Finite Group Databasebeta

← #30650105

Subgroups — #30650105

Order 16256.

348 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 113 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#22550431C211yesmaximal, characteristic, complemented
2#22550408C211yesmaximal, characteristic
3#22550186C212yesmaximal, complemented
4#22550168C211yesmaximal, characteristic, complemented
5#22550135C212yesmaximal, complemented
6#22550121C212yesmaximal
7#22550097C212yesmaximal, complemented
8#22550087C211yesmaximal, characteristic, complemented
9#22550081C211yesmaximal, characteristic, complemented
10#22549898C211yesmaximal, characteristic, complemented
11#476×C127C211yesmaximal, characteristic, complemented
12C2×Dic508C2212yes—
13C2×#14523236C2212yes—
14C2×#14523231C2212yes—
15#16861209C2211yescharacteristic
16#16861202—21no—
17#16861200—21no—
18#16861200C2211yescharacteristic
19#16861196C2211yescharacteristic
20#16861191—21no—
21#16861189C2212yes—
22#16861189C2211yescharacteristic
23#16861185—21no—
24#16861180C2211yescharacteristic
25Q8×Dic127—21no—
26C4×Dic254C2211yescharacteristic
27#16861174C2212yes—
28C4×C22⋊D127—21no—
29#16861168—21no—
30#16861167—21no—
31C42×D127C2211yescharacteristic
32#16861142C2212yes—
33C4⋊(C127⋊C8)C2211yescharacteristic
34Dic127⋊C8C2212yes—
35#16861134C2212yes—
36#16861132C2212yes—
37#16861132C2212yes—
38C4⋊Q8×C127C2211yescharacteristic
39C4×D4×C127C2211yescharacteristic
40#106×C127C2212yes—
41C4⋊C8×C127C2211yescharacteristic
42#102×C127C2212yes—
43C2×Dic254C2311yescharacteristic
44C2×Dic254C2312yes—
45C2×C22⋊D127—21no—
46C2×C4×D127D411yescharacteristic
47C2×C4×D127D411yescharacteristic
48C2×C4×D127C2311yescharacteristic
49Dic508—42no—
50#14523236—42no—
51#14523231—42no—
52C2×C127⋊C8C2312yes—
53C4⋊Dic127—22no—
54C4⋊Dic127—21no—
55C4⋊Dic127—21no—
56C4⋊Dic127—21no—
57C4⋊Dic127—21no—
58C4⋊Dic127C2311yescharacteristic
59Dic127⋊C4—22no—
60Dic127⋊C4—21no—
61Dic127⋊C4—21no—
62C22⋊Dic127—21no—
63C22⋊Dic127—21no—
64C22⋊Dic127—21no—
65(C2×D127)⋊C4—22no—
66(C2×D127)⋊C4—21no—
67C4×Dic127—22no—
68C4×Dic127—21no—
69C4×Dic127D411yescharacteristic
70C4×Dic127D411yescharacteristic
71C4×Dic127C2311yescharacteristic
72C2×Q8×C127—21no—
73C2×D4×C127C2311yescharacteristic
74C22×C4×C127—21no—
75C2×C8×C127C2312yes—
76C4⋊C4×C127—21no—
77C4⋊C4×C127C2311yescharacteristic
78C4⋊C4×C127C2312yes—
79C22⋊C4×C127—21no—
80C42×C127C2311yescharacteristic
81C22×D127C2×D411yescharacteristic
82Dic254—22no—
83Dic254—21no—
84Dic254—21no—
85Dic254—22no—
86C22⋊D127—42no—
87C2×Dic127—21no—
88C2×Dic127—22no—
89C2×Dic127—21no—
90C2×Dic127C2×D411yescharacteristic
91C2×Dic127C2×D411yescharacteristic
92C2×Dic127C2×D411yescharacteristic
93C4×D127—22no—
94C4×D127—21no—
95C4×D127—21no—
96C4×D127—41no—
97C127⋊C8—22no—
98C23×C127—21no—
99Q8×C127—42no—
100D4×C127—21no—
101D4×C127—21no—
102C2×C4×C127—21no—
103C2×C4×C127—22no—
104C2×C4×C127—42no—
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2411yescharacteristic
108C8×C127—22no—
109C22×C127—41no—
110C22×C127—41no—
111C22×C127—41no—
112C22×C127C22×D411yescharacteristic
113C2×D127—21no—
114C2×D127—21no—
115C2×D127—21no—
116C4×C127—21no—
117C4×C127—41no—
118C4×C127—42no—
119C4×C127#13712yes—
120C4×C127H(5,𝔽2)11yescharacteristic
121C4×C1272−1+411yescharacteristic
122Dic127—21no—
123Dic127—21no—
124Dic127—21no—
125Dic127—41no—
126Dic127—42no—
127Dic127—41no—
128C2×C127—81no—
129C2×C127#53711yescharacteristic
130C2×C127C2×#13711yescharacteristic
131C2×C127#57511yescharacteristic
132D127—41no—
133#3252—1271nomaximal, Sylow
134C127#325211yescharacteristic, complemented, Sylow
135#533—1271no—
136#529—1271no—
137#479—1272no—
138#477—1271no—
139#476D12711yescharacteristic, complemented
140#470—1272no—
141#468—1272no—
142#464—1272no—
143#462—1271no—
144#461—1271no—
145#423—1271no—
146C2×Q16—1272no—
147C2×SD16—1272no—
148C2×M16—1272no—
149C4⋊Q8—1271no—
150C4⋊Q8C2×D12711yescharacteristic
151#126—2541no—
152#125—1271no—
153#125—2541no—
154#124—1271no—
155#123—2541no—
156C22⋊Q8—1271no—
157C22⋊Q8—1272no—
158C22⋊Q8—2541no—
159#121—1271no—
160C4×Q8—1272no—
161C4×Q8—1271no—
162C4×Q8—2541no—
163C4×D4—2541no—
164C4×D4C2×D12711yescharacteristic
165#117—2541no—
166#117—2541no—
167C2×C42—1271no—
168#107—1272no—
169#106C2×D12712yes—
170C4⋊C8—1272no—
171C4⋊C8—1271no—
172C4⋊C8C2×D12711yescharacteristic
173#103—1272no—
174#103—1272no—
175#103—1272no—
176#102C2×D12712yes—
177C2×Q8—21no—
178C2×Q8—1271no—
179C2×Q8—1272no—
180C2×D4—2541no—
181C2×D4C22×D12711yescharacteristic
182C22×C4—21no—
183C22×C4—1271no—
184C22×C4—1271no—
185C22×C4—1271no—
186Q16—5082no—
187SD16—5082no—
188M16—5082no—
189C2×C8—1272no—
190C2×C8C22×D12712yes—
191C4⋊C4—21no—
192C4⋊C4—1271no—
193C4⋊C4—2541no—
194C4⋊C4—2542no—
195C4⋊C4—2541no—
196C4⋊C4—2541no—
197C4⋊C4—2542no—
198C4⋊C4—2541no—
199C4⋊C4—2541no—
200C4⋊C4—2541no—
201C4⋊C4C22×D12712yes—
202C4⋊C4C22×D12711yescharacteristic
203C22⋊C4—21no—
204C22⋊C4—2542no—
205C22⋊C4—2541no—
206C22⋊C4—2541no—
207C22⋊C4—2541no—
208C22⋊C4—2541no—
209C42—1271no—
210C42—1271no—
211C42—1271no—
212C42—2542no—
213C42—2541no—
214C42C22×D12711yescharacteristic
215C23—21no—
216C23—1271no—
217Q8—42no—
218Q8—2542no—
219Q8—2541no—
220Q8—2541no—
221Q8—2542no—
222D4—21no—
223D4—21no—
224D4—5082no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—42no—
228C2×C4—1271no—
229C2×C4—1271no—
230C2×C4—1271no—
231C2×C4—5081no—
232C2×C4—2541no—
233C2×C4—2541no—
234C2×C4—2542no—
235C2×C4—2542no—
236C2×C4—2541no—
237C2×C4—2541no—
238C2×C4D4×D12711yescharacteristic
239C2×C4D4×D12711yescharacteristic
240C2×C4C23×D12711yescharacteristic
241C8—22no—
242C8—2542no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—2541no—
247C22—2541no—
248C22—2541no—
249C22C2×D4×D12711yescharacteristic
250C4—21no—
251C4—41no—
252C4—42no—
253C4—5081no—
254C4—5081no—
255C4—5082no—
256C4—2541no—
257C4—2541no—
258C4—2541no—
259C4#1686123812yes—
260C4#1686125511yescharacteristic
261C4#1686125711yescharacteristic
262C2—81no—
263C2—5081no—
264C2#2255045611yescharacteristic
265C2C2×#1686123811yescharacteristic
266C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#476×C127C2
socleC22×C127C22×D4
prime radical#16861180C22
hypercenter#476D127
hypocenterC127#3252
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #22550431C2semidirect
3. #22550186C2semidirect
4. #22550168C2semidirect
5. #22550135C2semidirect
7. #22550097C2semidirect
8. #22550087C2semidirect
9. #22550081C2semidirect
10. #22549898C2semidirect
11. #476×C127C2semidirect
134. C127#3252semidirect
139. #476D127semidirect

Maximal subgroups

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

subgroupindex
#225504312
#225504082
#225501862
#225501682
#225501352
#225501212
#225500972
#225500872
#225500812
#225498982
#476×C1272
#3252127