Finite Group Databasebeta

← #30650647

Subgroups — #30650647

Order 16256.

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

normal subgroups only

This group has 386 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#22550226C211yesmaximal, characteristic, complemented
3#22550214C212yesmaximal, complemented
4#22550156C211yesmaximal, characteristic, complemented
5#22550115C212yesmaximal, complemented
6#22550083C211yesmaximal, characteristic, complemented
7#22550038C212yesmaximal, complemented
8C4×#14523236C211yesmaximal, characteristic, complemented
9D4×C127⋊C8C211yesmaximal, characteristic, complemented
10#22549872C212yesmaximal, complemented
11#547×C127C211yesmaximal, characteristic
12C2×#14523238C2212yes—
13C2×#14523236C2211yescharacteristic
14C22×C127⋊C8C2212yes—
15#16861209C2211yescharacteristic
16#16861202—22no—
17#16861196—21no—
18#16861188C2212yes—
19C4×Dic254C2211yescharacteristic
20#16861174—21no—
21#16861168C2212yes—
22#16861142C2212yes—
23#16861140C2211yescharacteristic
24C4⋊(C127⋊C8)C2211yescharacteristic
25#16861133C2212yes—
26#16861133C2212yes—
27#16861133C2212yes—
28#16861132C2211yescharacteristic
29#16861130C2212yes—
30#16861130C2211yescharacteristic
31C22⋊(C127⋊C8)C2212yes—
32C4×C127⋊C8C2211yescharacteristic
33C2×#41×C127C2212yes—
34C4⋊Q8×C127C2211yescharacteristic
35#123×C127—22no—
36C22⋊Q8×C127C2212yes—
37#121×C127—21no—
38C4×D4×C127C2211yescharacteristic
39C4×D4×C127C2211yescharacteristic
40C2×C4⋊C4×C127—21no—
41C2×Dic254—21no—
42C2×Dic254C2311yescharacteristic
43#14523238—42no—
44#14523236—22no—
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×C127C2312yes—
61C2×D4×C127—22no—
62C2×D4×C127D411yescharacteristic
63C2×D4×C127C2311yescharacteristic
64C22×C4×C127—21no—
65C22×C4×C127—22no—
66C22×C4×C127C2312yes—
67C4⋊C4×C127—21no—
68C4⋊C4×C127—22no—
69C4⋊C4×C127—21no—
70C4⋊C4×C127D411yescharacteristic
71C4⋊C4×C127C2311yescharacteristic
72C22⋊C4×C127—22no—
73C22⋊C4×C127—21no—
74C22⋊C4×C127D412yes—
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—21no—
86C23×C127C2×D412yes—
87Q8×C127—22no—
88Q8×C127—22no—
89D4×C127—22no—
90D4×C127—21no—
91D4×C127—44no—
92D4×C127#4112yes—
93C2×C4×C127—22no—
94C2×C4×C127—22no—
95C2×C4×C127—21no—
96C2×C4×C127—22no—
97C2×C4×C127—44no—
98C2×C4×C127—42no—
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D412yes—
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2411yescharacteristic
103C22×C127—22no—
104C22×C127—22no—
105C22×C127—22no—
106C22×C127—22no—
107C22×C127—41no—
108C22×C127—41no—
109C22×C127C22×D411yescharacteristic
110C4×C127—22no—
111C4×C127—21no—
112C4×C127—42no—
113C4×C127—41no—
114C4×C127#13512yes—
115C4×C127C2×#4111yescharacteristic
116C4×C127H(5,𝔽2)11yescharacteristic
117Dic127—41no—
118Dic127—42no—
119Dic127—41no—
120C2×C127—42no—
121C2×C127—42no—
122C2×C127#54611yescharacteristic
123C2×C127C2×#13511yescharacteristic
124C2×C127#57811yescharacteristic
125#3302—1271nomaximal, Sylow
126C127#330211yescharacteristic, complemented, Sylow
127#556—1271no—
128#547D12711yescharacteristic
129#487—1271no—
130#485—1272no—
131#474—1271no—
132#467—1272no—
133#462—1271no—
134#452—1272no—
135C4×SD16—1271no—
136C8×D4—1271no—
137#416—1272no—
138C2×#41C2×D12712yes—
139C2×Q16—1272no—
140C2×SD16—1271no—
141C22×C8—1272no—
142C4⋊Q8—1271no—
143C4⋊Q8C2×D12711yescharacteristic
144#126—2542no—
145#124—2541no—
146#123—22no—
147C22⋊Q8—1272no—
148C22⋊Q8C2×D12712yes—
149#121—21no—
150C4×Q8—1271no—
151C4×Q8—2541no—
152C4×D4C2×D12711yescharacteristic
153C4×D4C2×D12711yescharacteristic
154#117—1272no—
155C2×C4⋊C4—21no—
156#107—1272no—
157#106—1271no—
158C4⋊C8—1271no—
159#103—1272no—
160#103—1272no—
161#103—1272no—
162#103—1271no—
163#102—1271no—
164#102—1272no—
165C22⋊C8—1272no—
166C4×C8—1271no—
167#41—24no—
168#41—42no—
169C2×Q8—1271no—
170C2×Q8—2541no—
171C2×Q8C22×D12712yes—
172C2×D4—22no—
173C2×D4C22⋊D12711yescharacteristic
174C2×D4C22×D12711yescharacteristic
175C22×C4—21no—
176C22×C4—22no—
177C22×C4C22×D12712yes—
178Q16—5082no—
179SD16—2542no—
180C2×C8—1271no—
181C2×C8—1271no—
182C2×C8—1272no—
183C2×C8—5082no—
184C4⋊C4—21no—
185C4⋊C4—22no—
186C4⋊C4—21no—
187C4⋊C4—1271no—
188C4⋊C4—1272no—
189C4⋊C4—2541no—
190C4⋊C4—2542no—
191C4⋊C4—2541no—
192C4⋊C4C22⋊D12711yescharacteristic
193C4⋊C4C22×D12711yescharacteristic
194C22⋊C4—22no—
195C22⋊C4—21no—
196C22⋊C4—2542no—
197C22⋊C4—2542no—
198C22⋊C4C22⋊D12712yes—
199C42—2542no—
200C42—2541no—
201C42C22×D12711yescharacteristic
202C23—21no—
203C23C2×C22⋊D12712yes—
204Q8—22no—
205Q8—22no—
206Q8—5082no—
207Q8—2541no—
208Q8—2541no—
209D4—22no—
210D4—21no—
211D4—44no—
212D4#1452324712yes—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—2541no—
220C2×C4—2542no—
221C2×C4—2541no—
222C2×C4C2×C22⋊D12712yes—
223C2×C4C2×C22⋊D12711yescharacteristic
224C2×C4C2×C22⋊D12711yescharacteristic
225C2×C4C23×D12711yescharacteristic
226C8—2541no—
227C8—2541no—
228C8—2542no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—41no—
234C22—41no—
235C22C22×C22⋊D12711yescharacteristic
236C4—21no—
237C4—22no—
238C4—41no—
239C4—42no—
240C4—5082no—
241C4—5081no—
242C4—5081no—
243C4#1686122912yes—
244C4C2×#1452324711yescharacteristic
245C4#1686125411yescharacteristic
246C2—42no—
247C2—42no—
248C2#2255052011yescharacteristic
249C2C2×#1686122911yescharacteristic
250C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#547×C127C2
socleC22×C127C22×D4
hypercenter#547D127
hypocenterC127#3302
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #22550596C2semidirect
2. #22550226C2semidirect
3. #22550214C2semidirect
4. #22550156C2semidirect
5. #22550115C2semidirect
6. #22550083C2semidirect
7. #22550038C2semidirect
8. C4×#14523236C2semidirect
9. D4×C127⋊C8C2semidirect
10. #22549872C2semidirect
126. C127#3302semidirect

Maximal subgroups

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

subgroupindex
#225505962
#225502262
#225502142
#225501562
#225501152
#225500832
#225500382
C4×#145232362
D4×C127⋊C82
#225498722
#547×C1272
#3302127