Finite Group Databasebeta

← #46846302

Subgroups — #46846302

Order 34688.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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×#22576131C211yesmaximal, characteristic
2#117×D271C211yesmaximal, characteristic, complemented
3#31774220C212yesmaximal, complemented
4#31774219C212yesmaximal, complemented
5#31774214C212yesmaximal, complemented
6#31774209C212yesmaximal, complemented
7#31773940C211yesmaximal, characteristic, complemented
8#31773939C211yesmaximal, characteristic, complemented
9#31773902C211yesmaximal, characteristic, complemented
10#31773860C211yesmaximal, characteristic, complemented
11#418×C271C211yesmaximal, characteristic, complemented
12C22×C22⋊D271—21no—
13C22×C4×D271C2211yescharacteristic
14C2×#16864638C2211yescharacteristic
15#22576131—21no—
16#22576131—21no—
17#22576131C2212yes—
18#22576131C2212yes—
19#22576119C2211yescharacteristic
20#22576118C2211yescharacteristic, complemented
21#22576118C2211yescharacteristic, complemented
22#22576117—21no—
23#22576116—21no—
24C2×C4⋊Dic271C2211yescharacteristic
25C4⋊C4×D271C2211yescharacteristic, complemented
26C4⋊C4×D271C2211yescharacteristic, complemented
27C2×C22⋊Dic271—21no—
28C22⋊C4×D271—21no—
29C42×D271—21no—
30#22576094C2212yescomplemented
31#22576093C2212yescomplemented
32#22576092C2212yescomplemented
33#22576091C2212yescomplemented
34#22576081C2212yescomplemented
35#22576081C2212yescomplemented
36(C2×D271)⋊C8C2212yescomplemented
37(C2×D271)⋊C8C2212yescomplemented
38C22×D4×C271C2211yescharacteristic
39C2×M16×C271C2211yescharacteristic
40#117×C271C2211yescharacteristic
41#102×C271C2212yescomplemented
42#102×C271C2212yescomplemented
43C23×D271D411yescharacteristic
44C2×C22⋊D271—22no—
45C2×C22⋊D271—22no—
46C2×C22⋊D271—44no—
47C22×Dic271—21no—
48C22×Dic271D411yescharacteristic
49C2×C4×D271—21no—
50C2×C4×D271—21no—
51C2×C4×D271—22no—
52C2×C4×D271—22no—
53C2×C4×D271D411yescharacteristic
54C2×C4×D271D411yescharacteristic
55C2×C4×D271C2311yescharacteristic
56C2×C4×D271C2311yescharacteristic
57#16864638—22no—
58C2×C271⋊C8C2312yes—
59C4⋊Dic271—21no—
60C4⋊Dic271C2312yes—
61C4⋊Dic271C2311yescharacteristic
62C4⋊Dic271C2311yescharacteristic
63Dic271⋊C4—21no—
64Dic271⋊C4—21no—
65C22⋊Dic271—22no—
66C22⋊Dic271—22no—
67C22⋊Dic271—21no—
68(C2×D271)⋊C4—21no—
69(C2×D271)⋊C4—21no—
70C4×Dic271—21no—
71C4×Dic271—21no—
72C4×Dic271—21no—
73C24×C271—21no—
74C2×D4×C271—21no—
75C2×D4×C271—22no—
76C2×D4×C271—22no—
77C2×D4×C271C2312yes—
78C22×C4×C271C2311yescharacteristic
79M16×C271—22no—
80C2×C8×C271C2312yes—
81C4⋊C4×C271C2311yescharacteristic
82C4⋊C4×C271C2311yescharacteristic
83C22⋊C4×C271—21no—
84C42×C271—21no—
85C22×D271—22no—
86C22×D271—22no—
87C22×D271—41no—
88C22×D271C2×D411yescharacteristic
89C22×D271C2×D411yescharacteristic
90C22⋊D271—44no—
91C22⋊D271—44no—
92C2×Dic271—22no—
93C2×Dic271—21no—
94C2×Dic271—21no—
95C2×Dic271—22no—
96C2×Dic271—42no—
97C2×Dic271C2×D411yescharacteristic
98C2×Dic271C2×D411yescharacteristic
99C4×D271—22no—
100C4×D271—22no—
101C4×D271—21no—
102C4×D271—21no—
103C4×D271—21no—
104C4×D271—21no—
105C4×D271—42no—
106C4×D271—42no—
107C271⋊C8—22no—
108C23×C271—22no—
109C23×C271—42no—
110C23×C271—42no—
111C23×C271—42no—
112C23×C271C2×D411yescharacteristic
113D4×C271—22no—
114D4×C271—22no—
115D4×C271—42no—
116C2×C4×C271—21no—
117C2×C4×C271—21no—
118C2×C4×C271C2×D411yescharacteristic
119C2×C4×C271#4112yes—
120C2×C4×C271#4112yes—
121C2×C4×C271C2411yescharacteristic
122C8×C271—22no—
123C22×C271—21no—
124C22×C271—21no—
125C22×C271—42no—
126C22×C271—42no—
127C22×C271—42no—
128C22×C271—84no—
129C22×C271#12312yes—
130C22×C271C22×D411yescharacteristic
131C2×D271—21no—
132C2×D271—21no—
133C2×D271—21no—
134C2×D271—21no—
135C2×D271—21no—
136C2×D271—44no—
137C2×D271AGL(1,ℤ/8)12yes—
138C4×C271—41no—
139C4×C271—41no—
140C4×C271#12312yes—
141C4×C271C2×#4111yescharacteristic
142C4×C271C2×#4111yescharacteristic
143Dic271—21no—
144Dic271—21no—
145Dic271—41no—
146Dic271—42no—
147Dic271—41no—
148C2×C271—22no—
149C2×C271—82no—
150C2×C271C2×#12311yescharacteristic
151C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C271#57511yescharacteristic
153D271—22no—
154D271—41no—
155C271#309411yescharacteristic, complemented, Sylow
156#3094—2711nomaximal, Sylow
157C2×#121—2711no—
158C2×#117—2711no—
159#482—2712no—
160#482—2712no—
161#481—2712no—
162#480—2712no—
163#428—2711no—
164#428—2711no—
165#418—2711no—
166#418D27111yescharacteristic, complemented
167#407—2711no—
168C22×D4—5421no—
169C22×D4C2×D27111yescharacteristic
170C23×C4—2711no—
171C2×M16—2711no—
172C2×M16C2×D27111yescharacteristic
173#121—2712no—
174#121—2712no—
175#121—5421no—
176#121—5421no—
177#117—2711no—
178#117—2711no—
179#117—2711no—
180#117—5421no—
181#117—5421no—
182#117C2×D27111yescharacteristic
183C2×C4⋊C4—2711no—
184C2×C4⋊C4—2711no—
185C2×C4⋊C4—2711no—
186C2×C22⋊C4—5421no—
187C2×C22⋊C4—5421no—
188C2×C42—5421no—
189#107—2712no—
190#107—2712no—
191#106—2712no—
192#106—2712no—
193#102—2712no—
194#102—2712no—
195#102C2×D27112yescomplemented
196#102C2×D27112yescomplemented
197C22⋊C8—2712no—
198C22⋊C8—2712no—
199C24—21no—
200C24—2711no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—5422no—
205C2×D4—5422no—
206C2×D4—10844no—
207C2×D4C22×D27112yes—
208C22×C4—2711no—
209C22×C4—2711no—
210C22×C4—2711no—
211C22×C4—2711no—
212C22×C4—2711no—
213C22×C4—5421no—
214C22×C4—5421no—
215C22×C4—5422no—
216C22×C4—5422no—
217C22×C4—5421no—
218C22×C4C22×D27111yescharacteristic
219M16—22no—
220M16—5422no—
221C2×C8—2712no—
222C2×C8C22×D27112yes—
223C4⋊C4—2711no—
224C4⋊C4—2711no—
225C4⋊C4—2712no—
226C4⋊C4—5421no—
227C4⋊C4—5421no—
228C4⋊C4—5421no—
229C4⋊C4C22×D27111yescharacteristic
230C4⋊C4C22×D27111yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—5421no—
233C22⋊C4—5421no—
234C22⋊C4—5422no—
235C22⋊C4—5422no—
236C22⋊C4—5421no—
237C42—21no—
238C42—5421no—
239C42—5421no—
240C42—5421no—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—2711no—
246C23—2711no—
247C23—5422no—
248C23—5422no—
249C23—10841no—
250C23D4×D27111yescharacteristic
251D4—22no—
252D4—22no—
253D4—42no—
254D4—10844no—
255D4—10844no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—2711no—
259C2×C4—2711no—
260C2×C4—5422no—
261C2×C4—5422no—
262C2×C4—5422no—
263C2×C4—5421no—
264C2×C4—5421no—
265C2×C4—5421no—
266C2×C4—5421no—
267C2×C4—5421no—
268C2×C4—5421no—
269C2×C4—5422no—
270C2×C4—10842no—
271C2×C4—10842no—
272C2×C4—10842no—
273C2×C4D4×D27111yescharacteristic
274C2×C4#1686465312yes—
275C2×C4#1686465412yes—
276C2×C4C23×D27111yescharacteristic
277C8—22no—
278C8—5422no—
279C22—21no—
280C22—21no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—84no—
285C22—2712no—
286C22—5421no—
287C22—5421no—
288C22—5421no—
289C22—5421no—
290C22—5421no—
291C22—10844no—
292C22#2257614212yes—
293C22C2×D4×D27111yescharacteristic
294C4—41no—
295C4—41no—
296C4—5421no—
297C4—5421no—
298C4—10842no—
299C4—10841no—
300C4—10841no—
301C4#2257614212yes—
302C4C2×#1686465311yescharacteristic
303C4C2×#1686465411yescharacteristic
304C2—22no—
305C2—82no—
306C2—5422no—
307C2—10841no—
308C2C2×#2257614211yescharacteristic
309C2AGL(1,ℤ/8)×D27111yescharacteristic
310C2#3177469311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#418×C271C2
socleC22×C271C22×D4
prime radicalC2×#22576131C2
hypercenter#418D271
hypocenterC271#3094
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #117×D271C2semidirect
3. #31774220C2semidirect
4. #31774219C2semidirect
5. #31774214C2semidirect
6. #31774209C2semidirect
7. #31773940C2semidirect
8. #31773939C2semidirect
9. #31773902C2semidirect
10. #31773860C2semidirect
11. #418×C271C2semidirect
20. #22576118C22semidirect
21. #22576118C22semidirect
25. C4⋊C4×D271C22semidirect
26. C4⋊C4×D271C22semidirect
30. #22576094C22semidirect
31. #22576093C22semidirect
32. #22576092C22semidirect
33. #22576091C22semidirect
34. #22576081C22semidirect
35. #22576081C22semidirect
36. (C2×D271)⋊C8C22semidirect
37. (C2×D271)⋊C8C22semidirect
41. #102×C271C22semidirect
42. #102×C271C22semidirect
155. C271#3094semidirect
166. #418D271semidirect
195. #102C2×D271semidirect
196. #102C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761312
#117×D2712
#317742202
#317742192
#317742142
#317742092
#317739402
#317739392
#317739022
#317738602
#418×C2712
#3094271