Finite Group Databasebeta

← #46849507

Subgroups — #46849507

Order 34688.

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

normal subgroups only

This group has 360 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#31774711C211yesmaximal, characteristic, complemented
2#31774699C211yesmaximal, characteristic, complemented
3#31774339C212yesmaximal, complemented
4#31774337C211yesmaximal, characteristic, complemented
5#31774332C211yesmaximal, characteristic, complemented
6#31774331C212yesmaximal, complemented
7#31774328C212yesmaximal, complemented
8C2×#22576105C211yesmaximal, characteristic, complemented
9SD32×D271C212yesmaximal, complemented
10#31774313C211yesmaximal, characteristic, complemented
11#508×C271C211yesmaximal, characteristic, complemented
12#22576207—21no—
13#22576206—21no—
14#22576190—21no—
15#22576189—22no—
16#22576187—22no—
17#22576185—21no—
18#22576183C2212yes—
19#22576179C2212yescomplemented
20C2×Dic1084C2211yescharacteristic, complemented
21Q16×D271C2212yescomplemented
22C2×D2168C2211yescharacteristic
23D8×D271C2212yes—
24#22576166C2211yescharacteristic
25#22576106C2212yescomplemented
26#22576105C2212yescomplemented
27#22576105C2212yescomplemented
28#22576104C2212yescomplemented
29#22576103C2212yescomplemented
30#22576101C2212yescomplemented
31#22576100C2211yescharacteristic, complemented
32#22576099C2212yescomplemented
33C16×D271C2212yescomplemented
34#135×C271C2211yescharacteristic
35#135×C271C2211yescharacteristic, complemented
36Q32×C271C2211yescharacteristic, complemented
37SD32×C271C2211yescharacteristic, complemented
38SD32×C271C2211yescharacteristic, complemented
39D16×C271C2211yescharacteristic, complemented
40C2×C16×C271C2211yescharacteristic, complemented
41#16864655—22no—
42#16864654—41no—
43#16864654—41no—
44#16864654D411yescharacteristic
45#16864653—22no—
46#16864653—42no—
47C2×Dic542—21no—
48C2×Dic542—41no—
49Q8×D271—22no—
50C2×D1084—21no—
51C2×D1084—41no—
52D4×D271—22no—
53D4×D271—42no—
54Dic1084—21no—
55Dic1084C2312yes—
56#16864644—22no—
57#16864642—22no—
58#16864641—22no—
59D2168—21no—
60D2168C2312yes—
61#16864639—22no—
62#16864638D411yescharacteristic
63#16864637D412yes—
64C8×D271C2312yes—
65C271⋊C16C2312yes—
66#41×C271—21no—
67#41×C271—21no—
68Q16×C271C2311yescharacteristic
69Q16×C271C2311yescharacteristic
70SD16×C271—21no—
71SD16×C271—21no—
72D8×C271C2311yescharacteristic
73D8×C271C2311yescharacteristic
74C2×C8×C271C2311yescharacteristic
75C16×C271C2311yescharacteristic
76C16×C271C2311yescharacteristic
77C22×D271—41no—
78C22×D271—42no—
79Dic542—22no—
80Dic542—41no—
81Dic542—42no—
82Dic542D811yescharacteristic
83D1084—22no—
84D1084—42no—
85D1084—41no—
86D1084D811yescharacteristic
87C22⋊D271—42no—
88C22⋊D271—42no—
89C22⋊D271D812yes—
90C2×Dic271—41no—
91C2×Dic271—42no—
92C4×D271—42no—
93C4×D271—42no—
94C4×D271C2×D412yes—
95C271⋊C8C2×D412yes—
96Q8×C271—21no—
97Q8×C271—21no—
98D4×C271—21no—
99D4×C271—21no—
100D4×C271—41no—
101D4×C271—41no—
102C2×C4×C271—41no—
103C2×C4×C271—41no—
104C2×C4×C271C2×D411yescharacteristic
105C8×C271C2×D411yescharacteristic
106C8×C271C2411yescharacteristic
107C22×C271—41no—
108C22×C271—41no—
109C22×C271C2×D811yescharacteristic
110C2×D271—42no—
111C2×D271—162no—
112C2×D271—161no—
113C2×D271C2×D812yes—
114C4×C271—41no—
115C4×C271—41no—
116C4×C271C2×D811yescharacteristic
117C4×C271C22×D411yescharacteristic
118Dic271—42no—
119Dic271C2×D812yes—
120C2×C271—21no—
121C2×C271—81no—
122C2×C271—81no—
123C2×C271C22×D811yescharacteristic
124D271—22no—
125D271—82no—
126C271#341611yescharacteristic, complemented, Sylow
127#3416—2711nomaximal, Sylow
128#578—2711no—
129#576—2711no—
130#510—2711no—
131#510—2712no—
132#509—2711no—
133#509—2712no—
134#508—2712no—
135#508D27111yescharacteristic, complemented
136C2×SD32—2712no—
137C2×SD32—2711no—
138#504—2711no—
1392−1+4—5421no—
140H(5,𝔽2)—5421no—
141#137—5422no—
142#137—5421no—
143AGL(1,ℤ/8)—5422no—
144AGL(1,ℤ/8)—5421no—
145#135—2712no—
146#135—2712no—
147#135C2×D27111yescharacteristic
148#135C2×D27111yescharacteristic, complemented
149C2×Q16—2712no—
150C2×Q16—2711no—
151C2×D8—2712no—
152C2×D8—2711no—
153#131—2711no—
154Q32—2712no—
155Q32C2×D27111yescharacteristic, complemented
156SD32—2712no—
157SD32—2712no—
158SD32—2712no—
159SD32—2712no—
160SD32C2×D27111yescharacteristic, complemented
161SD32C2×D27111yescharacteristic, complemented
162D16—2712no—
163D16C2×D27111yescharacteristic, complemented
164M32—2711no—
165M32—2712no—
166C2×C16—2712no—
167C2×C16C2×D27111yescharacteristic, complemented
168#41—21no—
169#41—21no—
170#41—2711no—
171#41—5422no—
172#41—5422no—
173#41—10841no—
174#41—10841no—
175#41—10842no—
176C2×Q8—5422no—
177C2×Q8—5421no—
178C2×Q8—10841no—
179C2×D4—5422no—
180C2×D4—5421no—
181C2×D4—10841no—
182C2×D4—10842no—
183Q16—2712no—
184Q16—5422no—
185Q16—5421no—
186Q16C22×D27111yescharacteristic
187Q16C22×D27111yescharacteristic
188SD16—21no—
189SD16—21no—
190SD16—5422no—
191SD16—5422no—
192D8—2712no—
193D8—5422no—
194D8—5421no—
195D8C22×D27111yescharacteristic
196D8C22×D27111yescharacteristic
197M16—2711no—
198M16—2712no—
199C2×C8—2712no—
200C2×C8C22×D27111yescharacteristic
201C16—2712no—
202C16C22×D27111yescharacteristic
203C16C22×D27111yescharacteristic
204C23—10842no—
205C23—10841no—
206Q8—21no—
207Q8—21no—
208Q8—2711no—
209Q8—5422no—
210Q8—10842no—
211Q8—10841no—
212D4—21no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—2712no—
217D4—2711no—
218D4—5422no—
219D4—10842no—
220D4—10842no—
221D4—10842no—
222D4—10841no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—2712no—
226C2×C4—10842no—
227C2×C4—10842no—
228C2×C4—10842no—
229C2×C4—10841no—
230C2×C4D4×D27111yescharacteristic
231C8—2712no—
232C8D4×D27111yescharacteristic
233C8C23×D27111yescharacteristic
234C22—41no—
235C22—41no—
236C22—2712no—
237C22—10842no—
238C22—43361no—
239C22—43362no—
240C22D8×D27111yescharacteristic
241C4—41no—
242C4—41no—
243C4—2712no—
244C4—10842no—
245C4D8×D27111yescharacteristic
246C4C2×D4×D27111yescharacteristic
247C2—21no—
248C2—81no—
249C2—81no—
250C2—5422no—
251C2—21682no—
252C2C2×D8×D27111yescharacteristic
subgroupG/H
centerC2C2×D8×D271
commutator, cosocleC8×C271C24
FrattiniC8C23×D271
Fitting#508×C271C2
socleC2×C271C22×D8
hypercenter#508D271
hypocenterC271#3416
LCSG ▷ C8×C271 ▷ C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #31774711C2semidirect
2. #31774699C2semidirect
3. #31774339C2semidirect
4. #31774337C2semidirect
5. #31774332C2semidirect
6. #31774331C2semidirect
7. #31774328C2semidirect
8. C2×#22576105C2semidirect
9. SD32×D271C2semidirect
10. #31774313C2semidirect
11. #508×C271C2semidirect
19. #22576179C22semidirect
20. C2×Dic1084C22semidirect
21. Q16×D271C22semidirect
25. #22576106C22semidirect
26. #22576105C22semidirect
27. #22576105C22semidirect
28. #22576104C22semidirect
29. #22576103C22semidirect
30. #22576101C22semidirect
31. #22576100C22semidirect
32. #22576099C22semidirect
33. C16×D271C22semidirect
35. #135×C271C22semidirect
36. Q32×C271C22semidirect
37. SD32×C271C22semidirect
38. SD32×C271C22semidirect
39. D16×C271C22semidirect
40. C2×C16×C271C22semidirect
126. C271#3416semidirect
135. #508D271semidirect
148. #135C2×D271semidirect
155. Q32C2×D271semidirect
160. SD32C2×D271semidirect
161. SD32C2×D271semidirect
163. D16C2×D271semidirect
167. C2×C16C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317747112
#317746992
#317743392
#317743372
#317743322
#317743312
#317743282
C2×#225761052
SD32×D2712
#317743132
#508×C2712
#3416271