Finite Group Databasebeta

← #328×D127 #30637209

Subgroups — #328×D127 #30637209

Order 16256.

336 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C22⋊Q8×D127C211yesmaximal, characteristic
2C22⋊C8×D127C211yesmaximal, characteristic
3#95×D127C211yesmaximal, characteristic
4#22549614C211yesmaximal, characteristic, complemented
5#22549613C211yesmaximal, characteristic, complemented
6#22549612C211yesmaximal, characteristic, complemented
7#328×C127C211yesmaximal, characteristic, complemented
8C2×Q8×D127C411yescharacteristic, complemented
9C22×C4×D127—21no—
10C22×C4×D127C2211yescharacteristic
11C2×C8×D127—21no—
12#16861189C411yescharacteristic, complemented
13#16861188C2211yescharacteristic
14#16861187—21no—
15#16861186—21no—
16#16861185C411yescharacteristic, complemented
17C4⋊C4×D127—21no—
18C4⋊C4×D127C411yescharacteristic, complemented
19C22⋊C4×D127—21no—
20C22⋊(C127⋊C8)C2211yescharacteristic
21(C2×D127)⋊C8—21no—
22C127⋊#95—21no—
23C127⋊#95C2211yescharacteristic
24C22⋊Q8×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C23×D127D411yescharacteristic
28C2×Dic254—21no—
29C2×Dic254C2×C411yescharacteristic, complemented
30Q8×D127—42no—
31C22×Dic127—21no—
32C22×Dic127D411yescharacteristic
33C2×C4×D127—24no—
34C2×C4×D127—21no—
35C2×C4×D127—21no—
36C2×C4×D127—42no—
37C2×C4×D127—42no—
38C2×C4×D127—42no—
39C2×C4×D127C2×C411yescharacteristic
40C2×C4×D127C2×C411yescharacteristic
41C2×C4×D127D411yescharacteristic
42C2×C4×D127D411yescharacteristic
43C2×C127⋊C8—21no—
44C8×D127—42no—
45C4⋊Dic127—21no—
46C4⋊Dic127C2×C411yescharacteristic, complemented
47Dic127⋊C4—21no—
48Dic127⋊C4—21no—
49Dic127⋊C4—21no—
50C22⋊Dic127—21no—
51(C2×D127)⋊C4—21no—
52(C2×D127)⋊C4—21no—
53C2×Q8×C127C2×C411yescharacteristic, complemented
54C22×C4×C127—21no—
55C22×C4×C127C2311yescharacteristic
56C2×C8×C127—21no—
57C4⋊C4×C127—21no—
58C4⋊C4×C127C2×C411yescharacteristic, complemented
59C22⋊C4×C127—21no—
60C22×D127—22no—
61C22×D127—22no—
62C22×D127C22⋊C411yescharacteristic
63C22×D127C22⋊C411yescharacteristic
64C22×D127SD1612yes—
65C22×D127Q1612yes—
66Dic254—42no—
67Dic254—41no—
68C2×Dic127—22no—
69C2×Dic127—21no—
70C2×Dic127—21no—
71C2×Dic127—41no—
72C2×Dic127—41no—
73C2×Dic127—41no—
74C2×Dic127C22⋊C411yescharacteristic
75C2×Dic127C22⋊C411yescharacteristic
76C4×D127—22no—
77C4×D127—22no—
78C4×D127—41no—
79C4×D127—41no—
80C4×D127—48no—
81C4×D127—42no—
82C4×D127—42no—
83C127⋊C8—41no—
84C23×C127C2×D411yescharacteristic
85Q8×C127—41no—
86C2×C4×C127—22no—
87C2×C4×C127—21no—
88C2×C4×C127—21no—
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×C127C2×C22⋊C411yescharacteristic
98C22×C127C2×SD1611yescharacteristic
99C22×C127C2×Q1611yescharacteristic
100C2×D127—21no—
101C2×D127—21no—
102C2×D127—24no—
103C2×D127—24no—
104C2×D127#9912yes—
105C2×D127#10312yes—
106C2×D127#10312yes—
107C2×D127#10412yes—
108C4×C127—21no—
109C4×C127—21no—
110C4×C127—42no—
111C4×C127—41no—
112C4×C127—41no—
113Dic127—21no—
114Dic127—21no—
115Dic127—42no—
116Dic127—41no—
117Dic127—41no—
118C2×C127—21no—
119C2×C127—21no—
120C2×C127C2×#9911yescharacteristic
121C2×C127C2×#10311yescharacteristic
122C2×C127C2×#10411yescharacteristic
123D127—22no—
124D127#32814yesdirect product factor
125C2×#328—1271nomaximal, Sylow
126C127C2×#32811yescharacteristic, complemented, Sylow
127C2×C22⋊Q8—1271no—
128C2×C22⋊C8—1271no—
129C2×#95—1271no—
130#328—1271no—
131#328—1271no—
132#328—1271no—
133#328D12711yescharacteristic, direct product factor
134C22×Q8—1271no—
135C23×C4—1271no—
136C23×C4—2541no—
137C22×C8—2541no—
138C22⋊Q8—1271no—
139C22⋊Q8—1271no—
140C22⋊Q8—1271no—
141C22⋊Q8—2541no—
142C22⋊Q8—2541no—
143C22⋊Q8C2×D12711yescharacteristic
144C2×C4⋊C4—1271no—
145C2×C4⋊C4—2541no—
146C2×C22⋊C4—2541no—
147C22⋊C8—1271no—
148C22⋊C8—2541no—
149C22⋊C8C2×D12711yescharacteristic
150#95—1271no—
151#95—2541no—
152#95C2×D12711yescharacteristic
153C24—1271no—
154C2×Q8—1271no—
155C2×Q8—5082no—
156C2×Q8—2541no—
157C2×Q8C4×D12711yescharacteristic, complemented
158C22×C4—21no—
159C22×C4—1271no—
160C22×C4—1271no—
161C22×C4—1271no—
162C22×C4—1271no—
163C22×C4—1271no—
164C22×C4—5082no—
165C22×C4—5082no—
166C22×C4—5082no—
167C22×C4—2541no—
168C22×C4—2541no—
169C22×C4—2544no—
170C22×C4—2541no—
171C22×C4C22×D12711yescharacteristic
172C2×C8—21no—
173C2×C8—5082no—
174C2×C8—2541no—
175C4⋊C4—21no—
176C4⋊C4—1271no—
177C4⋊C4—2541no—
178C4⋊C4—2541no—
179C4⋊C4—2541no—
180C4⋊C4—2541no—
181C4⋊C4C4×D12711yescharacteristic, complemented
182C22⋊C4—21no—
183C22⋊C4—2541no—
184C22⋊C4—2541no—
185C22⋊C4—2541no—
186C23—1271no—
187C23—1272no—
188C23—1272no—
189C23—1271no—
190C23—2542no—
191C23—2542no—
192C23D4×D12711yescharacteristic
193Q8—41no—
194Q8—5081no—
195Q8—5082no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—1271no—
203C2×C4—1271no—
204C2×C4—5081no—
205C2×C4—5081no—
206C2×C4—5082no—
207C2×C4—5082no—
208C2×C4—5088no—
209C2×C4—5081no—
210C2×C4—5081no—
211C2×C4—5081no—
212C2×C4—2542no—
213C2×C4—2542no—
214C2×C4—2542no—
215C2×C4—2541no—
216C2×C4—2541no—
217C2×C4C2×C4×D12711yescharacteristic
218C2×C4D4×D12711yescharacteristic
219C8—41no—
220C8—5081no—
221C22—21no—
222C22—21no—
223C22—1272no—
224C22—1272no—
225C22—1272no—
226C22—1272no—
227C22—2544no—
228C22—2544no—
229C22—2541no—
230C22—2541no—
231C22C22⋊C4×D12711yescharacteristic
232C22SD16×D12711yescharacteristic
233C22Q16×D12711yescharacteristic
234C4—21no—
235C4—21no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—5082no—
240C4—5081no—
241C4—5081no—
242C4—2541no—
243C4—2541no—
244C2—21no—
245C2—21no—
246C2—1274no—
247C2—2542no—
248C2#99×D12711yescharacteristic
249C2#103×D12711yescharacteristic
250C2#104×D12711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#328×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×D127D4
cosocleC22×C4×C127C23
hypercenter#328D127
hypocenterC127C2×#328
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #22549614C2semidirect
5. #22549613C2semidirect
6. #22549612C2semidirect
7. #328×C127C2semidirect
8. C2×Q8×D127C4semidirect
12. #16861189C4semidirect
16. #16861185C4semidirect
18. C4⋊C4×D127C4semidirect
29. C2×Dic254C2×C4semidirect
46. C4⋊Dic127C2×C4semidirect
53. C2×Q8×C127C2×C4semidirect
58. C4⋊C4×C127C2×C4semidirect
124. D127#328direct (N × H)
126. C127C2×#328semidirect
133. #328D127direct (N × H)
157. C2×Q8C4×D127semidirect
181. C4⋊C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1272
C22⋊C8×D1272
#95×D1272
#225496142
#225496132
#225496122
#328×C1272
C2×#328127