Finite Group Databasebeta

← #30637654

Subgroups — #30637654

Order 16256.

316 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 51 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#22550501C211yesmaximal, characteristic, complemented
2#22550029C211yesmaximal, characteristic, complemented
3#22550026C211yesmaximal, characteristic, complemented
4#22549786C211yesmaximal, characteristic
5#22549612C211yesmaximal, characteristic, complemented
6#22549599C211yesmaximal, characteristic, complemented
7#328×C127C211yesmaximal, characteristic, complemented
8C22×D508—21no—
9C22×D508C2211yescharacteristic
10C2×#14523237—21no—
11C2×#14523235—21no—
12#16861189—21no—
13#16861188C2211yescharacteristic
14#16861187—21no—
15#16861187—21no—
16#16861186—21no—
17#16861185—21no—
18C2×(C2×D127)⋊C4—21no—
19C2×(C2×D127)⋊C4C2211yescharacteristic
20C22⋊C4×D127—21no—
21#16861131—21no—
22#16861129—21no—
23C22⋊(C127⋊C8)C2211yescharacteristic
24C22⋊Q8×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C23×D127—21no—
28C23×D127D411yescharacteristic
29C2×Dic254—21no—
30C2×Dic254D411yescharacteristic, complemented
31C2×D508—21no—
32C2×D508—21no—
33C2×D508—42no—
34C2×D508—42no—
35C2×D508—42no—
36C2×D508—42no—
37C22×Dic127D411yescharacteristic
38C2×C4×D127—21no—
39C2×C4×D127—21no—
40#14523237—42no—
41#14523235—42no—
42C2×C127⋊C8—21no—
43C4⋊Dic127—21no—
44C4⋊Dic127D411yescharacteristic, complemented
45Dic127⋊C4—21no—
46Dic127⋊C4—21no—
47Dic127⋊C4—21no—
48C22⋊Dic127—21no—
49(C2×D127)⋊C4—21no—
50(C2×D127)⋊C4—21no—
51(C2×D127)⋊C4—21no—
52(C2×D127)⋊C4—21no—
53(C2×D127)⋊C4—42no—
54C2×Q8×C127D411yescharacteristic, complemented
55C22×C4×C127—21no—
56C22×C4×C127C2311yescharacteristic
57C2×C8×C127—21no—
58C4⋊C4×C127—21no—
59C4⋊C4×C127D411yescharacteristic, complemented
60C22⋊C4×C127—21no—
61C22×D127—21no—
62C22×D127—21no—
63C22×D127—22no—
64C22×D127—41no—
65C22×D127—41no—
66C22×D127—41no—
67C22×D127—41no—
68C22×D127—41no—
69C22×D127—41no—
70C22×D127—41no—
71C22×D127—41no—
72C22×D127SD1612yescomplemented
73Dic254—41no—
74Dic254—81no—
75D508—22no—
76D508—22no—
77D508—41no—
78D508—41no—
79D508—48no—
80C2×Dic127—21no—
81C2×Dic127—21no—
82C2×Dic127—21no—
83C2×Dic127—41no—
84C4×D127—81no—
85C4×D127—81no—
86C127⋊C8—41no—
87C23×C127C2×D411yescharacteristic
88Q8×C127—41no—
89C2×C4×C127—22no—
90C2×C4×C127—21no—
91C2×C4×C127—21no—
92C2×C4×C127—41no—
93C2×C4×C127—41no—
94C2×C4×C127—41no—
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C8×C127—41no—
98C22×C127—21no—
99C22×C127—21no—
100C22×C127#12011yescharacteristic
101C22×C127C2×SD1611yescharacteristic
102C22×C127#13711yescharacteristic
103C2×D127—22no—
104C2×D127—41no—
105C2×D127—41no—
106C2×D127—42no—
107C2×D127—42no—
108C2×D127—42no—
109C2×D127—42no—
110C2×D127—81no—
111C2×D127—81no—
112C2×D127—81no—
113C2×D127—81no—
114C2×D127—81no—
115C4×C127—21no—
116C4×C127—21no—
117C4×C127—42no—
118C4×C127—41no—
119C4×C127—41no—
120Dic127—41no—
121Dic127—41no—
122Dic127—41no—
123C2×C127—21no—
124C2×C127—21no—
125C2×C127#44811yescharacteristic
126C2×C127#45311yescharacteristic
127C2×C127B(4,𝔽2)11yescharacteristic
128D127—42no—
129D127—82no—
130#1596—1271nomaximal, Sylow
131C127#159611yescharacteristic, complemented, Sylow
132#543—1271no—
133#450—1271no—
134#450—1271no—
135#392—1271no—
136#328—1271no—
137#328D12711yescharacteristic, complemented
138#323—1271no—
139C22×D4—1271no—
140C22×D4—2541no—
141C2×SD16—2541no—
142C2×SD16—2541no—
143C22⋊Q8—1271no—
144C22⋊Q8—2541no—
145C22⋊Q8—2541no—
146C22⋊Q8—2541no—
147C22⋊Q8—2541no—
148C22⋊Q8—2541no—
149C22⋊Q8C2×D12711yescharacteristic
150C2×C22⋊C4—1271no—
151C2×C22⋊C4—2541no—
152C2×C22⋊C4—2541no—
153#102—2541no—
154#102—2541no—
155C22⋊C8—1271no—
156C22⋊C8C2×D12711yescharacteristic
157#95C2×D12711yescharacteristic
158C24—1271no—
159C24—2541no—
160C2×Q8—1271no—
161C2×Q8—2541no—
162C2×Q8D50811yescharacteristic, complemented
163C2×D4—5082no—
164C2×D4—5082no—
165C2×D4—5082no—
166C2×D4—5082no—
167C2×D4—2541no—
168C2×D4—2541no—
169C22×C4—21no—
170C22×C4—1271no—
171C22×C4—2541no—
172C22×C4—2541no—
173C22×C4C22×D12711yescharacteristic
174SD16—5082no—
175SD16—5082no—
176C2×C8—21no—
177C2×C8—2541no—
178C4⋊C4—21no—
179C4⋊C4—1271no—
180C4⋊C4—2541no—
181C4⋊C4—2541no—
182C4⋊C4—2541no—
183C4⋊C4—2541no—
184C4⋊C4D50811yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—5082no—
187C22⋊C4—2541no—
188C22⋊C4—2541no—
189C22⋊C4—2541no—
190C22⋊C4—2541no—
191C22⋊C4—2541no—
192C23—1272no—
193C23—5081no—
194C23—5081no—
195C23—5081no—
196C23—5081no—
197C23—5081no—
198C23—5081no—
199C23—5081no—
200C23—5081no—
201C23—2542no—
202C23—2541no—
203C23—2541no—
204C23D4×D12711yescharacteristic
205Q8—41no—
206Q8—10161no—
207Q8—5081no—
208D4—5081no—
209D4—5081no—
210D4—5088no—
211D4—2542no—
212D4—2542no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—10161no—
220C2×C4—10161no—
221C2×C4—5081no—
222C2×C4—2541no—
223C2×C4—2541no—
224C2×C4—2541no—
225C2×C4D4×D12711yescharacteristic
226C2×C4C2×D50811yescharacteristic
227C8—41no—
228C8—5081no—
229C22—21no—
230C22—21no—
231C22—10161no—
232C22—10161no—
233C22—10161no—
234C22—10161no—
235C22—10161no—
236C22—5082no—
237C22—5082no—
238C22—5082no—
239C22—5082no—
240C22—5081no—
241C22—5081no—
242C22—2542no—
243C22#1686117911yescharacteristic
244C22SD16×D12711yescharacteristic
245C22#1686124111yescharacteristic
246C4—21no—
247C4—21no—
248C4—42no—
249C4—41no—
250C4—41no—
251C4—5081no—
252C4—5081no—
253C4—5081no—
254C2—21no—
255C2—21no—
256C2—10162no—
257C2—5082no—
258C2#2255001611yescharacteristic
259C2#2255004511yescharacteristic
260C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#328×C127C2
socleC22×C127#120
prime radical#22549786C2
hypercenter#328D127
hypocenterC127#1596
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #22550501C2semidirect
2. #22550029C2semidirect
3. #22550026C2semidirect
5. #22549612C2semidirect
6. #22549599C2semidirect
7. #328×C127C2semidirect
30. C2×Dic254D4semidirect
44. C4⋊Dic127D4semidirect
54. C2×Q8×C127D4semidirect
59. C4⋊C4×C127D4semidirect
72. C22×D127SD16semidirect
131. C127#1596semidirect
137. #328D127semidirect
162. C2×Q8D508semidirect
184. C4⋊C4D508semidirect

Maximal subgroups

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

subgroupindex
#225505012
#225500292
#225500262
#225497862
#225496122
#225495992
#328×C1272
#1596127