Finite Group Databasebeta

← D4×#14523235 #30650426

Subgroups — D4×#14523235 #30650426

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×#14523235C212yesmaximal, complemented
2D4×D508C211yesmaximal, characteristic
3#22550250C211yesmaximal, characteristic, complemented
4#22550107C212yesmaximal, complemented
5#22550078C211yesmaximal, characteristic, complemented
6#22550076C211yesmaximal, characteristic, complemented
7#22550026C212yesmaximal, complemented
8#22550015C212yesmaximal, complemented
9C4×#14523235C211yesmaximal, characteristic, complemented
10D4×C127⋊C8C211yesmaximal, characteristic, complemented
11D4×Q8×C127C211yesmaximal, characteristic, complemented
12C22×D508C2212yes—
13C2×D4×D127—21no—
14C2×#14523235—28no—
15C2×#14523235C2212yes—
16C2×#14523235C2211yescharacteristic
17C2×#14523235C2212yescomplemented
18C2×#14523235C2212yescomplemented
19C2×#14523235C2212yescomplemented
20C22×C127⋊C8C2212yescomplemented
21#16861206C2211yescharacteristic
22#16861184—21no—
23#16861183C2212yes—
24#16861179—22no—
25C4×D508C2211yescharacteristic
26#16861140C2211yescharacteristic, complemented
27C4⋊(C127⋊C8)C2211yescharacteristic, complemented
28#16861133C2212yescomplemented
29#16861133C2211yescharacteristic, complemented
30#16861129C2211yescharacteristic
31#16861129C2212yescomplemented
32C22⋊(C127⋊C8)C2212yescomplemented
33C4×C127⋊C8C2211yescharacteristic, complemented
34C22×Q8×C127C2212yescomplemented
35C4⋊Q8×C127—21no—
36C4⋊Q8×C127C2211yescharacteristic, complemented
37C22⋊Q8×C127—22no—
38C22⋊Q8×C127C2212yescomplemented
39C4×Q8×C127C2211yescharacteristic, complemented
40C4×D4×C127—21no—
41C4×D4×C127C2211yescharacteristic
42C23×D127—22no—
43C2×D508—24no—
44C2×D508—21no—
45C2×D508—22no—
46C2×D508—21no—
47C2×D508—42no—
48C2×D508C2312yes—
49C2×D508C2311yescharacteristic
50C2×C22⋊D127—22no—
51D4×D127—44no—
52C2×C4×D127—21no—
53#14523235—24no—
54#14523235—24no—
55#14523235—24no—
56#14523235—42no—
57#14523235D414yesdirect product factor
58C2×C127⋊C8—24no—
59C2×C127⋊C8C2312yes—
60C2×C127⋊C8C2311yescharacteristic
61C2×C127⋊C8C2311yescharacteristic
62C4⋊Dic127C2311yescharacteristic
63(C2×D127)⋊C4—22no—
64(C2×D127)⋊C4—21no—
65C2×Q8×C127—22no—
66C2×Q8×C127—24no—
67C2×Q8×C127—42no—
68C2×Q8×C127C2312yes—
69C2×Q8×C127C2311yescharacteristic
70C2×D4×C127D411yescharacteristic
71C22×C4×C127—22no—
72C22×C4×C127C2312yes—
73C4⋊C4×C127—21no—
74C4⋊C4×C127—21no—
75C4⋊C4×C127—21no—
76C4⋊C4×C127—22no—
77C4⋊C4×C127D411yescharacteristic
78C4⋊C4×C127C2311yescharacteristic
79C22⋊C4×C127—22no—
80C22⋊C4×C127D412yes—
81C42×C127—21no—
82C42×C127C2311yescharacteristic
83C22×D127—21no—
84C22×D127—22no—
85C22×D127—44no—
86C22×D127—44no—
87C22×D127—82no—
88D508—22no—
89D508—21no—
90D508—22no—
91D508—44no—
92D508—42no—
93D508—81no—
94D508C2×D412yes—
95C22⋊D127—44no—
96C2×Dic127—21no—
97C4×D127—42no—
98C127⋊C8—22no—
99C127⋊C8—21no—
100C127⋊C8C2×D412yescomplemented
101C23×C127C2×D412yes—
102Q8×C127—21no—
103Q8×C127—22no—
104Q8×C127—22no—
105Q8×C127—44no—
106Q8×C127C2×D412yescomplemented
107D4×C127SD1614yescomplemented
108C2×C4×C127—21no—
109C2×C4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127—21no—
112C2×C4×C127—22no—
113C2×C4×C127—44no—
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2×D412yes—
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2411yescharacteristic
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—22no—
131C4×C127—21no—
132C4×C127—42no—
133C4×C127—41no—
134C4×C127C2×SD1612yescomplemented
135C4×C127C22×D411yescharacteristic
136C4×C127H(5,𝔽2)11yescharacteristic
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×Q8D12711yescharacteristic, complemented
148D42—1271no—
149#492—1271no—
150#465—1272no—
151#461—1271no—
152#460—1271no—
153#450—1272no—
154#448—1272no—
155C4×SD16—1271no—
156C8×D4—1271no—
157C22×Q8C2×D12712yescomplemented
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×C8—1272no—
167C4⋊Q8—21no—
168C4⋊Q8C2×D12711yescharacteristic, complemented
169#127—1271no—
170C22⋊Q8—22no—
171C22⋊Q8C2×D12712yescomplemented
172#121—1272no—
173#121—2541no—
174#120—2542no—
175C4×Q8C2×D12711yescharacteristic, complemented
176C4×D4—21no—
177C4×D4—1271no—
178C4×D4C2×D12711yescharacteristic
179#106—1271no—
180C4⋊C8—1271no—
181#103—1271no—
182#103—1272no—
183#102—1271no—
184#102—1272no—
185C22⋊C8—1272no—
186C4×C8—1271no—
187C24—2542no—
188C2×Q8—22no—
189C2×Q8—24no—
190C2×Q8—42no—
191C2×Q8C22×D12712yes—
192C2×Q8C22×D12711yescharacteristic
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×D4C22⋊D12711yescharacteristic
203C22×C4—22no—
204C22×C4—2541no—
205C22×C4C22×D12712yes—
206SD16—1274no—
207SD16—5082no—
208SD16—2544no—
209SD16—2544no—
210SD16—2544no—
211C2×C8—1271no—
212C2×C8—1271no—
213C2×C8—1272no—
214C2×C8—2544no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—22no—
219C4⋊C4—1271no—
220C4⋊C4C22⋊D12711yescharacteristic
221C4⋊C4C22×D12711yescharacteristic
222C22⋊C4—22no—
223C22⋊C4—2541no—
224C22⋊C4—2542no—
225C22⋊C4C22⋊D12712yes—
226C42—21no—
227C42C22×D12711yescharacteristic
228C23—10162no—
229C23—5084no—
230C23—5084no—
231C23—2542no—
232C23—2541no—
233C23C2×C22⋊D12712yes—
234Q8—21no—
235Q8—22no—
236Q8—22no—
237Q8—44no—
238Q8D4×D12712yescomplemented
239D4—1272no—
240D4—10161no—
241D4—5084no—
242D4—5084no—
243D4—5082no—
244D4—2542no—
245D4—2542no—
246D4—2541no—
247D4#1452323514yesdirect product factor
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—44no—
254C2×C4—5082no—
255C2×C4—2541no—
256C2×C4C2×C22⋊D12711yescharacteristic
257C2×C4C2×C22⋊D12712yes—
258C2×C4C2×C22⋊D12711yescharacteristic
259C2×C4C23×D12711yescharacteristic
260C8—1272no—
261C8—2542no—
262C8—2541no—
263C22—22no—
264C22—22no—
265C22—10168no—
266C22—5082no—
267C22—5082no—
268C22—5082no—
269C22—5081no—
270C22—5081no—
271C22—2542no—
272C22C2×#1452323514yescomplemented
273C22C22×C22⋊D12711yescharacteristic
274C4—21no—
275C4—22no—
276C4—22no—
277C4—41no—
278C4—42no—
279C4—5081no—
280C4C2×#1452323512yescomplemented
281C4C2×D4×D12711yescharacteristic
282C4#1686125411yescharacteristic
283C2—24no—
284C2—10162no—
285C2—5082no—
286C2D4×C22⋊D12711yescharacteristic
287C2C22×#1452323511yescharacteristic
288C2#2255068411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingD4×Q8×C127C2
socleC22×C127C22×D4
prime radicalD4×D508C2
hypercenterD4×Q8D127
hypocenterC127D4×SD16
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14523235C2semidirect
3. #22550250C2semidirect
4. #22550107C2semidirect
5. #22550078C2semidirect
6. #22550076C2semidirect
7. #22550026C2semidirect
8. #22550015C2semidirect
9. C4×#14523235C2semidirect
10. D4×C127⋊C8C2semidirect
11. D4×Q8×C127C2semidirect
17. C2×#14523235C22semidirect
18. C2×#14523235C22semidirect
19. C2×#14523235C22semidirect
20. C22×C127⋊C8C22semidirect
26. #16861140C22semidirect
27. C4⋊(C127⋊C8)C22semidirect
28. #16861133C22semidirect
29. #16861133C22semidirect
31. #16861129C22semidirect
32. C22⋊(C127⋊C8)C22semidirect
33. C4×C127⋊C8C22semidirect
34. C22×Q8×C127C22semidirect
36. C4⋊Q8×C127C22semidirect
38. C22⋊Q8×C127C22semidirect
39. C4×Q8×C127C22semidirect
57. #14523235D4direct (N × H)
100. C127⋊C8C2×D4semidirect
106. Q8×C127C2×D4semidirect
107. D4×C127SD16semidirect
120. C22×C127C2×SD16semidirect
134. C4×C127C2×SD16semidirect
145. C127D4×SD16semidirect
147. D4×Q8D127semidirect
157. C22×Q8C2×D127semidirect
168. C4⋊Q8C2×D127semidirect
171. C22⋊Q8C2×D127semidirect
175. C4×Q8C2×D127semidirect
238. Q8D4×D127semidirect
247. D4#14523235direct (N × H)
272. C22C2×#14523235semidirect
280. C4C2×#14523235semidirect

Maximal subgroups

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

subgroupindex
C22×#145232352
D4×D5082
#225502502
#225501072
#225500782
#225500762
#225500262
#225500152
C4×#145232352
D4×C127⋊C82
D4×Q8×C1272
D4×SD16127