Finite Group Databasebeta

← #46846846

Subgroups — #46846846

Order 34688.

596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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×#22576187C212yesmaximal, complemented
2C2×#22576183C211yesmaximal, characteristic, complemented
3C2×D8×D271C211yesmaximal, characteristic, complemented
4#31774465C212yesmaximal, complemented
5#31774287C212yesmaximal, complemented
6#31774283C212yesmaximal, complemented
7#31774282C211yesmaximal, characteristic, complemented
8#31774274C211yesmaximal, characteristic, complemented
9#31774272C211yesmaximal, characteristic, complemented
10#31773851C211yesmaximal, characteristic, complemented
11#495×C271C211yesmaximal, characteristic, complemented
12C2×#16864655C2212yes—
13C2×D4×D271C2212yes—
14#22576187—28no—
15#22576183—24no—
16C2×#16864641C2212yescomplemented
17C2×#16864641C2212yescomplemented
18C2×D2168C2211yescharacteristic
19C2×D2168C2211yescharacteristic, complemented
20C2×D2168C2212yescomplemented
21C2×#16864639C2212yes—
22C2×#16864639C2212yescomplemented
23D8×D271—24no—
24C2×#16864637C2211yescharacteristic
25C2×C8×D271C2211yescharacteristic
26#22576157C2212yescomplemented
27#22576156C2212yescomplemented
28#22576149C2212yescomplemented
29#22576133—24no—
30#22576130—24no—
31#22576117C2211yescharacteristic
32#22576071C2211yescharacteristic, complemented
33#22576070C2211yescharacteristic, complemented
34C8×Dic271C2211yescharacteristic, complemented
35C2×Q16×C271C2211yescharacteristic, complemented
36C2×SD16×C271C2212yescomplemented
37C2×D8×C271C2211yescharacteristic
38#124×C271C2212yescomplemented
39C4×C8×C271C2211yescharacteristic, complemented
40C23×D271—22no—
41#16864655—24no—
42#16864655—24no—
43C2×D1084—22no—
44C2×D1084—24no—
45C2×D1084C2312yes—
46C2×D1084C2312yes—
47C2×C22⋊D271—22no—
48C2×C22⋊D271—24no—
49D4×D271—24no—
50D4×D271—24no—
51C2×C4×D271—22no—
52C2×C4×D271C2311yescharacteristic
53#16864641—24no—
54#16864641—24no—
55D2168—22no—
56D2168—22no—
57D2168—24no—
58#16864639—24no—
59#16864639—24no—
60#16864637D414yescomplemented
61C2×C271⋊C8C2311yescharacteristic
62C2×C271⋊C8C2311yescharacteristic
63C8×D271D414yescomplemented
64Dic271⋊C4D412yes—
65(C2×D271)⋊C4—24no—
66(C2×D271)⋊C4D412yes—
67C4×Dic271C2311yescharacteristic
68C2×Q8×C271C2312yes—
69C2×D4×C271C2312yes—
70Q16×C271—22no—
71SD16×C271—24no—
72D8×C271—22no—
73C2×C8×C271C2311yescharacteristic
74C2×C8×C271C2311yescharacteristic
75C22⋊C4×C271—24no—
76C42×C271C2311yescharacteristic
77C22×D271—22no—
78C22×D271—22no—
79C22×D271—44no—
80C22×D271—84no—
81C22×D271C2×D411yescharacteristic
82D1084—22no—
83D1084—22no—
84D1084—22no—
85D1084—22no—
86D1084—44no—
87D1084—84no—
88C22⋊D271—44no—
89C22⋊D271—84no—
90C2×Dic271C2×D412yes—
91C2×Dic271C2×D411yescharacteristic
92C4×D271—44no—
93C4×D271C2×D412yes—
94C4×D271C2×D412yes—
95C271⋊C8C2×D412yes—
96C271⋊C8C2×D412yes—
97C23×C271—22no—
98Q8×C271—22no—
99Q8×C271—22no—
100D4×C271—22no—
101D4×C271—22no—
102C2×C4×C271—22no—
103C2×C4×C271C2×D412yes—
104C2×C4×C271C2411yescharacteristic
105C8×C271C2×D412yes—
106C8×C271C2×D412yes—
107C22×C271—42no—
108C22×C271—44no—
109C22×C271C22×D411yescharacteristic
110C2×D271—22no—
111C2×D271—42no—
112C2×D271—42no—
113C2×D271—44no—
114C2×D271—44no—
115C2×D271—88no—
116C2×D271#12712yes—
117C4×C271—22no—
118C4×C271—42no—
119C4×C271C22×D411yescharacteristic
120C4×C271C22×D411yescharacteristic
121Dic271—22no—
122Dic271#12712yes—
123C2×C271—82no—
124C2×C271C2×#12711yescharacteristic
125C2×C271#57612yes—
126D271—22no—
127D271—82no—
128D271—82no—
129C271#315511yescharacteristic, complemented, Sylow
130#3155—2711nomaximal, Sylow
131C2×AGL(1,ℤ/8)—2712no—
132C2×#135—2711no—
133C22×D8—2711no—
134#535—2712no—
135#496—2712no—
136#496—2712no—
137#495—2711no—
138#495D27111yescharacteristic, complemented
139#493—2711no—
140#493—2711no—
141#405—2711no—
142C2×#41—2712no—
143C22×D4—2712no—
144AGL(1,ℤ/8)—5428no—
145#135—5424no—
146C2×Q16C2×D27111yescharacteristic, complemented
147C2×SD16—2712no—
148C2×SD16—2712no—
149C2×SD16C2×D27112yescomplemented
150C2×D8—2712no—
151C2×D8—2712no—
152C2×D8—2711no—
153C2×D8—2711no—
154C2×D8—2712no—
155C2×D8—5424no—
156C2×D8C2×D27111yescharacteristic
157C2×M16—2711no—
158C22×C8—2711no—
159#127—2712no—
160#127—2712no—
161#124—2712no—
162#124C2×D27112yescomplemented
163#121—5424no—
164#120—5424no—
165#117—2711no—
166#97—2711no—
167#97—2711no—
168C4×C8—2711no—
169C4×C8C2×D27111yescharacteristic, complemented
170C24—5422no—
171#41—5424no—
172#41—5424no—
173C2×Q8C22×D27112yes—
174C2×D4—2712no—
175C2×D4—2712no—
176C2×D4—5422no—
177C2×D4—5424no—
178C2×D4—5424no—
179C2×D4—5422no—
180C2×D4—5424no—
181C2×D4—5424no—
182C2×D4C22×D27112yes—
183C22×C4—2711no—
184C22×C4—5422no—
185Q16—22no—
186SD16—24no—
187SD16—5424no—
188SD16—5424no—
189D8—22no—
190D8—5424no—
191D8—5424no—
192D8—5422no—
193D8—5422no—
194D8—5424no—
195M16—2714no—
196C2×C8—2711no—
197C2×C8—2711no—
198C2×C8—2714no—
199C2×C8C22×D27111yescharacteristic
200C2×C8C22×D27111yescharacteristic
201C4⋊C4—2712no—
202C22⋊C4—24no—
203C22⋊C4—2712no—
204C22⋊C4—5424no—
205C42—2711no—
206C42C22×D27111yescharacteristic
207C23—22no—
208C23—2711no—
209C23—5422no—
210C23—5422no—
211C23—10844no—
212C23—21684no—
213Q8—22no—
214Q8—22no—
215D4—22no—
216D4—22no—
217D4—5422no—
218D4—5422no—
219D4—5422no—
220D4—5422no—
221D4—10844no—
222D4—10844no—
223D4—21684no—
224D4—21684no—
225C2×C4—22no—
226C2×C4—2711no—
227C2×C4—2712no—
228C2×C4—2712no—
229C2×C4—2712no—
230C2×C4—10844no—
231C2×C4D4×D27112yes—
232C2×C4C23×D27111yescharacteristic
233C8—2712no—
234C8—2712no—
235C8D4×D27112yes—
236C8D4×D27112yes—
237C22—42no—
238C22—44no—
239C22—2712no—
240C22—5422no—
241C22—10842no—
242C22—10842no—
243C22—10844no—
244C22—10844no—
245C22—21688no—
246C22C2×D4×D27111yescharacteristic
247C4—22no—
248C4—42no—
249C4—2712no—
250C4—5422no—
251C4C2×D4×D27111yescharacteristic
252C4C2×D4×D27111yescharacteristic
253C2—82no—
254C2—5422no—
255C2—21682no—
256C2—21682no—
257C2#127×D27111yescharacteristic
258C2#3177469912yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#495×C271C2
socleC22×C271C22×D4
hypercenter#495D271
hypocenterC271#3155
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#22576187C2semidirect
2. C2×#22576183C2semidirect
3. C2×D8×D271C2semidirect
4. #31774465C2semidirect
5. #31774287C2semidirect
6. #31774283C2semidirect
7. #31774282C2semidirect
8. #31774274C2semidirect
9. #31774272C2semidirect
10. #31773851C2semidirect
11. #495×C271C2semidirect
16. C2×#16864641C22semidirect
17. C2×#16864641C22semidirect
19. C2×D2168C22semidirect
20. C2×D2168C22semidirect
22. C2×#16864639C22semidirect
26. #22576157C22semidirect
27. #22576156C22semidirect
28. #22576149C22semidirect
32. #22576071C22semidirect
33. #22576070C22semidirect
34. C8×Dic271C22semidirect
35. C2×Q16×C271C22semidirect
36. C2×SD16×C271C22semidirect
38. #124×C271C22semidirect
39. C4×C8×C271C22semidirect
60. #16864637D4semidirect
63. C8×D271D4semidirect
129. C271#3155semidirect
138. #495D271semidirect
146. C2×Q16C2×D271semidirect
149. C2×SD16C2×D271semidirect
162. #124C2×D271semidirect
169. C4×C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761872
C2×#225761832
C2×D8×D2712
#317744652
#317742872
#317742832
#317742822
#317742742
#317742722
#317738512
#495×C2712
#3155271