Finite Group Databasebeta

← #46849504

Subgroups — #46849504

Order 34688.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#31774699C211yesmaximal, characteristic, complemented
2#31774698C211yesmaximal, characteristic, complemented
3#31774335C211yesmaximal, characteristic, complemented
4#31774334C212yesmaximal, complemented
5#31774331C212yesmaximal, complemented
6#31774328C212yesmaximal, complemented
7#31774327C211yesmaximal, characteristic, complemented
8C2×#22576101C211yesmaximal, characteristic, complemented
9D16×D271C212yesmaximal, complemented
10#31774315C211yesmaximal, characteristic, complemented
11#509×C271C211yesmaximal, characteristic, complemented
12#22576206—21no—
13#22576205—21no—
14#22576187—22no—
15#22576186—21no—
16#22576185—21no—
17#22576184—22no—
18#22576183C2212yescomplemented
19#22576182C2211yescharacteristic, complemented
20#22576179C2212yescomplemented
21C2×D2168C2211yescharacteristic, complemented
22D8×D271C2212yescomplemented
23D8×D271C2212yescomplemented
24#22576166C2211yescharacteristic
25#22576106C2211yescharacteristic, complemented
26#22576105C2212yescomplemented
27#22576104C2211yescharacteristic, complemented
28#22576103C2211yescharacteristic, complemented
29D4336C2212yescomplemented
30#22576101C2211yescharacteristic, complemented
31#22576101C2212yescomplemented
32#22576101C2212yescomplemented
33#22576099C2212yescomplemented
34C2×C271⋊C16C2211yescharacteristic, complemented
35C16×D271C2212yescomplemented
36#135×C271C2211yescharacteristic, complemented
37C2×D8×C271C2211yescharacteristic, complemented
38SD32×C271C2212yescomplemented
39D16×C271C2212yescomplemented
40M32×C271C2211yescharacteristic, complemented
41#16864655—22no—
42#16864654—21no—
43#16864654—41no—
44#16864654D411yescharacteristic
45#16864653—22no—
46#16864653—41no—
47C2×D1084—21no—
48C2×D1084—41no—
49C2×C22⋊D271—42no—
50D4×D271—22no—
51D4×D271—22no—
52D4×D271—41no—
53D4×D271—42no—
54Dic1084C2311yescharacteristic
55#16864643—21no—
56#16864642—22no—
57#16864641—22no—
58D2168—21no—
59D2168C2311yescharacteristic
60D2168C2312yes—
61#16864639—22no—
62#16864639—22no—
63#16864638D411yescharacteristic
64#16864637D412yes—
65C8×D271C2312yes—
66C271⋊C16C2311yescharacteristic
67C271⋊C16C2311yescharacteristic
68#41×C271—21no—
69C2×D4×C271—21no—
70Q16×C271C2311yescharacteristic
71SD16×C271—21no—
72D8×C271—21no—
73D8×C271C2311yescharacteristic
74D8×C271C2312yes—
75C2×C8×C271C2311yescharacteristic
76C16×C271C2312yes—
77C22×D271—41no—
78C22×D271—42no—
79C22×D271—42no—
80Dic542—21no—
81Dic542D811yescharacteristic
82D1084—21no—
83D1084—22no—
84D1084—42no—
85D1084—41no—
86D1084D811yescharacteristic
87C22⋊D271—41no—
88C22⋊D271—42no—
89C22⋊D271—42no—
90C22⋊D271—42no—
91C22⋊D271D812yes—
92C2×Dic271—42no—
93C4×D271—41no—
94C4×D271—42no—
95C4×D271C2×D412yes—
96C271⋊C8C2×D412yes—
97C23×C271—41no—
98Q8×C271—21no—
99D4×C271—22no—
100D4×C271—21no—
101D4×C271—41no—
102D4×C271—41no—
103C2×C4×C271—41no—
104C2×C4×C271C2×D411yescharacteristic
105C8×C271C2×D411yescharacteristic
106C8×C271C2411yescharacteristic
107C22×C271—41no—
108C22×C271—42no—
109C22×C271—161no—
110C22×C271C2×D811yescharacteristic
111C2×D271—41no—
112C2×D271—42no—
113C2×D271—162no—
114C2×D271—162no—
115C2×D271—161no—
116C2×D271C2×D812yes—
117C4×C271—41no—
118C4×C271C2×D811yescharacteristic
119C4×C271C22×D411yescharacteristic
120Dic271—41no—
121Dic271C2×D812yes—
122C2×C271—21no—
123C2×C271—81no—
124C2×C271—82no—
125C2×C271C22×D811yescharacteristic
126D271—22no—
127D271—81no—
128D271—82no—
129C271#341511yescharacteristic, complemented, Sylow
130#3415—2711nomaximal, Sylow
131#576—2711no—
132#576—2711no—
133#509—2712no—
134#509—2712no—
135#509—2711no—
136#509D27111yescharacteristic, complemented
137#508—2711no—
138#508—2712no—
139C2×D16—2711no—
140C2×D16—2712no—
141#504—2711no—
142H(5,𝔽2)—5421no—
143H(5,𝔽2)—5421no—
144AGL(1,ℤ/8)—5422no—
145AGL(1,ℤ/8)—5422no—
146AGL(1,ℤ/8)—5421no—
147AGL(1,ℤ/8)—5421no—
148#135—2712no—
149#135—2712no—
150#135—2711no—
151#135C2×D27111yescharacteristic, complemented
152C2×D8—2712no—
153C2×D8—2712no—
154C2×D8—2711no—
155C2×D8C2×D27111yescharacteristic, complemented
156#131—2711no—
157Q32—2711no—
158SD32—2711no—
159SD32—2711no—
160SD32—2712no—
161SD32C2×D27112yescomplemented
162D16—2712no—
163D16—2711no—
164D16—2712no—
165D16—2712no—
166D16C2×D27112yescomplemented
167M32—2712no—
168M32C2×D27111yescharacteristic, complemented
169C2×C16—2711no—
170C2×C16—2712no—
171#41—21no—
172#41—2711no—
173#41—5422no—
174#41—5422no—
175#41—5421no—
176#41—10841no—
177#41—10841no—
178C2×D4—21no—
179C2×D4—5422no—
180C2×D4—5422no—
181C2×D4—5421no—
182C2×D4—10841no—
183C2×D4—10842no—
184C2×D4—10841no—
185C2×D4—10842no—
186Q16—2711no—
187Q16C22×D27111yescharacteristic
188SD16—21no—
189SD16—5422no—
190SD16—5422no—
191SD16—5421no—
192D8—21no—
193D8—2712no—
194D8—2711no—
195D8—5422no—
196D8—5422no—
197D8—5421no—
198D8C22×D27112yes—
199D8C22×D27111yescharacteristic
200M16—2711no—
201M16—2712no—
202C2×C8—2712no—
203C2×C8C22×D27111yescharacteristic
204C16—2711no—
205C16—2711no—
206C16C22×D27112yes—
207C23—41no—
208C23—10842no—
209C23—10842no—
210C23—10841no—
211Q8—21no—
212Q8—2711no—
213Q8—5421no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—2712no—
219D4—2711no—
220D4—5421no—
221D4—5422no—
222D4—10842no—
223D4—10842no—
224D4—10842no—
225D4—10842no—
226D4—10841no—
227D4—10841no—
228C2×C4—41no—
229C2×C4—2712no—
230C2×C4—10842no—
231C2×C4—10842no—
232C2×C4—10841no—
233C2×C4D4×D27111yescharacteristic
234C8—2712no—
235C8D4×D27111yescharacteristic
236C8C23×D27111yescharacteristic
237C22—41no—
238C22—42no—
239C22—2712no—
240C22—161no—
241C22—10842no—
242C22—10841no—
243C22—43361no—
244C22—43362no—
245C22—43362no—
246C22D8×D27111yescharacteristic
247C4—41no—
248C4—2712no—
249C4—10841no—
250C4D8×D27111yescharacteristic
251C4C2×D4×D27111yescharacteristic
252C2—21no—
253C2—82no—
254C2—81no—
255C2—5422no—
256C2—21682no—
257C2—21681no—
258C2C2×D8×D27111yescharacteristic
subgroupG/H
centerC2C2×D8×D271
commutator, cosocleC8×C271C24
FrattiniC8C23×D271
Fitting#509×C271C2
socleC2×C271C22×D8
hypercenter#509D271
hypocenterC271#3415
LCSG ▷ C8×C271 ▷ C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #31774699C2semidirect
2. #31774698C2semidirect
3. #31774335C2semidirect
4. #31774334C2semidirect
5. #31774331C2semidirect
6. #31774328C2semidirect
7. #31774327C2semidirect
8. C2×#22576101C2semidirect
9. D16×D271C2semidirect
10. #31774315C2semidirect
11. #509×C271C2semidirect
18. #22576183C22semidirect
19. #22576182C22semidirect
20. #22576179C22semidirect
21. C2×D2168C22semidirect
22. D8×D271C22semidirect
23. D8×D271C22semidirect
25. #22576106C22semidirect
26. #22576105C22semidirect
27. #22576104C22semidirect
28. #22576103C22semidirect
29. D4336C22semidirect
30. #22576101C22semidirect
31. #22576101C22semidirect
32. #22576101C22semidirect
33. #22576099C22semidirect
34. C2×C271⋊C16C22semidirect
35. C16×D271C22semidirect
36. #135×C271C22semidirect
37. C2×D8×C271C22semidirect
38. SD32×C271C22semidirect
39. D16×C271C22semidirect
40. M32×C271C22semidirect
129. C271#3415semidirect
136. #509D271semidirect
151. #135C2×D271semidirect
155. C2×D8C2×D271semidirect
161. SD32C2×D271semidirect
166. D16C2×D271semidirect
168. M32C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317746992
#317746982
#317743352
#317743342
#317743312
#317743282
#317743272
C2×#225761012
D16×D2712
#317743152
#509×C2712
#3415271