Finite Group Databasebeta

← #30637683

Subgroups — #30637683

Order 16256.

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

normal subgroups only

This group has 308 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#22550434C211yesmaximal, characteristic
2#22550017C211yesmaximal, characteristic, complemented
3#22550014C211yesmaximal, characteristic, complemented
4#22549789C211yesmaximal, characteristic, complemented
5#22549609C211yesmaximal, characteristic, complemented
6#22549599C211yesmaximal, characteristic, complemented
7#327×C127C211yesmaximal, characteristic, complemented
8C22×Dic254C2211yescharacteristic, complemented
9C2×D4×D127—21no—
10C2×#14523237—21no—
11C2×#14523236—21no—
12#16861194—21no—
13#16861191—21no—
14#16861183C2211yescharacteristic
15#16861181—21no—
16#16861179—21no—
17#16861177—21no—
18C2×(C2×D127)⋊C4—21no—
19C2×(C2×D127)⋊C4C2211yescharacteristic
20#16861134—21no—
21#16861132—21no—
22C22⋊(C127⋊C8)C2211yescharacteristic, complemented
23C127⋊#95—21no—
24#121×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic, complemented
26#95×C127C2211yescharacteristic, complemented
27C23×D127—21no—
28C23×D127D411yescharacteristic
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—42no—
32C2×D508—21no—
33C2×D508D411yescharacteristic
34C2×C22⋊D127—21no—
35C2×C22⋊D127—21no—
36C2×C22⋊D127—21no—
37D4×D127—42no—
38D4×D127—42no—
39C22×Dic127—21no—
40C22×Dic127D411yescharacteristic
41C2×C4×D127—21no—
42#14523237—42no—
43#14523236—42no—
44C2×C127⋊C8—21no—
45C4⋊Dic127D411yescharacteristic
46Dic127⋊C4—21no—
47C22⋊Dic127—21no—
48(C2×D127)⋊C4—21no—
49(C2×D127)⋊C4—21no—
50(C2×D127)⋊C4—21no—
51(C2×D127)⋊C4—21no—
52(C2×D127)⋊C4—42no—
53C2×D4×C127—21no—
54C2×D4×C127D411yescharacteristic
55C22×C4×C127—21no—
56C22×C4×C127C2311yescharacteristic
57C2×C8×C127—21no—
58C4⋊C4×C127D411yescharacteristic
59C22⋊C4×C127—21no—
60C22×D127—21no—
61C22×D127—21no—
62C22×D127—21no—
63C22×D127—42no—
64C22×D127—41no—
65C22×D127—41no—
66C22×D127—81no—
67C22×D127—81no—
68C22×D127SD1612yescomplemented
69Dic254—22no—
70Dic254—22no—
71Dic254—41no—
72Dic254—41no—
73D508—41no—
74D508—42no—
75C22⋊D127—42no—
76C22⋊D127—41no—
77C22⋊D127—41no—
78C22⋊D127—81no—
79C2×Dic127—22no—
80C2×Dic127—21no—
81C2×Dic127—21no—
82C2×Dic127—41no—
83C2×Dic127—41no—
84C2×Dic127—41no—
85C4×D127—42no—
86C127⋊C8—41no—
87C23×C127—21no—
88C23×C127C2×D411yescharacteristic
89D4×C127—41no—
90D4×C127—41no—
91D4×C127—41no—
92C2×C4×C127—21no—
93C2×C4×C127—41no—
94C2×C4×C127—41no—
95C2×C4×C127—41no—
96C2×C4×C127—41no—
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C8×C127—41no—
100C22×C127—21no—
101C22×C127—21no—
102C22×C127—41no—
103C22×C127—41no—
104C22×C127—41no—
105C22×C127#12011yescharacteristic
106C22×C127C2×SD1611yescharacteristic
107C22×C127AGL(1,ℤ/8)11yescharacteristic
108C2×D127—22no—
109C2×D127—41no—
110C2×D127—41no—
111C2×D127—42no—
112C2×D127—41no—
113C2×D127—41no—
114C2×D127—41no—
115C2×D127—81no—
116C2×D127—84no—
117C4×C127—21no—
118C4×C127—21no—
119C4×C127—41no—
120C4×C127—81no—
121Dic127—41no—
122Dic127—41no—
123Dic127—42no—
124C2×C127—21no—
125C2×C127—21no—
126C2×C127—81no—
127C2×C127#45011yescharacteristic
128C2×C127#45511yescharacteristic
129C2×C127B(4,𝔽2)11yescharacteristic
130D127—42no—
131D127—81no—
132#1601—1271nomaximal, Sylow
133C127#160111yescharacteristic, complemented, Sylow
134#534—1271no—
135#448—1271no—
136#448—1271no—
137#393—1271no—
138#327—1271no—
139#327D12711yescharacteristic, complemented
140#323—1271no—
141C22×Q8—1271no—
142C22×D4—2541no—
143C2×SD16—2541no—
144C2×SD16—2541no—
145#123—2541no—
146#123—2541no—
147#121—1271no—
148#121—2541no—
149#121C2×D12711yescharacteristic
150#120—2541no—
151#120—2541no—
152C2×C22⋊C4—1271no—
153C2×C22⋊C4—2541no—
154#103—2541no—
155#103—2541no—
156C22⋊C8—1271no—
157C22⋊C8C2×D12711yescharacteristic, complemented
158#95—2541no—
159#95C2×D12711yescharacteristic, complemented
160C24—1271no—
161C24—2541no—
162C2×Q8—5082no—
163C2×Q8—2541no—
164C2×Q8—2541no—
165C2×D4—21no—
166C2×D4—1271no—
167C2×D4—5082no—
168C2×D4—5082no—
169C2×D4—2541no—
170C2×D4—2541no—
171C2×D4—2541no—
172C2×D4—2541no—
173C2×D4D50811yescharacteristic
174C22×C4—21no—
175C22×C4—1271no—
176C22×C4—2541no—
177C22×C4—2541no—
178C22×C4C22×D12711yescharacteristic
179SD16—5082no—
180SD16—5082no—
181C2×C8—21no—
182C2×C8—2541no—
183C4⋊C4—1271no—
184C4⋊C4—2541no—
185C4⋊C4D50811yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—5082no—
188C22⋊C4—2541no—
189C22⋊C4—2541no—
190C22⋊C4—2541no—
191C22⋊C4—2541no—
192C22⋊C4—2541no—
193C23—21no—
194C23—1272no—
195C23—10161no—
196C23—10161no—
197C23—5081no—
198C23—5081no—
199C23—5082no—
200C23—2541no—
201C23—2541no—
202C23—2541no—
203C23D4×D12711yescharacteristic
204Q8—5081no—
205Q8—5081no—
206Q8—2542no—
207Q8—2542no—
208D4—41no—
209D4—41no—
210D4—41no—
211D4—10161no—
212D4—5081no—
213D4—5082no—
214D4—5082no—
215D4—5081no—
216D4—5081no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—5082no—
223C2×C4—5081no—
224C2×C4—5081no—
225C2×C4—5081no—
226C2×C4—2542no—
227C2×C4—2541no—
228C2×C4—2541no—
229C2×C4D4×D12711yescharacteristic
230C2×C4C2×D50811yescharacteristic
231C8—41no—
232C8—5081no—
233C22—21no—
234C22—21no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—10161no—
239C22—10164no—
240C22—5082no—
241C22—5081no—
242C22—5081no—
243C22—5081no—
244C22—5081no—
245C22—5081no—
246C22—2542no—
247C22#1686117911yescharacteristic
248C22SD16×D12711yescharacteristic
249C22#1686123311yescharacteristic
250C4—21no—
251C4—21no—
252C4—41no—
253C4—81no—
254C4—5081no—
255C4—5081no—
256C4—5082no—
257C2—21no—
258C2—21no—
259C2—81no—
260C2—10161no—
261C2—5082no—
262C2#2255002811yescharacteristic
263C2#2255005511yescharacteristic
264C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#327×C127C2
socleC22×C127#120
prime radical#22550434C2
hypercenter#327D127
hypocenterC127#1601
LCSG ▷ C22×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. #22550017C2semidirect
3. #22550014C2semidirect
4. #22549789C2semidirect
5. #22549609C2semidirect
6. #22549599C2semidirect
7. #327×C127C2semidirect
8. C22×Dic254C22semidirect
22. C22⋊(C127⋊C8)C22semidirect
25. C22⋊C8×C127C22semidirect
26. #95×C127C22semidirect
68. C22×D127SD16semidirect
133. C127#1601semidirect
139. #327D127semidirect
157. C22⋊C8C2×D127semidirect
159. #95C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504342
#225500172
#225500142
#225497892
#225496092
#225495992
#327×C1272
#1601127