Finite Group Databasebeta

← #30649221

Subgroups — #30649221

Order 16256.

532 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#16861240C211yesmaximal, characteristic, complemented
2C2×#16861239C211yesmaximal, characteristic, complemented
3C2×#16861229C211yesmaximal, characteristic
4C22×#14523238C211yesmaximal, characteristic, complemented
5#22550452C211yesmaximal, characteristic, complemented
6#22550271C212yesmaximal, complemented
7#22550268C212yesmaximal, complemented
8#22550258C212yesmaximal, complemented
9#22550255C212yesmaximal, complemented
10#22549832C211yesmaximal, characteristic, complemented
11#536×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×Dic254C2211yescharacteristic
14#16861240—24no—
15#16861239—24no—
16#16861229—24no—
17C2×#14523238—24no—
18C2×#14523238C2212yes—
19C2×#14523238C2212yes—
20C2×#14523238C2211yescharacteristic
21C2×#14523238C2212yes—
22C2×#14523238C2212yes—
23C2×#14523236C2211yescharacteristic
24C2×#14523236C2212yes—
25C2×#14523235C2211yescharacteristic
26C2×#14523235C2212yes—
27C2×#14523233C2211yescharacteristic
28C2×#14523232C2211yescharacteristic
29C22×C127⋊C8C2211yescharacteristic
30#16861209C2212yes—
31#16861197C2212yes—
32#16861193—22no—
33#16861189—22no—
34#16861120C2212yes—
35C4×C127⋊C8C2212yes—
36C2×#41×C127C2211yescharacteristic
37C22×Q8×C127C2211yescharacteristic
38C4⋊Q8×C127C2212yes—
39#124×C127C2212yes—
40#123×C127—22no—
41C22⋊Q8×C127—22no—
42#117×C127C2211yescharacteristic
43#14523248—22no—
44#14523248—22no—
45C2×Dic254—21no—
46C2×Dic254—22no—
47C2×Dic254—22no—
48C2×Dic254C2312yes—
49C2×Dic254C2311yescharacteristic
50C2×D508C2311yescharacteristic
51C2×C22⋊D127—21no—
52C22×Dic127—21no—
53C2×C4×D127—21no—
54#14523238—24no—
55#14523238—24no—
56#14523238—22no—
57#14523238—24no—
58#14523238—24no—
59#14523236—22no—
60#14523236—24no—
61#14523235—22no—
62#14523235—24no—
63#14523233—22no—
64#14523232D414yes—
65C2×C127⋊C8D414yes—
66C2×C127⋊C8C2311yescharacteristic
67C2×C127⋊C8C2311yescharacteristic
68C2×C127⋊C8C2312yes—
69C4⋊Dic127—24no—
70(C2×D127)⋊C4—24no—
71#41×C127—22no—
72#41×C127—22no—
73C2×Q8×C127—21no—
74C2×Q8×C127—22no—
75C2×Q8×C127—22no—
76C2×Q8×C127C2312yes—
77C2×Q8×C127C2311yescharacteristic
78C2×D4×C127—21no—
79C2×D4×C127C2311yescharacteristic
80C22×C4×C127—21no—
81C22×C4×C127—21no—
82C22×C4×C127C2311yescharacteristic
83C4⋊C4×C127—24no—
84C4⋊C4×C127D412yes—
85C22⋊C4×C127—24no—
86C22⋊C4×C127D412yes—
87C42×C127C2312yes—
88C22×D127—21no—
89Dic254—22no—
90Dic254—21no—
91Dic254—22no—
92Dic254—21no—
93Dic254—42no—
94D508—21no—
95D508—21no—
96C22⋊D127—42no—
97C2×Dic127—22no—
98C2×Dic127—21no—
99C2×Dic127—42no—
100C4×D127—42no—
101C127⋊C8C2×D412yes—
102C127⋊C8C2×D412yes—
103C127⋊C8C2×D414yes—
104C23×C127—21no—
105C23×C127C2×D411yescharacteristic
106Q8×C127—22no—
107Q8×C127—22no—
108Q8×C127—21no—
109Q8×C127—21no—
110Q8×C127—42no—
111D4×C127—21no—
112D4×C127—21no—
113D4×C127—42no—
114C2×C4×C127—22no—
115C2×C4×C127—21no—
116C2×C4×C127—42no—
117C2×C4×C127—42no—
118C2×C4×C127C2×D412yes—
119C2×C4×C127C2×D412yes—
120C2×C4×C127C2×D414yes—
121C2×C4×C127C2×D411yescharacteristic
122C2×C4×C127C2411yescharacteristic
123C22×C127—22no—
124C22×C127—41no—
125C22×C127—42no—
126C22×C127#12712yes—
127C22×C127C22×D411yescharacteristic
128C2×D127—41no—
129C2×D127—42no—
130C4×C127—24no—
131C4×C127—41no—
132C4×C127—42no—
133C4×C127#12712yes—
134C4×C127C22×D411yescharacteristic
135C4×C127C22×D411yescharacteristic
136Dic127—41no—
137Dic127—42no—
138C2×C127—22no—
139C2×C127—81no—
140C2×C127C2×#12711yescharacteristic
141C2×C127#57812yes—
142D127—81no—
143#3156—1271nomaximal, Sylow
144C127#315611yescharacteristic, complemented, Sylow
145C2×#137—1271no—
146C2×#137—1271no—
147C2×#135—1271no—
148C22×Q16—1271no—
149#536—1271no—
150#536D12711yescharacteristic, complemented
151#497—1272no—
152#497—1272no—
153#495—1272no—
154#494—1272no—
155#405—1271no—
156C2×#41—1271no—
157C2×#41C2×D12711yescharacteristic
158C22×Q8—1271no—
159C22×Q8C2×D12711yescharacteristic
160#137—2544no—
161#137—2544no—
162#135—2544no—
163C2×Q16—1272no—
164C2×Q16—1272no—
165C2×Q16—1271no—
166C2×Q16—1272no—
167C2×Q16—1272no—
168C2×Q16—2544no—
169C2×SD16—1271no—
170C2×SD16—1272no—
171C2×SD16—1271no—
172C2×SD16—1272no—
173C2×D8—1271no—
174C2×M16—1271no—
175C22×C8—1271no—
176C4⋊Q8—1272no—
177C4⋊Q8C2×D12712yes—
178#124—1272no—
179#124C2×D12712yes—
180#123—22no—
181#123—2542no—
182C22⋊Q8—22no—
183C22⋊Q8—2542no—
184#117C2×D12711yescharacteristic
185#97—1272no—
186C4×C8—1272no—
187#41—22no—
188#41—22no—
189#41—2542no—
190#41—2542no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—1272no—
195C2×Q8—1271no—
196C2×Q8—2542no—
197C2×Q8—2542no—
198C2×Q8—2541no—
199C2×Q8C22×D12712yes—
200C2×Q8C22×D12711yescharacteristic
201C2×D4—21no—
202C2×D4—1271no—
203C2×D4—2541no—
204C2×D4C22×D12711yescharacteristic
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—2541no—
208C22×C4—2541no—
209C22×C4C22×D12711yescharacteristic
210Q16—2544no—
211Q16—2544no—
212Q16—2542no—
213Q16—2544no—
214Q16—2544no—
215SD16—2542no—
216SD16—2544no—
217SD16—2542no—
218SD16—2544no—
219D8—2542no—
220M16—1274no—
221C2×C8—1274no—
222C2×C8—1271no—
223C2×C8—1272no—
224C2×C8—1271no—
225C4⋊C4—24no—
226C4⋊C4—2544no—
227C4⋊C4C22⋊D12712yes—
228C22⋊C4—24no—
229C22⋊C4—2544no—
230C22⋊C4C22⋊D12712yes—
231C42C22×D12712yes—
232C23—21no—
233C23—2541no—
234C23C2×C22⋊D12711yescharacteristic
235Q8—22no—
236Q8—21no—
237Q8—22no—
238Q8—21no—
239Q8—42no—
240Q8—5082no—
241Q8—2542no—
242Q8—2541no—
243Q8—2542no—
244Q8—2541no—
245D4—21no—
246D4—21no—
247D4—42no—
248D4—5082no—
249D4—2541no—
250D4—2541no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—5082no—
256C2×C4—5082no—
257C2×C4—2542no—
258C2×C4—2541no—
259C2×C4D4×D12712yes—
260C2×C4D4×D12712yes—
261C2×C4C2×C22⋊D12714yes—
262C2×C4C2×C22⋊D12711yescharacteristic
263C2×C4C23×D12711yescharacteristic
264C8—1272no—
265C8—1272no—
266C8—1274no—
267C22—22no—
268C22—41no—
269C22—42no—
270C22—5081no—
271C22—5082no—
272C22#1686120512yes—
273C22C22×C22⋊D12711yescharacteristic
274C4—24no—
275C4—41no—
276C4—42no—
277C4—5081no—
278C4—5082no—
279C4#1686120512yes—
280C4C2×D4×D12711yescharacteristic
281C4C2×D4×D12711yescharacteristic
282C2—22no—
283C2—81no—
284C2—10161no—
285C2C2×#1686120511yescharacteristic
286C2#2255069112yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#536×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861229C2
hypercenter#536D127
hypocenterC127#3156
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
2. C2×#16861239C2semidirect
4. C22×#14523238C2semidirect
5. #22550452C2semidirect
6. #22550271C2semidirect
7. #22550268C2semidirect
8. #22550258C2semidirect
9. #22550255C2semidirect
10. #22549832C2semidirect
11. #536×C127C2semidirect
144. C127#3156semidirect
150. #536D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
C2×#168612392
C2×#168612292
C22×#145232382
#225504522
#225502712
#225502682
#225502582
#225502552
#225498322
#536×C1272
#3156127