Finite Group Databasebeta

← #30648467

Subgroups — #30648467

Order 16256.

446 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 145 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#22550674C211yesmaximal, characteristic
2C2×#16861242C211yesmaximal, characteristic, complemented
3C2×#16861239C211yesmaximal, characteristic, complemented
4#22550151C214yesmaximal, complemented
5#22550139C214yesmaximal, complemented
6#22549925C211yesmaximal, characteristic, complemented
7#22549858C211yesmaximal, characteristic, complemented
8C2×#16861127C211yesmaximal, characteristic, complemented
9#568×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11C22×Dic254C2211yescharacteristic
12#16861242—22no—
13#16861242—22no—
14#16861240—21no—
15#16861239—22no—
16#16861239C2214yes—
17#16861239C2212yes—
18#16861235—21no—
19#16861234C2212yes—
20#16861229—22no—
21C2×Dic508C2212yescomplemented
22C2×#14523238—21no—
23C2×#14523237C2212yescomplemented
24C2×#14523236—21no—
25C2×#14523232C2211yescharacteristic
26C2×#14523232C2211yescharacteristic
27#16861145C2214yescomplemented
28#16861127C2214yescomplemented
29#16861127C2212yescomplemented
30#16861125C2212yescomplemented
31C2×#41×C127C2211yescharacteristic
32#131×C127—22no—
33#131×C127C2214yes—
34C2×M16×C127—22no—
35C2×M16×C127C2211yescharacteristic
36C2×M16×C127C2211yescharacteristic
37#14523248—21no—
38#14523248—21no—
39#14523248—21no—
40#14523248—21no—
41C2×Dic254—21no—
42C2×Dic254—21no—
43C2×Dic254—21no—
44C2×Dic254—21no—
45C2×Dic254—21no—
46C2×Dic254C2312yes—
47C2×Dic254C2311yescharacteristic
48C2×D508C2311yescharacteristic
49C2×C22⋊D127—21no—
50C22×Dic127—21no—
51C2×C4×D127—21no—
52Dic508—24no—
53#14523238—24no—
54#14523238—22no—
55#14523237—24no—
56#14523236—24no—
57#14523236—22no—
58#14523235—22no—
59#14523233—22no—
60#14523232—21no—
61#14523232—21no—
62#14523232C2312yes—
63#14523232C2312yes—
64C2×C127⋊C8—21no—
65C2×C127⋊C8—21no—
66#41×C127—21no—
67#41×C127—21no—
68#41×C127D412yes—
69#41×C127C2312yes—
70C2×Q8×C127D411yescharacteristic
71C2×D4×C127—21no—
72C2×D4×C127D411yescharacteristic
73C22×C4×C127—21no—
74C22×C4×C127C2311yescharacteristic
75M16×C127—21no—
76M16×C127—24no—
77M16×C127D412yes—
78M16×C127D412yes—
79M16×C127C2312yes—
80C2×C8×C127—21no—
81C2×C8×C127—24no—
82C2×C8×C127C2312yes—
83C22×D127—21no—
84Dic254—22no—
85Dic254—21no—
86Dic254—22no—
87Dic254—21no—
88Dic254—41no—
89Dic254—41no—
90D508—21no—
91D508—21no—
92C22⋊D127—41no—
93C22⋊D127—41no—
94C2×Dic127—21no—
95C2×Dic127—22no—
96C2×Dic127—41no—
97C2×Dic127—41no—
98C4×D127—41no—
99C4×D127—41no—
100C127⋊C8—22no—
101C127⋊C8—22no—
102C23×C127—21no—
103C23×C127#4111yescharacteristic
104Q8×C127—21no—
105Q8×C127C2×D412yes—
106D4×C127—22no—
107D4×C127—22no—
108D4×C127—21no—
109D4×C127C2×D412yes—
110C2×C4×C127—21no—
111C2×C4×C127—22no—
112C2×C4×C127—22no—
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127#4111yescharacteristic
118C2×C4×C127C2411yescharacteristic
119C8×C127—22no—
120C8×C127C2×D414yes—
121C22×C127—22no—
122C22×C127—41no—
123C22×C127—81no—
124C22×C127#12111yescharacteristic
125C22×C127#12111yescharacteristic
126C22×C127C2×#4111yescharacteristic
127C2×D127—41no—
128C2×D127—81no—
129C4×C127—22no—
130C4×C127#12111yescharacteristic
131C4×C127#12111yescharacteristic
132C4×C127C22×D411yescharacteristic
133C4×C127C22×D411yescharacteristic
134Dic127—41no—
135Dic127—42no—
136C2×C127—21no—
137C2×C127—21no—
138C2×C127—21no—
139C2×C127—42no—
140C2×C127C2×#12111yescharacteristic
141D127—81no—
142#3069—1271nomaximal, Sylow
143C127#306911yescharacteristic, complemented, Sylow
144#575—1271no—
145C2×#137—1271no—
146C2×#137—1271no—
147#568D12711yescharacteristic, complemented
148#473—1274no—
149#471—1274no—
150#430—1271no—
151#413—1271no—
152C2×#101—1271no—
153C2×#41—1271no—
154C2×#41C2×D12711yescharacteristic
155C22×Q8—1271no—
156#137—1274no—
157#137—1272no—
158#137—2542no—
159#137—2541no—
160#137—2542no—
161#137—2542no—
162AGL(1,ℤ/8)—1272no—
163AGL(1,ℤ/8)—2541no—
164#135—2542no—
165C2×Q16—1272no—
166C2×Q16—2541no—
167C2×SD16—1272no—
168C2×SD16—2541no—
169#131—22no—
170#131C2×D12714yes—
171C2×M16—22no—
172C2×M16—1271no—
173C2×M16—1271no—
174C2×M16C2×D12711yescharacteristic
175C2×M16C2×D12711yescharacteristic
176#108—1274no—
177#101—1274no—
178#101—1272no—
179#100—1272no—
180#41—21no—
181#41—21no—
182#41—2541no—
183#41—2541no—
184#41—2541no—
185#41—2541no—
186#41D50812yes—
187#41C22×D12712yes—
188C2×Q8—1272no—
189C2×Q8—1271no—
190C2×Q8—2541no—
191C2×Q8—2541no—
192C2×Q8—2541no—
193C2×Q8—2541no—
194C2×Q8—2541no—
195C2×Q8D50811yescharacteristic
196C2×D4—21no—
197C2×D4—1271no—
198C2×D4—2541no—
199C2×D4D50811yescharacteristic
200C22×C4—21no—
201C22×C4—2541no—
202C22×C4—2541no—
203C22×C4C22×D12711yescharacteristic
204Q16—2544no—
205Q16—2542no—
206Q16—2544no—
207SD16—2542no—
208SD16—2544no—
209SD16—2542no—
210SD16—2544no—
211D8—2542no—
212M16—21no—
213M16—24no—
214M16—1272no—
215M16—1272no—
216M16—2541no—
217M16—2541no—
218M16C22⋊D12712yes—
219M16C22⋊D12712yes—
220M16C22×D12712yes—
221C2×C8—21no—
222C2×C8—24no—
223C2×C8—2541no—
224C2×C8—2541no—
225C2×C8C22×D12712yes—
226C23—21no—
227C23—2541no—
228C23#1452324811yescharacteristic
229Q8—21no—
230Q8—5081no—
231Q8—5081no—
232Q8—2542no—
233Q8—2541no—
234Q8—2542no—
235Q8—2541no—
236Q8C2×D50812yes—
237D4—22no—
238D4—22no—
239D4—21no—
240D4—5081no—
241D4—5081no—
242D4—2541no—
243D4—2541no—
244D4C2×D50812yes—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—5081no—
249C2×C4—5081no—
250C2×C4—5081no—
251C2×C4—5081no—
252C2×C4—2541no—
253C2×C4—2542no—
254C2×C4C2×C22⋊D12711yescharacteristic
255C2×C4C2×C22⋊D12711yescharacteristic
256C2×C4C2×D50811yescharacteristic
257C2×C4C2×D50811yescharacteristic
258C2×C4#1452324811yescharacteristic
259C2×C4C23×D12711yescharacteristic
260C8—22no—
261C8—2542no—
262C8—2542no—
263C8C2×C22⋊D12714yes—
264C22—22no—
265C22—41no—
266C22—81no—
267C22—10161no—
268C22—5081no—
269C22#1686118311yescharacteristic
270C22#1686118311yescharacteristic
271C22C2×#1452324811yescharacteristic
272C4—22no—
273C4—5081no—
274C4—5082no—
275C4#1686118311yescharacteristic
276C4#1686118311yescharacteristic
277C4C22×C22⋊D12711yescharacteristic
278C4C22×D50811yescharacteristic
279C2—21no—
280C2—21no—
281C2—21no—
282C2—42no—
283C2—10161no—
284C2C2×#1686118311yescharacteristic
subgroupG/H
centerC2C2×#16861183
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#568×C127C2
socleC2×C127C2×#121
prime radical#22550674C2
hypercenter#568D127
hypocenterC127#3069
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. C2×#16861242C2semidirect
3. C2×#16861239C2semidirect
4. #22550151C2semidirect
5. #22550139C2semidirect
6. #22549925C2semidirect
7. #22549858C2semidirect
8. C2×#16861127C2semidirect
9. #568×C127C2semidirect
21. C2×Dic508C22semidirect
23. C2×#14523237C22semidirect
27. #16861145C22semidirect
28. #16861127C22semidirect
29. #16861127C22semidirect
30. #16861125C22semidirect
143. C127#3069semidirect
147. #568D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
C2×#168612422
C2×#168612392
#225501512
#225501392
#225499252
#225498582
C2×#168611272
#568×C1272
#3069127