Finite Group Databasebeta

← #46848055

Subgroups — #46848055

Order 34688.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×SD16×D271C212yesmaximal, complemented
2Q8×D1084C211yesmaximal, characteristic, complemented
3#31774532C211yesmaximal, characteristic
4#31774269C211yesmaximal, characteristic, complemented
5#31774125C212yesmaximal, complemented
6#31774103C211yesmaximal, characteristic, complemented
7#31774049C212yesmaximal, complemented
8#31774034C212yesmaximal, complemented
9C4×#16864641C211yesmaximal, characteristic, complemented
10#31773962C211yesmaximal, characteristic, complemented
11#460×C271C211yesmaximal, characteristic, complemented
12C2×Q8×D271C2212yescomplemented
13C2×D4×D271—21no—
14C2×D4×D271C2212yes—
15C2×#16864643C2212yes—
16C2×#16864642C2212yescomplemented
17C2×#16864642C2212yescomplemented
18C2×#16864641C2211yescharacteristic
19SD16×D271—28no—
20C2×C8×D271C2212yescomplemented
21#22576160—21no—
22#22576160C2211yescharacteristic, complemented
23#22576140—22no—
24#22576140C2212yescomplemented
25#22576131—21no—
26#22576131C2212yes—
27#22576128—22no—
28C4×D1084—21no—
29C4×D1084C2211yescharacteristic
30C4×D1084C2211yescharacteristic
31#22576091C2211yescharacteristic, complemented
32#22576085C2212yescomplemented
33#22576084C2211yescharacteristic, complemented
34#22576080C2211yescharacteristic
35(C2×D271)⋊C8C2212yescomplemented
36C4×C271⋊C8C2211yescharacteristic, complemented
37C2×SD16×C271C2212yescomplemented
38#127×C271C2211yescharacteristic
39C4×Q8×C271C2211yescharacteristic, complemented
40C4⋊C8×C271C2211yescharacteristic, complemented
41#102×C271C2212yescomplemented
42C23×D271—22no—
43C2×Dic542—22no—
44C2×Dic542C2312yes—
45Q8×D271—24no—
46Q8×D271—42no—
47C2×D1084D411yescharacteristic
48C2×D1084C2311yescharacteristic
49C2×C22⋊D271—22no—
50C2×C22⋊D271—22no—
51D4×D271—24no—
52D4×D271—42no—
53D4×D271—44no—
54C2×C4×D271—22no—
55C2×C4×D271—21no—
56C2×C4×D271C2312yes—
57#16864643—24no—
58#16864642—24no—
59#16864642—42no—
60#16864641D414yescomplemented
61C2×C271⋊C8C2311yescharacteristic
62C2×C271⋊C8C2311yescharacteristic
63C8×D271—24no—
64C4⋊Dic271—22no—
65C4⋊Dic271—21no—
66C4⋊Dic271—21no—
67C4⋊Dic271D411yescharacteristic
68C4⋊Dic271C2311yescharacteristic
69C22⋊Dic271—22no—
70(C2×D271)⋊C4—22no—
71(C2×D271)⋊C4—21no—
72(C2×D271)⋊C4D412yes—
73C2×Q8×C271C2311yescharacteristic
74C2×D4×C271—21no—
75C2×D4×C271—21no—
76C2×D4×C271C2312yes—
77SD16×C271—24no—
78C2×C8×C271C2312yes—
79C4⋊C4×C271—21no—
80C4⋊C4×C271C2311yescharacteristic
81C42×C271—21no—
82C42×C271C2311yescharacteristic
83C22×D271—21no—
84C22×D271—44no—
85C22×D271—44no—
86C22×D271—82no—
87C22×D271C2×D412yes—
88Dic542—22no—
89Dic542—22no—
90Dic542—44no—
91D1084—21no—
92D1084SD1614yescomplemented
93D1084C2×D412yes—
94C22⋊D271—44no—
95C22⋊D271—44no—
96C2×Dic271—21no—
97C2×Dic271—22no—
98C2×Dic271C2×D412yes—
99C4×D271—22no—
100C4×D271—22no—
101C4×D271—44no—
102C4×D271—42no—
103C271⋊C8—21no—
104C271⋊C8C2×D412yescomplemented
105C23×C271—22no—
106Q8×C271—21no—
107Q8×C271C2×D412yescomplemented
108D4×C271—22no—
109D4×C271—22no—
110D4×C271—42no—
111D4×C271—81no—
112C2×C4×C271—21no—
113C2×C4×C271—21no—
114C2×C4×C271C2×D411yescharacteristic
115C2×C4×C271C2×D411yescharacteristic
116C2×C4×C271C2411yescharacteristic
117C8×C271—22no—
118C22×C271—42no—
119C22×C271—42no—
120C22×C271—42no—
121C22×C271C22×D411yescharacteristic
122C2×D271—22no—
123C2×D271—22no—
124C2×D271—22no—
125C2×D271—41no—
126C2×D271—41no—
127C2×D271—88no—
128C2×D271C2×SD1614yescomplemented
129C4×C271—22no—
130C4×C271—21no—
131C4×C271—41no—
132C4×C271C2×SD1612yescomplemented
133C4×C271C22×D411yescharacteristic
134C4×C271H(5,𝔽2)11yescharacteristic
135Dic271—22no—
136Dic271—42no—
137Dic271—41no—
138C2×C271—82no—
139C2×C271D4211yescharacteristic
140C2×C271C22×SD1611yescharacteristic
141C2×C271#57711yescharacteristic
142D271—24no—
143D271—42no—
144C271D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—2711nomaximal, Sylow
146C22×SD16—2712no—
147D4×Q8—2711no—
148D42—2711no—
149#492—2711no—
150#465—2712no—
151#461—2711no—
152#460D27111yescharacteristic, complemented
153#450—2712no—
154#448—2712no—
155C4×SD16—2711no—
156C8×D4—2711no—
157C22×Q8—2712no—
158C22×D4—2712no—
159C22×D4—5421no—
160C2×SD16—2711no—
161C2×SD16—2712no—
162C2×SD16—2712no—
163C2×SD16—2712no—
164C2×SD16—5428no—
165C2×SD16C2×D27112yescomplemented
166C22×C8—2712no—
167C4⋊Q8—2711no—
168C4⋊Q8—5421no—
169#127C2×D27111yescharacteristic
170C22⋊Q8—2712no—
171C22⋊Q8—5422no—
172#121—2712no—
173#121—5421no—
174#120—5422no—
175C4×Q8C2×D27111yescharacteristic, complemented
176C4×D4—2711no—
177C4×D4—2711no—
178C4×D4—5421no—
179#106—2711no—
180C4⋊C8C2×D27111yescharacteristic, complemented
181#103—2711no—
182#103—2712no—
183#102—2711no—
184#102C2×D27112yescomplemented
185C22⋊C8—2712no—
186C4×C8—2711no—
187C24—5422no—
188C2×Q8—2712no—
189C2×Q8—5422no—
190C2×Q8—5424no—
191C2×Q8—10842no—
192C2×Q8C22×D27111yescharacteristic
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—2711no—
196C2×D4—2711no—
197C2×D4—5422no—
198C2×D4—5424no—
199C2×D4—5422no—
200C2×D4—10842no—
201C2×D4—10844no—
202C2×D4C22×D27112yes—
203C22×C4—2712no—
204C22×C4—5422no—
205C22×C4—5421no—
206SD16—24no—
207SD16—2714no—
208SD16—5424no—
209SD16—5424no—
210SD16—10842no—
211C2×C8—2711no—
212C2×C8—2711no—
213C2×C8—5424no—
214C2×C8C22×D27112yes—
215C4⋊C4—21no—
216C4⋊C4—2711no—
217C4⋊C4—2711no—
218C4⋊C4—5422no—
219C4⋊C4—5421no—
220C4⋊C4—5421no—
221C4⋊C4C22×D27111yescharacteristic
222C22⋊C4—2712no—
223C22⋊C4—5422no—
224C22⋊C4—5422no—
225C22⋊C4—5421no—
226C42—21no—
227C42C22×D27111yescharacteristic
228C23—22no—
229C23—2712no—
230C23—5421no—
231C23—10844no—
232C23—10844no—
233C23—21682no—
234Q8—21no—
235Q8—5422no—
236Q8—5422no—
237Q8—10844no—
238Q8D4×D27112yescomplemented
239D4—22no—
240D4—22no—
241D4—42no—
242D4—81no—
243D4—2714no—
244D4—2712no—
245D4—5421no—
246D4—10844no—
247D4—10844no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—2712no—
251C2×C4—5422no—
252C2×C4—5422no—
253C2×C4—5421no—
254C2×C4—5422no—
255C2×C4—10844no—
256C2×C4—10842no—
257C2×C4D4×D27111yescharacteristic
258C2×C4D4×D27111yescharacteristic
259C2×C4C23×D27111yescharacteristic
260C8—22no—
261C8—2712no—
262C8—5421no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—2714no—
267C22—5422no—
268C22—5422no—
269C22—5422no—
270C22—10841no—
271C22—10841no—
272C22—21688no—
273C22C2×D4×D27111yescharacteristic
274C4—22no—
275C4—21no—
276C4—41no—
277C4—5422no—
278C4—10842no—
279C4—10841no—
280C4SD16×D27112yescomplemented
281C4C2×D4×D27111yescharacteristic
282C4#2257620611yescharacteristic
283C2—82no—
284C2—5424no—
285C2—10842no—
286C2#3177453111yescharacteristic
287C2C2×SD16×D27111yescharacteristic
288C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#460×C271C2
socleC22×C271C22×D4
prime radical#31774532C2
hypercenter#460D271
hypocenterC271D4×SD16
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D271C2semidirect
2. Q8×D1084C2semidirect
4. #31774269C2semidirect
5. #31774125C2semidirect
6. #31774103C2semidirect
7. #31774049C2semidirect
8. #31774034C2semidirect
9. C4×#16864641C2semidirect
10. #31773962C2semidirect
11. #460×C271C2semidirect
12. C2×Q8×D271C22semidirect
16. C2×#16864642C22semidirect
17. C2×#16864642C22semidirect
20. C2×C8×D271C22semidirect
22. #22576160C22semidirect
24. #22576140C22semidirect
31. #22576091C22semidirect
32. #22576085C22semidirect
33. #22576084C22semidirect
35. (C2×D271)⋊C8C22semidirect
36. C4×C271⋊C8C22semidirect
37. C2×SD16×C271C22semidirect
39. C4×Q8×C271C22semidirect
40. C4⋊C8×C271C22semidirect
41. #102×C271C22semidirect
60. #16864641D4semidirect
92. D1084SD16semidirect
104. C271⋊C8C2×D4semidirect
107. Q8×C271C2×D4semidirect
128. C2×D271C2×SD16semidirect
132. C4×C271C2×SD16semidirect
144. C271D4×SD16semidirect
152. #460D271semidirect
165. C2×SD16C2×D271semidirect
175. C4×Q8C2×D271semidirect
180. C4⋊C8C2×D271semidirect
184. #102C2×D271semidirect
238. Q8D4×D271semidirect
280. C4SD16×D271semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D2712
Q8×D10842
#317745322
#317742692
#317741252
#317741032
#317740492
#317740342
C4×#168646412
#317739622
#460×C2712
D4×SD16271