Finite Group Databasebeta

← #30651319

Subgroups — #30651319

Order 16256.

386 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 123 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#22550530C211yesmaximal, characteristic, complemented
2#22550272C211yesmaximal, characteristic, complemented
3#22550200C212yesmaximal, complemented
4#22550192C212yesmaximal, complemented
5#22550163C211yesmaximal, characteristic, complemented
6#22550156C211yesmaximal, characteristic, complemented
7#22550127C212yesmaximal, complemented
8C4×#14523233C211yesmaximal, characteristic, complemented
9#22549954C211yesmaximal, characteristic, complemented
10#22549919C212yesmaximal, complemented
11#555×C127C211yesmaximal, characteristic, complemented
12C2×#14523247—21no—
13C2×#14523233C2211yescharacteristic
14C2×#14523232C2212yes—
15#16861209C2211yescharacteristic
16#16861193—22no—
17#16861189—21no—
18#16861183C2212yes—
19C4×D508C2211yescharacteristic
20C2×C4⋊Dic127C2212yes—
21#16861142C2212yes—
22#16861142C2211yescharacteristic
23#16861140C2212yes—
24#16861140C2212yes—
25#16861140C2212yes—
26C4⋊(C127⋊C8)C2211yescharacteristic
27#16861130C2212yes—
28#16861130C2211yescharacteristic
29#16861129C2212yes—
30#16861129C2211yescharacteristic
31C22⋊(C127⋊C8)C2212yes—
32C4×C127⋊C8C2211yescharacteristic
33C4⋊Q8×C127C2211yescharacteristic
34#124×C127—22no—
35#121×C127—21no—
36#121×C127C2212yes—
37C4×D4×C127C2211yescharacteristic
38C4×D4×C127C2211yescharacteristic
39#117×C127—21no—
40#117×C127C2212yes—
41#14523247—44no—
42C2×Dic254—21no—
43C2×D508C2311yescharacteristic
44C2×C22⋊D127—22no—
45C22×Dic127—22no—
46C2×C4×D127—21no—
47#14523233—22no—
48#14523232—24no—
49C2×C127⋊C8C2312yes—
50C2×C127⋊C8C2311yescharacteristic
51C2×C127⋊C8C2311yescharacteristic
52C4⋊Dic127—22no—
53C4⋊Dic127—21no—
54C4⋊Dic127C2312yes—
55C4⋊Dic127C2311yescharacteristic
56(C2×D127)⋊C4—22no—
57(C2×D127)⋊C4—21no—
58C2×Q8×C127—21no—
59C2×D4×C127—22no—
60C2×D4×C127D411yescharacteristic
61C2×D4×C127C2311yescharacteristic
62C22×C4×C127—21no—
63C22×C4×C127C2312yes—
64C4⋊C4×C127—21no—
65C4⋊C4×C127D411yescharacteristic
66C4⋊C4×C127C2311yescharacteristic
67C4⋊C4×C127C2312yes—
68C22⋊C4×C127—22no—
69C22⋊C4×C127—22no—
70C22⋊C4×C127—21no—
71C22⋊C4×C127D412yes—
72C42×C127—22no—
73C42×C127C2311yescharacteristic
74C22×D127—21no—
75Dic254—42no—
76D508—21no—
77D508—21no—
78C22⋊D127—44no—
79C2×Dic127—21no—
80C2×Dic127—22no—
81C2×Dic127—44no—
82C4×D127—42no—
83C127⋊C8—22no—
84C127⋊C8—21no—
85C127⋊C8#4112yescomplemented
86C23×C127—21no—
87C23×C127C2×D412yes—
88Q8×C127—81no—
89D4×C127—22no—
90D4×C127—21no—
91D4×C127—21no—
92D4×C127—82no—
93C2×C4×C127—22no—
94C2×C4×C127—22no—
95C2×C4×C127—21no—
96C2×C4×C127—22no—
97C2×C4×C127—42no—
98C2×C4×C127C2×D412yes—
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2411yescharacteristic
102C22×C127—22no—
103C22×C127—22no—
104C22×C127—22no—
105C22×C127—41no—
106C22×C127—41no—
107C22×C127—41no—
108C22×C127C22×D411yescharacteristic
109C2×D127—41no—
110C2×D127—41no—
111C2×D127—41no—
112C4×C127—22no—
113C4×C127—21no—
114C4×C127—41no—
115C4×C127—42no—
116C4×C127AGL(1,ℤ/8)12yes—
117C4×C127C2×#4111yescharacteristic
118C4×C1272−1+411yescharacteristic
119Dic127—41no—
120Dic127—42no—
121C2×C127—42no—
122C2×C127—81no—
123C2×C127#54711yescharacteristic
124C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
125C2×C127#57711yescharacteristic
126D127—81no—
127#3354—1271nomaximal, Sylow
128C127#335411yescharacteristic, complemented, Sylow
129#555D12711yescharacteristic, complemented
130#547—1271no—
131#498—1271no—
132#482—1272no—
133#481—1272no—
134#476—1271no—
135#474—1271no—
136#469—1272no—
137C4×D8—1271no—
138#436—1271no—
139#428—1272no—
140C2×#41—2541no—
141C2×D8—1271no—
142C2×M16—1272no—
143C4⋊Q8—1271no—
144C4⋊Q8C2×D12711yescharacteristic
145#124—22no—
146#123—2542no—
147C22⋊Q8—2541no—
148#121—21no—
149#121—1272no—
150#121C2×D12712yes—
151C4×D4—1271no—
152C4×D4C2×D12711yescharacteristic
153C4×D4C2×D12711yescharacteristic
154#117—21no—
155#117C2×D12712yes—
156C2×C4⋊C4—1272no—
157#107—1271no—
158#107—1272no—
159#106—1272no—
160#106—1272no—
161#106—1272no—
162C4⋊C8—1271no—
163#102—1271no—
164#102—1272no—
165#102—1271no—
166#102—1272no—
167C22⋊C8—1272no—
168C4×C8—1271no—
169#41—5084no—
170C2×Q8—21no—
171C2×Q8—2541no—
172C2×D4—22no—
173C2×D4—1271no—
174C2×D4—2542no—
175C2×D4C22⋊D12711yescharacteristic
176C2×D4C22×D12711yescharacteristic
177C22×C4—21no—
178C22×C4—2542no—
179C22×C4—2541no—
180C22×C4C22×D12712yes—
181D8—2542no—
182M16—2544no—
183C2×C8—1271no—
184C2×C8—1271no—
185C2×C8—1272no—
186C4⋊C4—21no—
187C4⋊C4—1271no—
188C4⋊C4—1272no—
189C4⋊C4—2542no—
190C4⋊C4—2541no—
191C4⋊C4C22⋊D12711yescharacteristic
192C4⋊C4C22×D12712yes—
193C4⋊C4C22×D12711yescharacteristic
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—21no—
197C22⋊C4—2541no—
198C22⋊C4—2542no—
199C22⋊C4C22⋊D12712yes—
200C42—22no—
201C42C22×D12711yescharacteristic
202C23—21no—
203C23—2541no—
204C23C2×C22⋊D12712yes—
205Q8—81no—
206Q8—5082no—
207D4—22no—
208D4—21no—
209D4—21no—
210D4—82no—
211D4—5084no—
212D4—2541no—
213D4—2541no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—42no—
219C2×C4—5084no—
220C2×C4—5082no—
221C2×C4—2541no—
222C2×C4—2542no—
223C2×C4C2×C22⋊D12712yes—
224C2×C4C2×C22⋊D12711yescharacteristic
225C2×C4C2×C22⋊D12711yescharacteristic
226C2×C4C23×D12711yescharacteristic
227C8—1272no—
228C8—2542no—
229C8—2541no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—5081no—
237C22—5081no—
238C22—5081no—
239C22C22×C22⋊D12711yescharacteristic
240C4—21no—
241C4—22no—
242C4—41no—
243C4—42no—
244C4—5081no—
245C4—5082no—
246C4#1686123412yes—
247C4#41×D12711yescharacteristic
248C4#1686125611yescharacteristic
249C2—42no—
250C2—81no—
251C2—10161no—
252C2#2255052511yescharacteristic
253C2C2×#1686123411yescharacteristic
254C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#555×C127C2
socleC22×C127C22×D4
hypercenter#555D127
hypocenterC127#3354
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #22550530C2semidirect
2. #22550272C2semidirect
3. #22550200C2semidirect
4. #22550192C2semidirect
5. #22550163C2semidirect
6. #22550156C2semidirect
7. #22550127C2semidirect
8. C4×#14523233C2semidirect
9. #22549954C2semidirect
10. #22549919C2semidirect
11. #555×C127C2semidirect
85. C127⋊C8#41semidirect
128. C127#3354semidirect
129. #555D127semidirect

Maximal subgroups

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

subgroupindex
#225505302
#225502722
#225502002
#225501922
#225501632
#225501562
#225501272
C4×#145232332
#225499542
#225499192
#555×C1272
#3354127