Finite Group Databasebeta

← #30637075

Subgroups — #30637075

Order 16256.

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

normal subgroups only

This group has 280 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
1C2×#16861191C211yesmaximal, characteristic
2#22549838C211yesmaximal, characteristic, complemented
3C22⋊C8×D127C211yesmaximal, characteristic, complemented
4#22549601C211yesmaximal, characteristic, complemented
5#22549601C211yesmaximal, characteristic, complemented
6#22549600C211yesmaximal, characteristic, complemented
7#323×C127C211yesmaximal, characteristic, complemented
8C22×C22⋊D127C411yescharacteristic
9C22×C4×D127C2211yescharacteristic
10C2×#14523231—21no—
11C2×C8×D127—21no—
12#16861191—21no—
13#16861191—21no—
14#16861191—21no—
15#16861191—21no—
16C2×C4⋊Dic127C411yescharacteristic
17C2×Dic127⋊C4C2211yescharacteristic
18C2×C22⋊Dic127C411yescharacteristic
19C2×(C2×D127)⋊C4C411yescharacteristic
20C22⋊(C127⋊C8)C2211yescharacteristic
21C22⋊(C127⋊C8)C2211yescharacteristic
22(C2×D127)⋊C8—21no—
23(C2×D127)⋊C8—21no—
24C2×C22⋊C4×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26C22⋊C8×C127C2211yescharacteristic
27C23×D127C2×C411yescharacteristic
28C2×C22⋊D127—21no—
29C2×C22⋊D127—21no—
30C2×C22⋊D127—42no—
31C22×Dic127C2×C411yescharacteristic
32C22×Dic127C2×C411yescharacteristic
33C22×Dic127C2×C411yescharacteristic
34C2×C4×D127—42no—
35C2×C4×D127D411yescharacteristic
36C2×C4×D127D411yescharacteristic
37C2×C4×D127D411yescharacteristic
38C2×C4×D127D411yescharacteristic
39#14523231—42no—
40C2×C127⋊C8—21no—
41C2×C127⋊C8—21no—
42C8×D127—42no—
43C4⋊Dic127—21no—
44C4⋊Dic127—21no—
45Dic127⋊C4—21no—
46Dic127⋊C4—21no—
47C22⋊Dic127—21no—
48C22⋊Dic127—21no—
49(C2×D127)⋊C4—21no—
50(C2×D127)⋊C4—21no—
51C24×C127C2×C411yescharacteristic
52C22×C4×C127C2×C411yescharacteristic
53C22×C4×C127C2311yescharacteristic
54C2×C8×C127—21no—
55C2×C8×C127—21no—
56C22⋊C4×C127—21no—
57C22⋊C4×C127—21no—
58C22×D127—21no—
59C22×D127—21no—
60C22×D127—22no—
61C22×D127—41no—
62C22×D127C22⋊C411yescharacteristic
63C22×D127C22⋊C411yescharacteristic
64C22⋊D127—42no—
65C22⋊D127—42no—
66C2×Dic127—21no—
67C2×Dic127—21no—
68C2×Dic127—21no—
69C2×Dic127—21no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C2×Dic127C22⋊C411yescharacteristic
73C2×Dic127C22⋊C411yescharacteristic
74C4×D127—22no—
75C4×D127—22no—
76C4×D127—41no—
77C4×D127—41no—
78C127⋊C8—41no—
79C127⋊C8—41no—
80C23×C127—21no—
81C23×C127—21no—
82C23×C127—21no—
83C23×C127—41no—
84C23×C127—41no—
85C23×C127C22×C411yescharacteristic
86C2×C4×C127—21no—
87C2×C4×C127—41no—
88C2×C4×C127—41no—
89C2×C4×C127C2×D411yescharacteristic
90C2×C4×C127C2×D411yescharacteristic
91C8×C127—41no—
92C8×C127—41no—
93C22×C127—21no—
94C22×C127—21no—
95C22×C127—41no—
96C22×C127—41no—
97C22×C127—41no—
98C22×C127—81no—
99C22×C127—81no—
100C22×C127C2×C22⋊C411yescharacteristic
101C22×C127#13111yescharacteristic
102C22×C127#13111yescharacteristic
103C2×D127—21no—
104C2×D127—21no—
105C2×D127—21no—
106C2×D127—21no—
107C2×D127—21no—
108C2×D127—42no—
109C2×D127—42no—
110C2×D127#9912yes—
111C4×C127—21no—
112C4×C127—21no—
113C4×C127—41no—
114Dic127—21no—
115Dic127—21no—
116Dic127—41no—
117Dic127—41no—
118C2×C127—21no—
119C2×C127—21no—
120C2×C127—81no—
121C2×C127#40811yescharacteristic
122C2×C127C2×#9911yescharacteristic
123C2×C127#41311yescharacteristic
124D127—22no—
125D127—41no—
126#1468—1271nomaximal, Sylow
127C127#146811yescharacteristic, complemented, Sylow
128C2×#123—1271no—
129#407—1271no—
130C2×C22⋊C8—1271no—
131#324—1271no—
132#324—1271no—
133#323—1271no—
134#323D12711yescharacteristic, complemented
135C22×D4—1271no—
136C23×C4—1271no—
137C2×M16—2541no—
138C22×C8—2541no—
139#123—2541no—
140#123—2541no—
141#123—2541no—
142#123—2541no—
143C2×C4⋊C4—1271no—
144C2×C4⋊C4—1271no—
145C2×C22⋊C4—1271no—
146C2×C22⋊C4—1271no—
147C2×C22⋊C4C2×D12711yescharacteristic
148C22⋊C8—1271no—
149C22⋊C8—1271no—
150C22⋊C8—2541no—
151C22⋊C8—2541no—
152C22⋊C8C2×D12711yescharacteristic
153C22⋊C8C2×D12711yescharacteristic
154C24—1271no—
155C24C4×D12711yescharacteristic
156C2×D4—5082no—
157C2×D4—2541no—
158C2×D4—2541no—
159C22×C4—1271no—
160C22×C4—1271no—
161C22×C4—1271no—
162C22×C4—1271no—
163C22×C4—1271no—
164C22×C4—1271no—
165C22×C4—1271no—
166C22×C4—5082no—
167C22×C4C4×D12711yescharacteristic
168C22×C4C22×D12711yescharacteristic
169M16—5082no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—5082no—
173C2×C8—2541no—
174C2×C8—2541no—
175C4⋊C4—2541no—
176C4⋊C4—2541no—
177C4⋊C4—2541no—
178C4⋊C4—2541no—
179C22⋊C4—21no—
180C22⋊C4—21no—
181C22⋊C4—2541no—
182C22⋊C4—2541no—
183C22⋊C4—2541no—
184C22⋊C4—2541no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—41no—
189C23—41no—
190C23—1271no—
191C23—1271no—
192C23—5081no—
193C23—2542no—
194C23—2541no—
195C23—2541no—
196C23C2×C4×D12711yescharacteristic
197D4—5082no—
198D4—5082no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—1271no—
203C2×C4—1271no—
204C2×C4—5081no—
205C2×C4—5081no—
206C2×C4—5081no—
207C2×C4—5081no—
208C2×C4—2542no—
209C2×C4—2542no—
210C2×C4—2541no—
211C2×C4—2541no—
212C2×C4—2541no—
213C2×C4—2541no—
214C2×C4D4×D12711yescharacteristic
215C2×C4D4×D12711yescharacteristic
216C8—41no—
217C8—41no—
218C8—5081no—
219C8—5081no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—81no—
226C22—81no—
227C22—1272no—
228C22—5082no—
229C22—5082no—
230C22—2541no—
231C22—2541no—
232C22—2541no—
233C22—2541no—
234C22—2541no—
235C22C22⋊C4×D12711yescharacteristic
236C22#1686121511yescharacteristic
237C22#1686121711yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—5081no—
242C4—5081no—
243C4—2541no—
244C4—2541no—
245C2—21no—
246C2—21no—
247C2—81no—
248C2—5081no—
249C2—2542no—
250C2#2254984111yescharacteristic
251C2#99×D12711yescharacteristic
252C2#2254985711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#323×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×C22⋊D127C4
cosocleC22×C4×C127C23
hypercenter#323D127
hypocenterC127#1468
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #22549838C2semidirect
3. C22⋊C8×D127C2semidirect
4. #22549601C2semidirect
5. #22549601C2semidirect
6. #22549600C2semidirect
7. #323×C127C2semidirect
127. C127#1468semidirect
134. #323D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611912
#225498382
C22⋊C8×D1272
#225496012
#225496002
#323×C1272
#1468127