaximal, characteristic, complemented
5#22550103C212yesmaximal, complemented
6#22550073C211yesmaximal, characteristic, complemented
7#22550038C212yesmaximal, complemented
8C4×#14523238C211yesmaximal, characteristic, complemented
9D4×C127⋊C8C211yesmaximal, characteristic, complemented
10#22549872C212yesmaximal, complemented
11#550×C127C211yesmaximal, characteristic
12C2×#14523238C2211yescharacteristic
13C2×#14523236C2212yes—
14C22×C127⋊C8C2212yes—
15#16861209C2211yescharacteristic
16#16861202—22no—
17#16861196—21no—
18#16861188C2212yes—
19C4×Dic254C2211yescharacteristic
20#16861174—21no—
21#16861168C2212yes—
22#16861142C2211yescharacteristic
23#16861140C2212yes—
24C4⋊(C127⋊C8)C2211yescharacteristic
25#16861133C2212yes—
26#16861133C2211yescharacteristic
27#16861132C2211yescharacteristic
28#16861130C2212yes—
29#16861130C2212yes—
30#16861130C2212yes—
31C22⋊(C127⋊C8)C2212yes—
32C4×C127⋊C8C2211yescharacteristic
33C2×#41×C127C2212yes—
34#127×C127—21no—
35#127×C127C2211yescharacteristic
36#121×C127—22no—
37#121×C127C2212yes—
38C4×Q8×C127C2211yescharacteristic
39C4×D4×C127—21no—
40C4×D4×C127C2211yescharacteristic
41C2×Dic254—21no—
42C2×Dic254C2311yescharacteristic
43#14523238—22no—
44#14523236—42no—
45C2×C127⋊C8—42no—
46C2×C127⋊C8C2311yescharacteristic
47C2×C127⋊C8C2312yes—
48C2×C127⋊C8C2311yescharacteristic
49C4⋊Dic127—21no—
50C4⋊Dic127C2312yes—
51C4⋊Dic127C2311yescharacteristic
52Dic127⋊C4—22no—
53Dic127⋊C4—21no—
54C22⋊Dic127—22no—
55C22⋊Dic127—22no—
56C4×Dic127—21no—
57C4×Dic127—22no—
58#41×C127—24no—
59#41×C127—42no—
60C2×Q8×C127C2311yescharacteristic
61C2×D4×C127—22no—
62C2×D4×C127—22no—
63C2×D4×C127—21no—
64C2×D4×C127—21no—
65C2×D4×C127D411yescharacteristic
66C2×D4×C127C2312yes—
67C22×C4×C127—22no—
68C22×C4×C127C2312yes—
69C4⋊C4×C127—21no—
70C4⋊C4×C127D411yescharacteristic
71C4⋊C4×C127C2311yescharacteristic
72C22⋊C4×C127—22no—
73C22⋊C4×C127D412yes—
74C42×C127—21no—
75C42×C127C2311yescharacteristic
76Dic254—21no—
77Dic254—21no—
78Dic254—42no—
79C2×Dic127—21no—
80C2×Dic127—21no—
81C2×Dic127—22no—
82C127⋊C8—21no—
83C127⋊C8—21no—
84C127⋊C8—22no—
85C23×C127—22no—
86C23×C127C2×D412yes—
87Q8×C127—21no—
88Q8×C127#4112yes—
89D4×C127—22no—
90D4×C127—22no—
91D4×C127—22no—
92D4×C127—44no—
93D4×C127—42no—
94D4×C127—81no—
95D4×C127—82no—
96C2×C4×C127—21no—
97C2×C4×C127—22no—
98C2×C4×C127—22no—
99C2×C4×C127—21no—
100C2×C4×C127—44no—
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2×D412yes—
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2411yescharacteristic
105C22×C127—22no—
106C22×C127—22no—
107C22×C127—22no—
108C22×C127—42no—
109C22×C127—42no—
110C22×C127—42no—
111C22×C127C22×D411yescharacteristic
112C4×C127—22no—
113C4×C127—22no—
114C4×C127—21no—
115C4×C127—41no—
116C4×C127#13512yes—
117C4×C127C2×#4111yescharacteristic
118C4×C127H(5,𝔽2)11yescharacteristic
119Dic127—41no—
120Dic127—42no—
121Dic127—41no—
122C2×C127—42no—
123C2×C127—82no—
124C2×C127#54611yescharacteristic
125C2×C127C2×#13511yescharacteristic
126C2×C127#57711yescharacteristic
127#3304—1271nomaximal, Sylow
128C127#330411yescharacteristic, complemented, Sylow
129#556—1271no—
130#550D12711yescharacteristic
131#486—1271no—
132#482—1272no—
133#475—1271no—
134#465—1272no—
135#460—1271no—
136#452—1272no—
137C4×Q16—1271no—
138C8×D4—1271no—
139#416—1272no—
140C2×#41C2×D12712yes—
141C2×Q16—1271no—
142C2×SD16—1272no—
143C22×C8—1272no—
144C4⋊Q8—1271no—
145#127—21no—
146#127C2×D12711yescharacteristic
147#126—2542no—
148#124—2541no—
149C22⋊Q8—1272no—
150#121—22no—
151#121C2×D12712yes—
152C4×Q8—1271no—
153C4×Q8—2541no—
154C4×Q8C2×D12711yescharacteristic
155C4×D4—21no—
156C4×D4C2×D12711yescharacteristic
157#117—1272no—
158#107—1271no—
159#106—1272no—
160C4⋊C8—1271no—
161#103—1271no—
162#103—1272no—
163#103—1271no—
164#102—1272no—
165#102—1272no—
166#102—1272no—
167C22⋊C8—1272no—
168C4×C8—1271no—
169#41—24no—
170#41—42no—
171C2×Q8—1271no—
172C2×Q8—2541no—
173C2×Q8C22×D12711yescharacteristic
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4C22⋊D12711yescharacteristic
179C2×D4C22×D12712yes—
180C22×C4—22no—
181C22×C4C22×D12712yes—
182Q16—2542no—
183SD16—5082no—
184C2×C8—1271no—
185C2×C8—1271no—
186C2×C8—1272no—
187C2×C8—5082no—
188C4⋊C4—21no—
189C4⋊C4—1272no—
190C4⋊C4—1271no—
191C4⋊C4—2541no—
192C4⋊C4—2542no—
193C4⋊C4—2541no—
194C4⋊C4C22⋊D12711yescharacteristic
195C4⋊C4C22×D12711yescharacteristic
196C22⋊C4—22no—
197C22⋊C4—2542no—
198C22⋊C4—2542no—
199C22⋊C4C22⋊D12712yes—
200C42—21no—
201C42—2542no—
202C42—2541no—
203C42C22×D12711yescharacteristic
204C23—22no—
205C23C2×C22⋊D12712yes—
206Q8—21no—
207Q8—5082no—
208Q8—2541no—
209Q8—2541no—
210Q8#1452324712yes—
211D4—22no—
212D4—22no—
213D4—22no—
214D4—44no—
215D4—42no—
216D4—81no—
217D4—82no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—44no—
223C2×C4—2541no—
224C2×C4—2542no—
225C2×C4—2541no—
226C2×C4C2×C22⋊D12712yes—
227C2×C4C2×C22⋊D12711yescharacteristic
228C2×C4C2×C22⋊D12711yescharacteristic
229C2×C4C23×D12711yescharacteristic
230C8—2541no—
231C8—2541no—
232C8—2542no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22C22×C22⋊D12711yescharacteristic
240C4—21no—
241C4—22no—
242C4—22no—
243C4—41no—
244C4—5082no—
245C4—5081no—
246C4—5081no—
247C4#1686122912yes—
248C4C2×#1452324711yescharacteristic
249C4#1686125411yescharacteristic
250C2—42no—
251C2—82no—
252C2#2255052011yescharacteristic
253C2C2×#1686122911yescharacteristic
254C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#550×C127C2
socleC22×C127C22×D4
hypercenter#550D127
hypocenterC127#3304
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #22550596C2semidirect
2. #22550222C2semidirect
3. #22550197C2semidirect
4. #22550161C2semidirect
5. #22550103C2semidirect
6. #22550073C2semidirect
7. #22550038C2semidirect
8. C4×#14523238C2semidirect
9. D4×C127⋊C8C2semidirect
10. #22549872C2semidirect
128. C127#3304semidirect

Maximal subgroups

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

subgroupindex
#225505962
#225502222
#225501972
#225501612
#225501032
#225500732
#225500382
C4×#145232382
D4×C127⋊C82
#225498722
#550×C1272
#3304127