Finite Group Databasebeta

← D4×#14523237 #30650428

Subgroups — D4×#14523237 #30650428

Order 16256.

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

normal subgroups only

This group has 546 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
1C22×#14523237C212yesmaximal, complemented
2D4×Dic254C211yesmaximal, characteristic, complemented
3D4×D508C211yesmaximal, characteristic
4#22550249C211yesmaximal, characteristic, complemented
5#22550105C212yesmaximal, complemented
6#22550082C211yesmaximal, characteristic, complemented
7#22550077C211yesmaximal, characteristic, complemented
8#22550029C212yesmaximal, complemented
9#22550017C212yesmaximal, complemented
10C4×#14523237C211yesmaximal, characteristic, complemented
11C8×D4×C127C211yesmaximal, characteristic, complemented
12C22×Dic254C2212yescomplemented
13C22×D508C2212yes—
14C2×D4×D127—21no—
15C2×#14523237—28no—
16C2×#14523237C2212yes—
17C2×#14523237C2211yescharacteristic
18C2×#14523237C2212yescomplemented
19C2×#14523237C2212yescomplemented
20C2×#14523237C2212yescomplemented
21#16861209C2211yescharacteristic, complemented
22#16861208—21no—
23#16861206C2211yescharacteristic
24#16861188C2212yescomplemented
25#16861186—22no—
26#16861184—21no—
27#16861183C2212yes—
28#16861179—22no—
29C4×Dic254C2211yescharacteristic, complemented
30C4×D508C2211yescharacteristic
31D4×Dic127—21no—
32#16861141C2211yescharacteristic, complemented
33#16861134C2211yescharacteristic, complemented
34#16861134C2212yescomplemented
35#16861131C2211yescharacteristic
36#16861131C2212yescomplemented
37C22×C8×C127C2212yescomplemented
38C4×D4×C127C2211yescharacteristic
39C4⋊C8×C127C2211yescharacteristic, complemented
40C22⋊C8×C127C2212yescomplemented
41C4×C8×C127C2211yescharacteristic, complemented
42C23×D127—22no—
43C2×Dic254—24no—
44C2×Dic254—22no—
45C2×Dic254—42no—
46C2×Dic254C2312yes—
47C2×Dic254C2311yescharacteristic
48C2×D508—24no—
49C2×D508—21no—
50C2×D508—22no—
51C2×D508—21no—
52C2×D508—42no—
53C2×D508C2312yes—
54C2×D508C2311yescharacteristic
55C2×C22⋊D127—22no—
56D4×D127—44no—
57C22×Dic127—22no—
58C2×C4×D127—21no—
59#14523237—24no—
60#14523237—24no—
61#14523237—24no—
62#14523237—42no—
63#14523237D414yesdirect product factor
64C4⋊Dic127—21no—
65C4⋊Dic127—21no—
66C4⋊Dic127C2311yescharacteristic
67C4⋊Dic127C2311yescharacteristic
68Dic127⋊C4—21no—
69Dic127⋊C4—22no—
70C22⋊Dic127—22no—
71(C2×D127)⋊C4—22no—
72(C2×D127)⋊C4—21no—
73C4×Dic127—21no—
74C2×D4×C127D411yescharacteristic
75C22×C4×C127C2312yes—
76C2×C8×C127—24no—
77C2×C8×C127C2312yes—
78C2×C8×C127C2311yescharacteristic
79C2×C8×C127C2311yescharacteristic
80C4⋊C4×C127D411yescharacteristic
81C22⋊C4×C127D412yes—
82C42×C127C2311yescharacteristic
83C22×D127—21no—
84C22×D127—22no—
85C22×D127—44no—
86C22×D127—44no—
87C22×D127—82no—
88Dic254—21no—
89Dic254—22no—
90Dic254—22no—
91Dic254—44no—
92Dic254C2×D412yescomplemented
93D508—22no—
94D508—21no—
95D508—22no—
96D508—44no—
97D508—42no—
98D508—81no—
99D508C2×D412yes—
100C22⋊D127—44no—
101C2×Dic127—21no—
102C2×Dic127—21no—
103C2×Dic127—21no—
104C2×Dic127—22no—
105C2×Dic127—44no—
106C4×D127—42no—
107C23×C127C2×D412yes—
108D4×C127SD1614yescomplemented
109C2×C4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2×D412yes—
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127C2411yescharacteristic
115C8×C127—22no—
116C8×C127—21no—
117C8×C127C2×D412yescomplemented
118C22×C127—22no—
119C22×C127—22no—
120C22×C127C2×SD1614yescomplemented
121C22×C127C22×D411yescharacteristic
122C2×D127—22no—
123C2×D127—41no—
124C2×D127—41no—
125C2×D127—42no—
126C2×D127—42no—
127C2×D127—42no—
128C2×D127—88no—
129C4×C127—22no—
130C4×C127—21no—
131C4×C127C2×SD1612yescomplemented
132C4×C127C22×D411yescharacteristic
133C4×C127H(5,𝔽2)11yescharacteristic
134Dic127—22no—
135Dic127—42no—
136Dic127—41no—
137Dic127—41no—
138C2×C127—24no—
139C2×C127D4211yescharacteristic
140C2×C127C22×SD1611yescharacteristic
141C2×C127#57711yescharacteristic
142D127—42no—
143D127—82no—
144D4×SD16—1271nomaximal, Sylow
145C127D4×SD1611yescharacteristic, complemented, Sylow
146C22×SD16—1272no—
147D4×Q8—1271no—
148D42—1271no—
149#492—1271no—
150#465—1272no—
151#461—1271no—
152#460—1271no—
153#450—1272no—
154#448—1272no—
155C4×SD16—1271no—
156C8×D4D12711yescharacteristic, complemented
157C22×Q8—1272no—
158C22×D4—1272no—
159C22×D4—2541no—
160C2×SD16—1271no—
161C2×SD16—1272no—
162C2×SD16—1272no—
163C2×SD16—1272no—
164C2×SD16—1272no—
165C2×SD16—2548no—
166C22×C8C2×D12712yescomplemented
167C4⋊Q8—1271no—
168C4⋊Q8—2541no—
169#127—1271no—
170C22⋊Q8—1272no—
171C22⋊Q8—2542no—
172#121—1272no—
173#121—2541no—
174#120—2542no—
175C4×Q8—1271no—
176C4×D4—1271no—
177C4×D4—2541no—
178C4×D4C2×D12711yescharacteristic
179#106—1271no—
180C4⋊C8C2×D12711yescharacteristic, complemented
181#103—1271no—
182#103—1272no—
183#102—1271no—
184#102—1272no—
185C22⋊C8C2×D12712yescomplemented
186C4×C8C2×D12711yescharacteristic, complemented
187C24—2542no—
188C2×Q8—1271no—
189C2×Q8—1272no—
190C2×Q8—5082no—
191C2×Q8—2544no—
192C2×Q8—2542no—
193C2×D4—1271no—
194C2×D4—1272no—
195C2×D4—5082no—
196C2×D4—5084no—
197C2×D4—2544no—
198C2×D4—2542no—
199C2×D4—2541no—
200C2×D4—2542no—
201C2×D4—2541no—
202C2×D4D50811yescharacteristic
203C22×C4—2542no—
204C22×C4—2541no—
205C22×C4C22×D12712yes—
206SD16—1274no—
207SD16—5082no—
208SD16—2544no—
209SD16—2544no—
210SD16—2544no—
211C2×C8—24no—
212C2×C8C22×D12711yescharacteristic
213C2×C8C22×D12712yes—
214C2×C8C22×D12711yescharacteristic
215C4⋊C4—1271no—
216C4⋊C4—1271no—
217C4⋊C4—2541no—
218C4⋊C4—2542no—
219C4⋊C4—2541no—
220C4⋊C4—2541no—
221C4⋊C4D50811yescharacteristic
222C22⋊C4—2542no—
223C22⋊C4—2541no—
224C22⋊C4—2542no—
225C22⋊C4D50812yes—
226C42—2541no—
227C42C22×D12711yescharacteristic
228C23—10162no—
229C23—5084no—
230C23—5084no—
231C23—2542no—
232C23—2541no—
233C23C2×D50812yes—
234Q8—1272no—
235Q8—5084no—
236Q8—2542no—
237Q8—2542no—
238Q8—2541no—
239D4—1272no—
240D4—10161no—
241D4—5084no—
242D4—5084no—
243D4—5082no—
244D4—2542no—
245D4—2542no—
246D4—2541no—
247D4#1452323714yesdirect product factor
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—5084no—
251C2×C4—5082no—
252C2×C4—2541no—
253C2×C4—2541no—
254C2×C4—2542no—
255C2×C4—2541no—
256C2×C4C2×D50811yescharacteristic
257C2×C4C2×D50812yes—
258C2×C4C2×D50811yescharacteristic
259C2×C4C23×D12711yescharacteristic
260C8—22no—
261C8—21no—
262C8D4×D12712yescomplemented
263C22—22no—
264C22—22no—
265C22—10168no—
266C22—5082no—
267C22—5082no—
268C22—5082no—
269C22—5081no—
270C22—5081no—
271C22—2542no—
272C22C2×#1452323714yescomplemented
273C22C22×D50811yescharacteristic
274C4—21no—
275C4—22no—
276C4—5081no—
277C4—5081no—
278C4—5082no—
279C4—2542no—
280C4C2×#1452323712yescomplemented
281C4C2×D4×D12711yescharacteristic
282C4#1686125511yescharacteristic
283C2—24no—
284C2—10162no—
285C2—5082no—
286C2D4×D50811yescharacteristic
287C2C22×#1452323711yescharacteristic
288C2#2255068811yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC8×D4×C127C2
socleC22×C127C22×D4
prime radicalD4×D508C2
hypercenterC8×D4D127
hypocenterC127D4×SD16
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C22×#14523237C2semidirect
2. D4×Dic254C2semidirect
4. #22550249C2semidirect
5. #22550105C2semidirect
6. #22550082C2semidirect
7. #22550077C2semidirect
8. #22550029C2semidirect
9. #22550017C2semidirect
10. C4×#14523237C2semidirect
11. C8×D4×C127C2semidirect
12. C22×Dic254C22semidirect
18. C2×#14523237C22semidirect
19. C2×#14523237C22semidirect
20. C2×#14523237C22semidirect
21. #16861209C22semidirect
24. #16861188C22semidirect
29. C4×Dic254C22semidirect
32. #16861141C22semidirect
33. #16861134C22semidirect
34. #16861134C22semidirect
36. #16861131C22semidirect
37. C22×C8×C127C22semidirect
39. C4⋊C8×C127C22semidirect
40. C22⋊C8×C127C22semidirect
41. C4×C8×C127C22semidirect
63. #14523237D4direct (N × H)
92. Dic254C2×D4semidirect
108. D4×C127SD16semidirect
117. C8×C127C2×D4semidirect
120. C22×C127C2×SD16semidirect
131. C4×C127C2×SD16semidirect
145. C127D4×SD16semidirect
156. C8×D4D127semidirect
166. C22×C8C2×D127semidirect
180. C4⋊C8C2×D127semidirect
185. C22⋊C8C2×D127semidirect
186. C4×C8C2×D127semidirect
247. D4#14523237direct (N × H)
262. C8D4×D127semidirect
272. C22C2×#14523237semidirect
280. C4C2×#14523237semidirect

Maximal subgroups

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

subgroupindex
C22×#145232372
D4×Dic2542
D4×D5082
#225502492
#225501052
#225500822
#225500772
#225500292
#225500172
C4×#145232372
C8×D4×C1272
D4×SD16127