Finite Group Databasebeta

← #46848059

Subgroups — #46848059

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
2#31774562C211yesmaximal, characteristic, complemented
3#31774532C211yesmaximal, characteristic
4#31774123C212yesmaximal, complemented
5#31774099C211yesmaximal, characteristic, complemented
6#31774094C211yesmaximal, characteristic, complemented
7#31774046C212yesmaximal, complemented
8#31774034C212yesmaximal, complemented
9C4×#16864643C211yesmaximal, characteristic, complemented
10C8×D1084C211yesmaximal, characteristic, complemented
11#492×C271C211yesmaximal, characteristic, complemented
12C2×Q8×D271C2212yescomplemented
13C2×D4×D271—21no—
14C2×D4×D271C2212yes—
15C2×#16864643C2211yescharacteristic
16C2×#16864642C2212yescomplemented
17C2×#16864641C2212yes—
18SD16×D271—28no—
19C2×C8×D271C2212yescomplemented
20#22576159—21no—
21#22576138—22no—
22#22576136C2212yescomplemented
23#22576131—21no—
24#22576131C2212yes—
25#22576128—22no—
26C4×Dic542C2211yescharacteristic, complemented
27C4×D1084C2211yescharacteristic
28C4×D1084C2211yescharacteristic
29#22576122—21no—
30#22576092C2211yescharacteristic, complemented
31C4⋊(C271⋊C8)C2211yescharacteristic, complemented
32#22576085C2211yescharacteristic, complemented
33#22576084C2212yescomplemented
34#22576082C2211yescharacteristic
35#22576081C2212yescomplemented
36(C2×D271)⋊C8C2212yescomplemented
37C2×SD16×C271C2212yescomplemented
38C2×SD16×C271C2212yescomplemented
39C4⋊Q8×C271C2211yescharacteristic, complemented
40#127×C271C2211yescharacteristic
41C4×C8×C271C2211yescharacteristic, complemented
42C23×D271—22no—
43C2×Dic542—22no—
44C2×Dic542C2311yescharacteristic
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#16864643D414yescomplemented
58#16864642—24no—
59#16864641—24no—
60C2×C271⋊C8C2312yes—
61C8×D271—24no—
62C4⋊Dic271D411yescharacteristic
63C4⋊Dic271C2311yescharacteristic
64C4⋊Dic271C2311yescharacteristic
65Dic271⋊C4—22no—
66Dic271⋊C4—21no—
67C22⋊Dic271—22no—
68(C2×D271)⋊C4—22no—
69(C2×D271)⋊C4—21no—
70(C2×D271)⋊C4D412yes—
71C4×Dic271—21no—
72C2×Q8×C271C2312yes—
73C2×D4×C271—21no—
74C2×D4×C271—21no—
75C2×D4×C271C2312yes—
76SD16×C271—24no—
77SD16×C271—42no—
78C2×C8×C271C2311yescharacteristic
79C2×C8×C271C2311yescharacteristic
80C4⋊C4×C271—21no—
81C4⋊C4×C271—21no—
82C42×C271C2311yescharacteristic
83C22×D271—21no—
84C22×D271—44no—
85C22×D271—44no—
86C22×D271—82no—
87C22×D271C2×D412yes—
88Dic542—21no—
89Dic542—44no—
90Dic542C2×D412yescomplemented
91D1084—21no—
92D1084SD1614yescomplemented
93D1084C2×D412yes—
94C22⋊D271—44no—
95C22⋊D271—44no—
96C2×Dic271—21no—
97C2×Dic271—21no—
98C2×Dic271—21no—
99C2×Dic271C2×D412yes—
100C4×D271—22no—
101C4×D271—22no—
102C4×D271—44no—
103C4×D271—42no—
104C271⋊C8—22no—
105C23×C271—22no—
106Q8×C271—22no—
107Q8×C271—22no—
108D4×C271—22no—
109D4×C271—22no—
110D4×C271—42no—
111D4×C271—81no—
112C2×C4×C271—22no—
113C2×C4×C271C2×D411yescharacteristic
114C2×C4×C271C2×D411yescharacteristic
115C2×C4×C271C2411yescharacteristic
116C8×C271—21no—
117C8×C271C2×D412yescomplemented
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—21no—
130C4×C271—42no—
131C4×C271C2×SD1612yescomplemented
132C4×C271C22×D411yescharacteristic
133C4×C271H(5,𝔽2)11yescharacteristic
134Dic271—22no—
135Dic271—22no—
136Dic271—41no—
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#492D27111yescharacteristic, complemented
150#465—2712no—
151#461—2711no—
152#460—2711no—
153#450—2712no—
154#448—2712no—
155C4×SD16—2711no—
156C8×D4—2711no—
157C22×Q8—2712no—
158C22×D4—2712no—
159C22×D4—5421no—
160C2×SD16—2712no—
161C2×SD16—2712no—
162C2×SD16—2711no—
163C2×SD16—5428no—
164C2×SD16C2×D27112yescomplemented
165C2×SD16C2×D27112yescomplemented
166C22×C8—2712no—
167C4⋊Q8—5421no—
168C4⋊Q8C2×D27111yescharacteristic, complemented
169#127C2×D27111yescharacteristic
170C22⋊Q8—2712no—
171C22⋊Q8—5422no—
172#121—2712no—
173#121—5421no—
174#120—5422no—
175C4×Q8—2711no—
176C4×D4—2711no—
177C4×D4—2711no—
178C4×D4—5421no—
179#106—2711no—
180C4⋊C8—2711no—
181#103—2712no—
182#103—2711no—
183#102—2712no—
184#102—2711no—
185C22⋊C8—2712no—
186C4×C8C2×D27111yescharacteristic, complemented
187C24—5422no—
188C2×Q8—2711no—
189C2×Q8—5422no—
190C2×Q8—5424no—
191C2×Q8—10842no—
192C2×Q8C22×D27112yes—
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—42no—
208SD16—2714no—
209SD16—5424no—
210SD16—5424no—
211C2×C8—2712no—
212C2×C8—5424no—
213C2×C8C22×D27111yescharacteristic
214C2×C8C22×D27111yescharacteristic
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—2711no—
218C4⋊C4—2711no—
219C4⋊C4—2711no—
220C4⋊C4—5422no—
221C4⋊C4—5421no—
222C22⋊C4—2712no—
223C22⋊C4—5422no—
224C22⋊C4—5422no—
225C22⋊C4—5421no—
226C42—5421no—
227C42C22×D27111yescharacteristic
228C23—22no—
229C23—2712no—
230C23—5421no—
231C23—10844no—
232C23—10844no—
233C23—21682no—
234Q8—22no—
235Q8—22no—
236Q8—2712no—
237Q8—5421no—
238Q8—10844no—
239D4—22no—
240D4—22no—
241D4—42no—
242D4—81no—
243D4—2714no—
244D4—2712no—
245D4—5421no—
246D4—10844no—
247D4—10844no—
248C2×C4—22no—
249C2×C4—2712no—
250C2×C4—5421no—
251C2×C4—5422no—
252C2×C4—5422no—
253C2×C4—5421no—
254C2×C4—5421no—
255C2×C4—10844no—
256C2×C4—10842no—
257C2×C4D4×D27111yescharacteristic
258C2×C4D4×D27111yescharacteristic
259C2×C4C23×D27111yescharacteristic
260C8—21no—
261C8—5422no—
262C8D4×D27112yescomplemented
263C22—42no—
264C22—42no—
265C22—42no—
266C22—2714no—
267C22—5422no—
268C22—5422no—
269C22—5422no—
270C22—10841no—
271C22—10841no—
272C22—21688no—
273C22C2×D4×D27111yescharacteristic
274C4—21no—
275C4—42no—
276C4—5422no—
277C4—5422no—
278C4—10841no—
279C4—10841no—
280C4SD16×D27112yescomplemented
281C4C2×D4×D27111yescharacteristic
282C4#2257620511yescharacteristic
283C2—82no—
284C2—5424no—
285C2—10842no—
286C2#3177453211yescharacteristic
287C2C2×SD16×D27111yescharacteristic
288C2#3177470311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#492×C271C2
socleC22×C271C22×D4
prime radical#31774532C2
hypercenter#492D271
hypocenterC271D4×SD16
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D271C2semidirect
2. #31774562C2semidirect
4. #31774123C2semidirect
5. #31774099C2semidirect
6. #31774094C2semidirect
7. #31774046C2semidirect
8. #31774034C2semidirect
9. C4×#16864643C2semidirect
10. C8×D1084C2semidirect
11. #492×C271C2semidirect
12. C2×Q8×D271C22semidirect
16. C2×#16864642C22semidirect
19. C2×C8×D271C22semidirect
22. #22576136C22semidirect
26. C4×Dic542C22semidirect
30. #22576092C22semidirect
31. C4⋊(C271⋊C8)C22semidirect
32. #22576085C22semidirect
33. #22576084C22semidirect
35. #22576081C22semidirect
36. (C2×D271)⋊C8C22semidirect
37. C2×SD16×C271C22semidirect
38. C2×SD16×C271C22semidirect
39. C4⋊Q8×C271C22semidirect
41. C4×C8×C271C22semidirect
57. #16864643D4semidirect
90. Dic542C2×D4semidirect
92. D1084SD16semidirect
117. C8×C271C2×D4semidirect
128. C2×D271C2×SD16semidirect
131. C4×C271C2×SD16semidirect
144. C271D4×SD16semidirect
149. #492D271semidirect
164. C2×SD16C2×D271semidirect
165. C2×SD16C2×D271semidirect
168. C4⋊Q8C2×D271semidirect
186. C4×C8C2×D271semidirect
262. C8D4×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
#317745622
#317745322
#317741232
#317740992
#317740942
#317740462
#317740342
C4×#168646432
C8×D10842
#492×C2712
D4×SD16271