Finite Group Databasebeta

← #30637270

Subgroups — #30637270

Order 16256.

286 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 63 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#22550386C211yesmaximal, characteristic
2#22549838C211yesmaximal, characteristic, complemented
3#22549735C211yesmaximal, characteristic, complemented
4#22549614C211yesmaximal, characteristic, complemented
5#22549612C211yesmaximal, characteristic, complemented
6#22549610C211yesmaximal, characteristic, complemented
7#327×C127C211yesmaximal, characteristic, complemented
8C2×#14523247C411yescharacteristic, complemented
9C22×C4×D127C2211yescharacteristic
10C2×#14523231—21no—
11#16861191—21no—
12#16861189C411yescharacteristic, complemented
13#16861188C2211yescharacteristic
14#16861180C411yescharacteristic, complemented
15#16861177—21no—
16D4×Dic127—21no—
17(C22⋊D127)⋊C4—21no—
18#16861167C411yescharacteristic, complemented
19C2×Dic127⋊C4—21no—
20C2×(C2×D127)⋊C4—21no—
21C22⋊(C127⋊C8)C2211yescharacteristic
22(C2×D127)⋊C8—21no—
23C127⋊#95C2211yescharacteristic
24#121×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C23×D127D411yescharacteristic
28#14523247—42no—
29C2×Dic254C2×C411yescharacteristic
30C2×C22⋊D127—21no—
31C2×C22⋊D127—21no—
32C22×Dic127—21no—
33C22×Dic127—21no—
34C22×Dic127D411yescharacteristic
35C2×C4×D127—42no—
36C2×C4×D127C2×C411yescharacteristic
37C2×C4×D127C2×C411yescharacteristic
38C2×C4×D127D411yescharacteristic
39C2×C4×D127D411yescharacteristic
40#14523231—42no—
41C2×C127⋊C8—21no—
42C4⋊Dic127—21no—
43C4⋊Dic127C2×C411yescharacteristic
44Dic127⋊C4—21no—
45Dic127⋊C4—42no—
46C22⋊Dic127—21no—
47C22⋊Dic127—21no—
48(C2×D127)⋊C4—21no—
49(C2×D127)⋊C4—21no—
50(C2×D127)⋊C4—42no—
51C4×Dic127—21no—
52C2×D4×C127—21no—
53C2×D4×C127C2×C411yescharacteristic
54C22×C4×C127—21no—
55C22×C4×C127C2311yescharacteristic
56C2×C8×C127—21no—
57C4⋊C4×C127C2×C411yescharacteristic
58C22⋊C4×C127—21no—
59C22×D127—21no—
60C22×D127—21no—
61C22×D127—41no—
62C22×D127—41no—
63C22×D127C22⋊C411yescharacteristic
64C22×D127C22⋊C411yescharacteristic
65Dic254—41no—
66C22⋊D127—42no—
67C22⋊D127—42no—
68C2×Dic127—22no—
69C2×Dic127—21no—
70C2×Dic127—21no—
71C2×Dic127—42no—
72C2×Dic127—41no—
73C2×Dic127—41no—
74C2×Dic127—41no—
75C2×Dic127C22⋊C411yescharacteristic
76C2×Dic127C22⋊C411yescharacteristic
77C4×D127—22no—
78C4×D127—22no—
79C4×D127—41no—
80C4×D127—41no—
81C127⋊C8—41no—
82C23×C127—21no—
83C23×C127C2×D411yescharacteristic
84D4×C127—41no—
85D4×C127—41no—
86D4×C127—41no—
87C2×C4×C127—21no—
88C2×C4×C127—41no—
89C2×C4×C127—41no—
90C2×C4×C127—41no—
91C2×C4×C127—41no—
92C2×C4×C127C22×C411yescharacteristic
93C2×C4×C127C2×D411yescharacteristic
94C8×C127—41no—
95C22×C127—21no—
96C22×C127—21no—
97C22×C127—41no—
98C22×C127—41no—
99C22×C127—41no—
100C22×C127C2×C22⋊C411yescharacteristic
101C22×C127AGL(1,ℤ/8)11yescharacteristic
102C22×C127#13711yescharacteristic
103C2×D127—21no—
104C2×D127—21no—
105C2×D127—21no—
106C2×D127—21no—
107C2×D127—21no—
108C2×D127—21no—
109C2×D127—42no—
110C2×D127—42no—
111C4×C127—21no—
112C4×C127—21no—
113C4×C127—41no—
114C4×C127—81no—
115Dic127—21no—
116Dic127—41no—
117Dic127—41no—
118Dic127—81no—
119Dic127#9912yes—
120C2×C127—21no—
121C2×C127—21no—
122C2×C127—81no—
123C2×C127C2×#9911yescharacteristic
124C2×C127#41711yescharacteristic
125C2×C127#42111yescharacteristic
126D127—41no—
127D127—41no—
128#1513—1271nomaximal, Sylow
129C127#151311yescharacteristic, complemented, Sylow
130#525—1271no—
131#407—1271no—
132#380—1271no—
133#328—1271no—
134#328—1271no—
135#327—1271no—
136#327D12711yescharacteristic, complemented
137C2×#41—1271no—
138C23×C4—1271no—
139C2×M16—2541no—
140#123—2541no—
141C22⋊Q8—1271no—
142C22⋊Q8—1271no—
143#121—1271no—
144#121C2×D12711yescharacteristic
145#120—2541no—
146C4×D4—2541no—
147C4×D4—2541no—
148#117—1271no—
149C2×C4⋊C4—2541no—
150C2×C22⋊C4—2541no—
151C22⋊C8—1271no—
152C22⋊C8—2541no—
153C22⋊C8C2×D12711yescharacteristic
154#95—1271no—
155#95C2×D12711yescharacteristic
156C24—1271no—
157#41—5082no—
158C2×Q8—1271no—
159C2×D4—21no—
160C2×D4—2541no—
161C2×D4—2541no—
162C2×D4C4×D12711yescharacteristic
163C22×C4—21no—
164C22×C4—1271no—
165C22×C4—1271no—
166C22×C4—1271no—
167C22×C4—1271no—
168C22×C4—1271no—
169C22×C4—5082no—
170C22×C4—2541no—
171C22×C4—2541no—
172C22×C4C22×D12711yescharacteristic
173M16—5082no—
174C2×C8—21no—
175C2×C8—2541no—
176C4⋊C4—1271no—
177C4⋊C4—5082no—
178C4⋊C4—2541no—
179C4⋊C4—2541no—
180C4⋊C4C4×D12711yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—5082no—
183C22⋊C4—2541no—
184C22⋊C4—2541no—
185C22⋊C4—2541no—
186C22⋊C4—2541no—
187C42—2541no—
188C23—21no—
189C23—1271no—
190C23—1271no—
191C23—5081no—
192C23—5081no—
193C23—2541no—
194C23—2541no—
195C23D4×D12711yescharacteristic
196Q8—5081no—
197D4—41no—
198D4—41no—
199D4—41no—
200D4—5082no—
201D4—5082no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—1271no—
208C2×C4—1271no—
209C2×C4—5081no—
210C2×C4—5081no—
211C2×C4—5082no—
212C2×C4—5081no—
213C2×C4—5081no—
214C2×C4—5081no—
215C2×C4—2542no—
216C2×C4—2542no—
217C2×C4—2542no—
218C2×C4—2541no—
219C2×C4—2541no—
220C2×C4C2×C4×D12711yescharacteristic
221C2×C4D4×D12711yescharacteristic
222C8—41no—
223C8—5081no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—5082no—
230C22—5082no—
231C22—2541no—
232C22—2541no—
233C22—2541no—
234C22—2541no—
235C22—2541no—
236C22—2541no—
237C22C22⋊C4×D12711yescharacteristic
238C22#1686123311yescharacteristic
239C22#1686123811yescharacteristic
240C4—21no—
241C4—21no—
242C4—41no—
243C4—81no—
244C4—1272no—
245C4—10161no—
246C4—5081no—
247C4—5081no—
248C4—2541no—
249C2—21no—
250C2—21no—
251C2—81no—
252C2—5081no—
253C2—5081no—
254C2#99×D12711yescharacteristic
255C2#2254987411yescharacteristic
256C2#2254989111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#327×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550386C2
cosocleC22×C4×C127C23
hypercenter#327D127
hypocenterC127#1513
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #22549838C2semidirect
3. #22549735C2semidirect
4. #22549614C2semidirect
5. #22549612C2semidirect
6. #22549610C2semidirect
7. #327×C127C2semidirect
8. C2×#14523247C4semidirect
12. #16861189C4semidirect
14. #16861180C4semidirect
18. #16861167C4semidirect
129. C127#1513semidirect
136. #327D127semidirect

Maximal subgroups

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

subgroupindex
#225503862
#225498382
#225497352
#225496142
#225496122
#225496102
#327×C1272
#1513127