Finite Group Databasebeta

← #30650104

Subgroups — #30650104

Order 16256.

348 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 121 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#22550433C211yesmaximal, characteristic, complemented
2#22550410C211yesmaximal, characteristic
3#22550184C212yesmaximal, complemented
4#22550172C211yesmaximal, characteristic, complemented
5#22550167C211yesmaximal, characteristic, complemented
6#22550134C212yesmaximal, complemented
7#22550122C212yesmaximal
8#22550099C212yesmaximal, complemented
9#22550087C211yesmaximal, characteristic, complemented
10#22550082C211yesmaximal, characteristic, complemented
11#423×C127C211yesmaximal, characteristic, complemented
12C2×Dic508C2212yes—
13C2×#14523237C2212yes—
14#16861209C2211yescharacteristic
15#16861209C2211yescharacteristic
16#16861203—21no—
17#16861201—21no—
18#16861201C2211yescharacteristic
19#16861197C2211yescharacteristic
20#16861192—21no—
21#16861188—21no—
22#16861188C2212yes—
23#16861188C2211yescharacteristic
24#16861183C2211yescharacteristic
25C4×Dic254—21no—
26C4×Dic254C2211yescharacteristic
27C4×Dic254C2212yes—
28C4×D508C2211yescharacteristic
29C4×C22⋊D127—21no—
30#16861168—21no—
31#16861166—21no—
32#16861143C2212yes—
33#16861141C2212yes—
34#16861134C2212yes—
35#16861134C2212yes—
36#16861134C2212yes—
37#16861131C2212yes—
38C2×M16×C127C2212yes—
39C2×C42×C127C2211yescharacteristic
40C4⋊C8×C127C2211yescharacteristic
41C4⋊C8×C127C2211yescharacteristic
42C4⋊C8×C127C2212yes—
43C2×Dic254—21no—
44C2×Dic254C2311yescharacteristic
45C2×Dic254C2312yes—
46C2×D508C2311yescharacteristic
47C2×C22⋊D127—21no—
48C2×C4×D127—21no—
49Dic508—42no—
50#14523237—42no—
51C4⋊Dic127—21no—
52C4⋊Dic127—21no—
53C4⋊Dic127—21no—
54C4⋊Dic127C2311yescharacteristic
55C4⋊Dic127C2312yes—
56C4⋊Dic127C2311yescharacteristic
57Dic127⋊C4—22no—
58Dic127⋊C4—22no—
59Dic127⋊C4—21no—
60Dic127⋊C4—21no—
61Dic127⋊C4—21no—
62Dic127⋊C4—21no—
63C22⋊Dic127—22no—
64C22⋊Dic127—21no—
65(C2×D127)⋊C4—21no—
66(C2×D127)⋊C4—21no—
67(C2×D127)⋊C4—21no—
68(C2×D127)⋊C4—21no—
69C4×Dic127—21no—
70C4×Dic127—22no—
71C22×C4×C127D411yescharacteristic
72C22×C4×C127D411yescharacteristic
73C22×C4×C127C2311yescharacteristic
74M16×C127—42no—
75C2×C8×C127C2312yes—
76C2×C8×C127C2312yes—
77C42×C127D411yescharacteristic
78C42×C127D411yescharacteristic
79C42×C127C2311yescharacteristic
80C42×C127C2311yescharacteristic
81C22×D127—21no—
82Dic254—22no—
83Dic254—21no—
84Dic254—21no—
85Dic254—22no—
86Dic254—42no—
87D508—21no—
88D508—21no—
89C22⋊D127—42no—
90C2×Dic127—21no—
91C2×Dic127—21no—
92C2×Dic127—22no—
93C2×Dic127—22no—
94C2×Dic127—21no—
95C4×D127—42no—
96C23×C127C2×D411yescharacteristic
97C2×C4×C127—22no—
98C2×C4×C127—21no—
99C2×C4×C127—21no—
100C2×C4×C127—41no—
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2411yescharacteristic
107C8×C127—22no—
108C8×C127—22no—
109C22×C127—21no—
110C22×C127—21no—
111C22×C127—21no—
112C22×C127C22×D411yescharacteristic
113C2×D127—41no—
114C2×D127—41no—
115C2×D127—41no—
116C4×C127—21no—
117C4×C127—21no—
118C4×C127—21no—
119C4×C127—21no—
120C4×C127#13712yes—
121C4×C127H(5,𝔽2)11yescharacteristic
122C4×C1272−1+411yescharacteristic
123Dic127—41no—
124Dic127—42no—
125Dic127—41no—
126Dic127—41no—
127Dic127—42no—
128C2×C127—41no—
129C2×C127#53711yescharacteristic
130C2×C127C2×#13711yescharacteristic
131C2×C127#57511yescharacteristic
132D127—81no—
133#3252—1271nomaximal, Sylow
134C127#325211yescharacteristic, complemented, Sylow
135#533—1271no—
136#529—1271no—
137#479—1272no—
138#477—1271no—
139#476—1271no—
140#470—1272no—
141#468—1272no—
142#464—1272no—
143#462—1271no—
144#461—1271no—
145#423D12711yescharacteristic, complemented
146C2×Q16—1272no—
147C2×SD16—1272no—
148C2×M16C2×D12712yes—
149C4⋊Q8—1271no—
150C4⋊Q8—1271no—
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—1271no—
161C4×Q8—1272no—
162C4×Q8—2541no—
163C4×D4—1271no—
164C4×D4—2541no—
165#117—2541no—
166#117—2541no—
167C2×C42C2×D12711yescharacteristic
168#107—1272no—
169#106—1272no—
170C4⋊C8C2×D12711yescharacteristic
171C4⋊C8C2×D12711yescharacteristic
172C4⋊C8C2×D12712yes—
173#103—1272no—
174#103—1272no—
175#103—1272no—
176#102—1272no—
177C2×Q8—1271no—
178C2×Q8—1272no—
179C2×Q8—2541no—
180C2×D4—1271no—
181C2×D4—2541no—
182C22×C4—2541no—
183C22×C4D50811yescharacteristic
184C22×C4D50811yescharacteristic
185C22×C4C22×D12711yescharacteristic
186Q16—5082no—
187SD16—5082no—
188M16—42no—
189C2×C8C22×D12712yes—
190C2×C8C22×D12712yes—
191C4⋊C4—1272no—
192C4⋊C4—1271no—
193C4⋊C4—1271no—
194C4⋊C4—2541no—
195C4⋊C4—2541no—
196C4⋊C4—2541no—
197C4⋊C4—2542no—
198C4⋊C4—2541no—
199C4⋊C4—2542no—
200C4⋊C4—2541no—
201C4⋊C4—2541no—
202C4⋊C4—2541no—
203C22⋊C4—2542no—
204C22⋊C4—2541no—
205C22⋊C4—2541no—
206C22⋊C4—2541no—
207C22⋊C4—2541no—
208C22⋊C4—2541no—
209C42—2541no—
210C42—2542no—
211C42D50811yescharacteristic
212C42D50811yescharacteristic
213C42C22×D12711yescharacteristic
214C42C22×D12711yescharacteristic
215C23—2541no—
216C23C2×D50811yescharacteristic
217Q8—5082no—
218Q8—2542no—
219Q8—2541no—
220Q8—2541no—
221Q8—2542no—
222D4—5082no—
223D4—2541no—
224D4—2541no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—41no—
229C2×C4—5082no—
230C2×C4—2541no—
231C2×C4—2541no—
232C2×C4—2542no—
233C2×C4—2542no—
234C2×C4—2541no—
235C2×C4C2×D50811yescharacteristic
236C2×C4C2×D50811yescharacteristic
237C2×C4C2×D50811yescharacteristic
238C2×C4C2×D50811yescharacteristic
239C2×C4C2×D50811yescharacteristic
240C2×C4C23×D12711yescharacteristic
241C8—22no—
242C8—22no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—5081no—
247C22—5081no—
248C22—5081no—
249C22C22×D50811yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—5081no—
255C4—5081no—
256C4—5082no—
257C4—5081no—
258C4—5082no—
259C4#1686124212yes—
260C4#1686125411yescharacteristic
261C4#1686125711yescharacteristic
262C2—41no—
263C2—10161no—
264C2#2255045711yescharacteristic
265C2C2×#1686124211yescharacteristic
266C2#2255067611yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#423×C127C2
socleC22×C127C22×D4
prime radical#16861183C22
hypercenter#423D127
hypocenterC127#3252
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
1. #22550433C2semidirect
3. #22550184C2semidirect
4. #22550172C2semidirect
5. #22550167C2semidirect
6. #22550134C2semidirect
8. #22550099C2semidirect
9. #22550087C2semidirect
10. #22550082C2semidirect
11. #423×C127C2semidirect
134. C127#3252semidirect
145. #423D127semidirect

Maximal subgroups

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

subgroupindex
#225504332
#225504102
#225501842
#225501722
#225501672
#225501342
#225501222
#225500992
#225500872
#225500822
#423×C1272
#3252127