Finite Group Databasebeta

← D4×#14523236 #30650422

Subgroups — D4×#14523236 #30650422

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×#14523236C212yesmaximal, complemented
2D4×Dic254C211yesmaximal, characteristic, complemented
3#22550248C211yesmaximal, characteristic, complemented
4#22550103C212yesmaximal, complemented
5#22550081C211yesmaximal, characteristic, complemented
6#22550073C211yesmaximal, characteristic, complemented
7#22550027C212yesmaximal, complemented
8#22550014C212yesmaximal, complemented
9C4×#14523236C211yesmaximal, characteristic, complemented
10D4×C127⋊C8C211yesmaximal, characteristic, complemented
11D42×C127C211yesmaximal, characteristic
12C22×Dic254C2212yescomplemented
13C2×#14523236—28no—
14C2×#14523236C2212yes—
15C2×#14523236C2211yescharacteristic
16C2×#14523236C2212yescomplemented
17C2×#14523236C2212yescomplemented
18C2×#14523236C2212yescomplemented
19C22×C127⋊C8C2212yescomplemented
20#16861209C2211yescharacteristic, complemented
21#16861208—21no—
22#16861188C2212yescomplemented
23#16861186—22no—
24C4×Dic254C2211yescharacteristic, complemented
25D4×Dic127—21no—
26#16861140C2211yescharacteristic, complemented
27C4⋊(C127⋊C8)C2211yescharacteristic, complemented
28#16861132C2212yescomplemented
29#16861132C2211yescharacteristic, complemented
30#16861130C2211yescharacteristic
31#16861130C2212yescomplemented
32C22⋊(C127⋊C8)C2212yescomplemented
33C4×C127⋊C8C2211yescharacteristic, complemented
34C22×D4×C127—21no—
35C22×D4×C127C2212yes—
36#127×C127C2211yescharacteristic
37#121×C127—21no—
38#121×C127C2212yes—
39#120×C127—22no—
40C4×D4×C127C2211yescharacteristic
41C4×D4×C127C2211yescharacteristic
42C2×Dic254—24no—
43C2×Dic254—22no—
44C2×Dic254—42no—
45C2×Dic254C2312yes—
46C2×Dic254C2311yescharacteristic
47C22×Dic127—22no—
48#14523236—24no—
49#14523236—24no—
50#14523236—24no—
51#14523236—42no—
52#14523236D414yesdirect product factor
53C2×C127⋊C8—24no—
54C2×C127⋊C8C2312yes—
55C2×C127⋊C8C2311yescharacteristic
56C2×C127⋊C8C2311yescharacteristic
57C4⋊Dic127—21no—
58C4⋊Dic127—21no—
59C4⋊Dic127C2311yescharacteristic
60Dic127⋊C4—21no—
61Dic127⋊C4—22no—
62C22⋊Dic127—22no—
63C4×Dic127—21no—
64C24×C127—22no—
65C2×D4×C127—24no—
66C2×D4×C127—22no—
67C2×D4×C127—22no—
68C2×D4×C127—21no—
69C2×D4×C127—21no—
70C2×D4×C127—42no—
71C2×D4×C127—44no—
72C2×D4×C127D411yescharacteristic
73C2×D4×C127C2312yes—
74C2×D4×C127C2311yescharacteristic
75C22×C4×C127—21no—
76C22×C4×C127C2312yes—
77C4⋊C4×C127D411yescharacteristic
78C4⋊C4×C127C2311yescharacteristic
79C22⋊C4×C127—21no—
80C22⋊C4×C127—22no—
81C22⋊C4×C127D412yes—
82C42×C127C2311yescharacteristic
83Dic254—21no—
84Dic254—22no—
85Dic254—22no—
86Dic254—44no—
87Dic254C2×D412yescomplemented
88C2×Dic127—21no—
89C2×Dic127—21no—
90C2×Dic127—22no—
91C2×Dic127—44no—
92C127⋊C8—22no—
93C127⋊C8—21no—
94C127⋊C8C2×D412yescomplemented
95C23×C127—21no—
96C23×C127—22no—
97C23×C127—44no—
98C23×C127—44no—
99C23×C127—82no—
100C23×C127C2×D412yes—
101D4×C127—22no—
102D4×C127—21no—
103D4×C127—22no—
104D4×C127—44no—
105D4×C127—44no—
106D4×C127—42no—
107D4×C127—81no—
108D4×C127SD1614yescomplemented
109D4×C127C2×D412yes—
110C2×C4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127—21no—
113C2×C4×C127—42no—
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2×D412yes—
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2411yescharacteristic
118C22×C127—22no—
119C22×C127—22no—
120C22×C127—22no—
121C22×C127—41no—
122C22×C127—41no—
123C22×C127—42no—
124C22×C127—42no—
125C22×C127—42no—
126C22×C127—88no—
127C22×C127C2×SD1614yescomplemented
128C22×C127C22×D411yescharacteristic
129C4×C127—22no—
130C4×C127—21no—
131C4×C127—41no—
132C4×C127C2×SD1612yescomplemented
133C4×C127C22×D411yescharacteristic
134C4×C127H(5,𝔽2)11yescharacteristic
135Dic127—22no—
136Dic127—42no—
137Dic127—41no—
138C2×C127—24no—
139C2×C127—42no—
140C2×C127—82no—
141C2×C127D4211yescharacteristic
142C2×C127C22×SD1611yescharacteristic
143C2×C127#57711yescharacteristic
144D4×SD16—1271nomaximal, Sylow
145C127D4×SD1611yescharacteristic, complemented, Sylow
146C22×SD16—1272no—
147D4×Q8—1271no—
148D42D12711yescharacteristic
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×Q8—1272no—
158C22×D4—21no—
159C22×D4C2×D12712yes—
160C2×SD16—1271no—
161C2×SD16—1272no—
162C2×SD16—1272no—
163C2×SD16—1272no—
164C2×SD16—1272no—
165C2×SD16—2548no—
166C22×C8—1272no—
167C4⋊Q8—1271no—
168C4⋊Q8—2541no—
169#127C2×D12711yescharacteristic
170C22⋊Q8—1272no—
171C22⋊Q8—2542no—
172#121—21no—
173#121C2×D12712yes—
174#120—22no—
175C4×Q8—1271no—
176C4×D4—2541no—
177C4×D4C2×D12711yescharacteristic
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—22no—
188C2×Q8—1271no—
189C2×Q8—1272no—
190C2×Q8—5082no—
191C2×Q8—2544no—
192C2×Q8—2542no—
193C2×D4—24no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—42no—
199C2×D4—44no—
200C2×D4C22⋊D12711yescharacteristic
201C2×D4C22×D12712yes—
202C2×D4C22×D12711yescharacteristic
203C22×C4—21no—
204C22×C4—2542no—
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—1271no—
216C4⋊C4—2541no—
217C4⋊C4—2542no—
218C4⋊C4—2541no—
219C4⋊C4—2541no—
220C4⋊C4C22⋊D12711yescharacteristic
221C4⋊C4C22×D12711yescharacteristic
222C22⋊C4—21no—
223C22⋊C4—22no—
224C22⋊C4—2542no—
225C22⋊C4C22⋊D12712yes—
226C42—2541no—
227C42C22×D12711yescharacteristic
228C23—21no—
229C23—22no—
230C23—44no—
231C23—44no—
232C23—82no—
233C23C2×C22⋊D12712yes—
234Q8—1272no—
235Q8—5084no—
236Q8—2542no—
237Q8—2542no—
238Q8—2541no—
239D4—22no—
240D4—21no—
241D4—22no—
242D4—44no—
243D4—44no—
244D4—42no—
245D4—81no—
246D4#1452323614yesdirect product factor
247D4D4×D12712yes—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—5084no—
253C2×C4—2541no—
254C2×C4—2542no—
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—22no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—88no—
272C22C2×#1452323614yescomplemented
273C22C22×C22⋊D12711yescharacteristic
274C4—21no—
275C4—22no—
276C4—41no—
277C4—5081no—
278C4—5082no—
279C4—2542no—
280C4C2×#1452323612yescomplemented
281C4C2×D4×D12711yescharacteristic
282C4#1686125411yescharacteristic
283C2—24no—
284C2—42no—
285C2—82no—
286C2D4×C22⋊D12711yescharacteristic
287C2C22×#1452323611yescharacteristic
288C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radicalD42×C127C2
socleC22×C127C22×D4
hypercenterD42D127
hypocenterC127D4×SD16
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C22×#14523236C2semidirect
2. D4×Dic254C2semidirect
3. #22550248C2semidirect
4. #22550103C2semidirect
5. #22550081C2semidirect
6. #22550073C2semidirect
7. #22550027C2semidirect
8. #22550014C2semidirect
9. C4×#14523236C2semidirect
10. D4×C127⋊C8C2semidirect
12. C22×Dic254C22semidirect
16. C2×#14523236C22semidirect
17. C2×#14523236C22semidirect
18. C2×#14523236C22semidirect
19. C22×C127⋊C8C22semidirect
20. #16861209C22semidirect
22. #16861188C22semidirect
24. C4×Dic254C22semidirect
26. #16861140C22semidirect
27. C4⋊(C127⋊C8)C22semidirect
28. #16861132C22semidirect
29. #16861132C22semidirect
31. #16861130C22semidirect
32. C22⋊(C127⋊C8)C22semidirect
33. C4×C127⋊C8C22semidirect
52. #14523236D4direct (N × H)
87. Dic254C2×D4semidirect
94. C127⋊C8C2×D4semidirect
108. D4×C127SD16semidirect
127. C22×C127C2×SD16semidirect
132. C4×C127C2×SD16semidirect
145. C127D4×SD16semidirect
246. D4#14523236direct (N × H)
272. C22C2×#14523236semidirect
280. C4C2×#14523236semidirect

Maximal subgroups

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

subgroupindex
C22×#145232362
D4×Dic2542
#225502482
#225501032
#225500812
#225500732
#225500272
#225500142
C4×#145232362
D4×C127⋊C82
D42×C1272
D4×SD16127