Finite Group Databasebeta

← #30637666

Subgroups — #30637666

Order 16256.

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

normal subgroups only

This group has 310 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#22550453C211yesmaximal, characteristic, complemented
2#22550012C211yesmaximal, characteristic, complemented
3#22550009C211yesmaximal, characteristic, complemented
4#22549794C211yesmaximal, characteristic, complemented
5#22549609C211yesmaximal, characteristic, complemented
6#22549601C211yesmaximal, characteristic, complemented
7#327×C127C211yesmaximal, characteristic, complemented
8C2×#14523247—21no—
9C22×D508C2211yescharacteristic
10C2×D1016—21no—
11C2×#14523233—21no—
12#16861187—21no—
13#16861186—21no—
14#16861183C2211yescharacteristic
15#16861182—21no—
16#16861181—21no—
17#16861181—21no—
18C2×Dic127⋊C4C2211yescharacteristic
19C2×(C2×D127)⋊C4—21no—
20C2×(C2×D127)⋊C4—21no—
21#16861131—21no—
22#16861129—21no—
23C22⋊(C127⋊C8)C2211yescharacteristic
24#121×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C23×D127—21no—
28#14523247—42no—
29#14523247—42no—
30C2×Dic254—21no—
31C2×D508—21no—
32C2×D508—21no—
33C2×D508—42no—
34C2×D508D411yescharacteristic
35C2×C22⋊D127—21no—
36C2×C22⋊D127—21no—
37C2×C22⋊D127—21no—
38C22×Dic127—21no—
39C22×Dic127D411yescharacteristic
40C22×Dic127D411yescharacteristic
41C2×C4×D127—21no—
42D1016—42no—
43#14523233—42no—
44C2×C127⋊C8—21no—
45C4⋊Dic127D411yescharacteristic
46Dic127⋊C4—21no—
47Dic127⋊C4—21no—
48Dic127⋊C4—21no—
49C22⋊Dic127—21no—
50(C2×D127)⋊C4—21no—
51(C2×D127)⋊C4—21no—
52(C2×D127)⋊C4—42no—
53(C2×D127)⋊C4—42no—
54C2×D4×C127—21no—
55C2×D4×C127D411yescharacteristic
56C22×C4×C127—21no—
57C22×C4×C127C2311yescharacteristic
58C2×C8×C127—21no—
59C4⋊C4×C127D411yescharacteristic
60C22⋊C4×C127—21no—
61C22×D127—22no—
62C22×D127—21no—
63C22×D127—41no—
64C22×D127—41no—
65C22×D127—41no—
66C22×D127—41no—
67C22×D127—41no—
68C22×D127—41no—
69Dic254—42no—
70D508—22no—
71D508—22no—
72D508—41no—
73D508—41no—
74D508—41no—
75C22⋊D127—42no—
76C22⋊D127—41no—
77C22⋊D127—41no—
78C22⋊D127—81no—
79C2×Dic127—21no—
80C2×Dic127—21no—
81C2×Dic127—21no—
82C2×Dic127—21no—
83C2×Dic127—42no—
84C2×Dic127—41no—
85C2×Dic127D812yescomplemented
86C4×D127—42no—
87C127⋊C8—41no—
88C23×C127—21no—
89C23×C127C2×D411yescharacteristic
90D4×C127—41no—
91D4×C127—41no—
92D4×C127—41no—
93C2×C4×C127—21no—
94C2×C4×C127—41no—
95C2×C4×C127—41no—
96C2×C4×C127—41no—
97C2×C4×C127—41no—
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C8×C127—41no—
101C22×C127—21no—
102C22×C127—21no—
103C22×C127—41no—
104C22×C127—41no—
105C22×C127—41no—
106C22×C127#12011yescharacteristic
107C22×C127C2×D811yescharacteristic
108C22×C127AGL(1,ℤ/8)11yescharacteristic
109C2×D127—42no—
110C2×D127—42no—
111C2×D127—41no—
112C2×D127—41no—
113C2×D127—41no—
114C2×D127—42no—
115C2×D127—81no—
116C2×D127—81no—
117C2×D127—81no—
118C2×D127—81no—
119C4×C127—21no—
120C4×C127—21no—
121C4×C127—41no—
122C4×C127—81no—
123Dic127—22no—
124Dic127—41no—
125Dic127—41no—
126C2×C127—21no—
127C2×C127—21no—
128C2×C127—81no—
129C2×C127#44711yescharacteristic
130C2×C127#45411yescharacteristic
131C2×C127B(4,𝔽2)11yescharacteristic
132D127—81no—
133D127—82no—
134#1598—1271nomaximal, Sylow
135C127#159811yescharacteristic, complemented, Sylow
136#537—1271no—
137#447—1271no—
138#447—1271no—
139#394—1271no—
140#327—1271no—
141#327D12711yescharacteristic, complemented
142#324—1271no—
143C2×#41—2541no—
144C22×D4—1271no—
145C2×D8—2541no—
146C2×D8—2541no—
147C22⋊Q8—2541no—
148C22⋊Q8—2541no—
149#121—1271no—
150#121—2541no—
151#121—2541no—
152#121—2541no—
153#121C2×D12711yescharacteristic
154C2×C4⋊C4—1271no—
155C2×C22⋊C4—2541no—
156C2×C22⋊C4—2541no—
157#102—2541no—
158#102—2541no—
159C22⋊C8—1271no—
160C22⋊C8C2×D12711yescharacteristic
161#95C2×D12711yescharacteristic
162C24—2541no—
163#41—5082no—
164#41—5082no—
165C2×Q8—2541no—
166C2×D4—21no—
167C2×D4—1271no—
168C2×D4—5082no—
169C2×D4—2541no—
170C2×D4—2541no—
171C2×D4—2541no—
172C2×D4—2541no—
173C2×D4—2541no—
174C2×D4D50811yescharacteristic
175C22×C4—21no—
176C22×C4—1271no—
177C22×C4—1271no—
178C22×C4—2541no—
179C22×C4—2541no—
180C22×C4C22×D12711yescharacteristic
181D8—5082no—
182D8—5082no—
183C2×C8—21no—
184C2×C8—2541no—
185C4⋊C4—1271no—
186C4⋊C4—2541no—
187C4⋊C4—2541no—
188C4⋊C4—2541no—
189C4⋊C4D50811yescharacteristic
190C22⋊C4—21no—
191C22⋊C4—5082no—
192C22⋊C4—5082no—
193C22⋊C4—2541no—
194C22⋊C4—2541no—
195C22⋊C4—2541no—
196C23—21no—
197C23—5081no—
198C23—5081no—
199C23—5081no—
200C23—5081no—
201C23—5081no—
202C23—5081no—
203C23—2542no—
204C23—2541no—
205C23D4×D12711yescharacteristic
206Q8—5082no—
207D4—41no—
208D4—41no—
209D4—41no—
210D4—10161no—
211D4—5081no—
212D4—5081no—
213D4—5081no—
214D4—5081no—
215D4—5081no—
216D4—5082no—
217D4—2542no—
218D4—2542no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—1272no—
225C2×C4—5082no—
226C2×C4—5082no—
227C2×C4—5081no—
228C2×C4—2541no—
229C2×C4—2541no—
230C2×C4—2541no—
231C2×C4—2541no—
232C2×C4D4×D12711yescharacteristic
233C2×C4C2×D50811yescharacteristic
234C8—41no—
235C8—5081no—
236C22—21no—
237C22—21no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—10161no—
242C22—10161no—
243C22—10161no—
244C22—10161no—
245C22—5082no—
246C22—5082no—
247C22—5081no—
248C22—5081no—
249C22—5082no—
250C22—5081no—
251C22#1686117911yescharacteristic
252C22D8×D12711yescharacteristic
253C22#1686123611yescharacteristic
254C4—21no—
255C4—21no—
256C4—41no—
257C4—81no—
258C4—5081no—
259C4—5081no—
260C4—2542no—
261C2—21no—
262C2—21no—
263C2—81no—
264C2—10162no—
265C2—10161no—
266C2#2255001111yescharacteristic
267C2#2255005011yescharacteristic
268C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#327×C127C2
socleC22×C127#120
hypercenter#327D127
hypocenterC127#1598
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
1. #22550453C2semidirect
2. #22550012C2semidirect
3. #22550009C2semidirect
4. #22549794C2semidirect
5. #22549609C2semidirect
6. #22549601C2semidirect
7. #327×C127C2semidirect
85. C2×Dic127D8semidirect
135. C127#1598semidirect
141. #327D127semidirect

Maximal subgroups

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

subgroupindex
#225504532
#225500122
#225500092
#225497942
#225496092
#225496012
#327×C1272
#1598127