Finite Group Databasebeta

← #46847983

Subgroups — #46847983

Order 34688.

556 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 125 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×#22576184C212yesmaximal, complemented
2#31774567C211yesmaximal, characteristic, complemented
3#31774532C211yesmaximal, characteristic, complemented
4#31774147C212yesmaximal, complemented
5#31774094C211yesmaximal, characteristic, complemented
6#31774089C211yesmaximal, characteristic, complemented
7#31774039C212yesmaximal, complemented
8#31774029C212yesmaximal, complemented
9C4×#16864643C211yesmaximal, characteristic, complemented
10#31773976C211yesmaximal, characteristic, complemented
11#493×C271C211yesmaximal, characteristic, complemented
12C2×#16864653C2212yescomplemented
13C2×D4×D271—21no—
14C2×D4×D271C2212yes—
15#22576184—28no—
16C2×#16864643C2211yescharacteristic
17C2×#16864642C2212yescomplemented
18C2×#16864639C2212yes—
19C2×#16864637C2212yescomplemented
20#22576156—21no—
21#22576132—22no—
22#22576131—21no—
23#22576131C2212yes—
24#22576131C2212yescomplemented
25#22576128—22no—
26C4×Dic542C2211yescharacteristic, complemented
27C4×D1084C2211yescharacteristic
28C4×D1084C2211yescharacteristic
29D4×Dic271—21no—
30#22576092C2211yescharacteristic, complemented
31C4⋊(C271⋊C8)C2211yescharacteristic, complemented
32#22576085C2211yescharacteristic, complemented
33#22576082C2211yescharacteristic
34#22576081C2212yescomplemented
35#22576081C2212yescomplemented
36(C2×D271)⋊C8C2212yescomplemented
37C2×D8×C271C2212yescomplemented
38C2×D8×C271C2212yescomplemented
39#127×C271C2211yescharacteristic
40#127×C271C2211yescharacteristic, complemented
41C4×C8×C271C2211yescharacteristic, complemented
42C23×D271—22no—
43#16864653—24no—
44#16864653—42no—
45C2×Dic542C2311yescharacteristic
46C2×D1084D411yescharacteristic
47C2×D1084C2311yescharacteristic
48C2×C22⋊D271—22no—
49C2×C22⋊D271—22no—
50C2×C22⋊D271—22no—
51C2×C22⋊D271—22no—
52D4×D271—24no—
53D4×D271—42no—
54D4×D271—44no—
55C22×Dic271—22no—
56C2×C4×D271—21no—
57C2×C4×D271C2312yes—
58#16864643D414yescomplemented
59#16864642—24no—
60#16864639—24no—
61#16864637—24no—
62C2×C271⋊C8C2312yes—
63C4⋊Dic271D411yescharacteristic
64C4⋊Dic271C2311yescharacteristic
65C4⋊Dic271C2311yescharacteristic
66Dic271⋊C4—21no—
67C22⋊Dic271—22no—
68C22⋊Dic271—22no—
69(C2×D271)⋊C4—21no—
70(C2×D271)⋊C4D412yes—
71C4×Dic271—21no—
72C2×D4×C271—21no—
73C2×D4×C271—21no—
74C2×D4×C271—21no—
75C2×D4×C271—21no—
76C2×D4×C271C2312yes—
77C2×D4×C271C2312yes—
78D8×C271—24no—
79D8×C271—42no—
80C2×C8×C271C2311yescharacteristic
81C2×C8×C271C2311yescharacteristic
82C42×C271C2311yescharacteristic
83C22×D271—21no—
84C22×D271—44no—
85C22×D271—44no—
86C22×D271—82no—
87C22×D271C2×D412yes—
88Dic542—21no—
89Dic542C2×D412yescomplemented
90D1084—22no—
91D1084—21no—
92D1084C2×D412yes—
93C22⋊D271—44no—
94C22⋊D271—44no—
95C22⋊D271—44no—
96C22⋊D271—82no—
97C2×Dic271—21no—
98C2×Dic271—21no—
99C2×Dic271—21no—
100C2×Dic271—44no—
101C2×Dic271C2×D412yes—
102C4×D271—22no—
103C4×D271—22no—
104C4×D271—42no—
105C271⋊C8—22no—
106C23×C271—22no—
107C23×C271—22no—
108D4×C271—22no—
109D4×C271—22no—
110D4×C271—22no—
111D4×C271—22no—
112D4×C271—42no—
113D4×C271—42no—
114D4×C271—81no—
115D4×C271—81no—
116C2×C4×C271C2×D411yescharacteristic
117C2×C4×C271C2×D411yescharacteristic
118C2×C4×C271C2411yescharacteristic
119C8×C271—21no—
120C8×C271C2×D412yescomplemented
121C22×C271—42no—
122C22×C271—42no—
123C22×C271—42no—
124C22×C271—42no—
125C22×C271—42no—
126C22×C271—42no—
127C22×C271C22×D411yescharacteristic
128C2×D271—22no—
129C2×D271—22no—
130C2×D271—22no—
131C2×D271—22no—
132C2×D271—41no—
133C2×D271—41no—
134C2×D271—88no—
135C4×C271—21no—
136C4×C271AGL(1,ℤ/8)12yescomplemented
137C4×C271C22×D411yescharacteristic
138C4×C271H(5,𝔽2)11yescharacteristic
139Dic271—22no—
140Dic271—22no—
141Dic271—41no—
142Dic271—41no—
143C2×C271—82no—
144C2×C271—82no—
145C2×C271D4211yescharacteristic
146C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C271#57611yescharacteristic
148D271—42no—
149D271—42no—
150C271#327411yescharacteristic, complemented, Sylow
151#3274—2711nomaximal, Sylow
152C2×AGL(1,ℤ/8)—2712no—
153#550—2711no—
154D42—2711no—
155#493D27111yescharacteristic, complemented
156#469—2712no—
157#460—2711no—
158#459—2711no—
159#449—2712no—
160#447—2712no—
161C4×SD16—2711no—
162#436—2711no—
163C2×#41—2712no—
164C22×D4—2712no—
165C22×D4—5421no—
166AGL(1,ℤ/8)—5428no—
167C2×SD16—2712no—
168C2×SD16—2711no—
169C2×D8—2712no—
170C2×D8C2×D27112yescomplemented
171C2×D8C2×D27112yescomplemented
172C2×M16—2712no—
173#127—5421no—
174#127C2×D27111yescharacteristic
175#127C2×D27111yescharacteristic, complemented
176#121—2712no—
177#121—2712no—
178#121—5422no—
179#121—5421no—
180#120—5422no—
181C4×Q8—2711no—
182C4×D4—2711no—
183C4×D4—2711no—
184C4×D4—5421no—
185#106—2711no—
186C4⋊C8—2711no—
187#103—2711no—
188#102—2712no—
189#102—2712no—
190#102—2711no—
191C22⋊C8—2712no—
192C4×C8C2×D27111yescharacteristic, complemented
193C24—5422no—
194#41—5424no—
195#41—10842no—
196C2×Q8—2711no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—2711no—
202C2×D4—2711no—
203C2×D4—5422no—
204C2×D4—5424no—
205C2×D4—5422no—
206C2×D4—5422no—
207C2×D4—5422no—
208C2×D4—10842no—
209C2×D4—10844no—
210C2×D4C22×D27112yes—
211C2×D4C22×D27112yes—
212C22×C4—2712no—
213C22×C4—5422no—
214C22×C4—5421no—
215SD16—2714no—
216SD16—5424no—
217D8—24no—
218D8—42no—
219D8—5424no—
220M16—5424no—
221C2×C8—2712no—
222C2×C8C22×D27111yescharacteristic
223C2×C8C22×D27111yescharacteristic
224C4⋊C4—2711no—
225C4⋊C4—2711no—
226C4⋊C4—2711no—
227C4⋊C4—5421no—
228C22⋊C4—2712no—
229C22⋊C4—5422no—
230C22⋊C4—5422no—
231C22⋊C4—5421no—
232C42—5421no—
233C42C22×D27111yescharacteristic
234C23—22no—
235C23—22no—
236C23—2712no—
237C23—5421no—
238C23—10844no—
239C23—10844no—
240C23—21682no—
241Q8—2712no—
242Q8—5421no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—22no—
247D4—42no—
248D4—42no—
249D4—81no—
250D4—81no—
251D4—2712no—
252D4—5422no—
253D4—5421no—
254D4—10844no—
255D4—10844no—
256D4—10844no—
257D4—21682no—
258C2×C4—2712no—
259C2×C4—5421no—
260C2×C4—5422no—
261C2×C4—5422no—
262C2×C4—5421no—
263C2×C4—5421no—
264C2×C4—10844no—
265C2×C4—10842no—
266C2×C4D4×D27111yescharacteristic
267C2×C4D4×D27111yescharacteristic
268C2×C4C23×D27111yescharacteristic
269C8—21no—
270C8—5422no—
271C8D4×D27112yescomplemented
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—5422no—
279C22—5422no—
280C22—5422no—
281C22—5422no—
282C22—10841no—
283C22—10841no—
284C22—21688no—
285C22C2×D4×D27111yescharacteristic
286C4—21no—
287C4—5422no—
288C4—5422no—
289C4—10841no—
290C4—10841no—
291C4#2257618412yescomplemented
292C4C2×D4×D27111yescharacteristic
293C4#2257620511yescharacteristic
294C2—82no—
295C2—82no—
296C2—10842no—
297C2—10842no—
298C2#3177453211yescharacteristic
299C2C2×#2257618411yescharacteristic
300C2#3177469811yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#493×C271C2
socleC22×C271C22×D4
hypercenter#493D271
hypocenterC271#3274
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#22576184C2semidirect
2. #31774567C2semidirect
3. #31774532C2semidirect
4. #31774147C2semidirect
5. #31774094C2semidirect
6. #31774089C2semidirect
7. #31774039C2semidirect
8. #31774029C2semidirect
9. C4×#16864643C2semidirect
10. #31773976C2semidirect
11. #493×C271C2semidirect
12. C2×#16864653C22semidirect
17. C2×#16864642C22semidirect
19. C2×#16864637C22semidirect
24. #22576131C22semidirect
26. C4×Dic542C22semidirect
30. #22576092C22semidirect
31. C4⋊(C271⋊C8)C22semidirect
32. #22576085C22semidirect
34. #22576081C22semidirect
35. #22576081C22semidirect
36. (C2×D271)⋊C8C22semidirect
37. C2×D8×C271C22semidirect
38. C2×D8×C271C22semidirect
40. #127×C271C22semidirect
41. C4×C8×C271C22semidirect
58. #16864643D4semidirect
89. Dic542C2×D4semidirect
120. C8×C271C2×D4semidirect
136. C4×C271AGL(1,ℤ/8)semidirect
150. C271#3274semidirect
155. #493D271semidirect
170. C2×D8C2×D271semidirect
171. C2×D8C2×D271semidirect
175. #127C2×D271semidirect
192. C4×C8C2×D271semidirect
271. C8D4×D271semidirect
291. C4#22576184semidirect

Maximal subgroups

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

subgroupindex
C2×#225761842
#317745672
#317745322
#317741472
#317740942
#317740892
#317740392
#317740292
C4×#168646432
#317739762
#493×C2712
#3274271