Finite Group Databasebeta

← #46846813

Subgroups — #46846813

Order 34688.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#22576184C212yesmaximal, complemented
3#31774446C211yesmaximal, characteristic, complemented
4#127×D271C211yesmaximal, characteristic, complemented
5#31774287C211yesmaximal, characteristic, complemented
6#31774286C211yesmaximal, characteristic, complemented
7#31774284C211yesmaximal, characteristic, complemented
8#31774272C212yesmaximal, complemented
9#31774268C212yesmaximal, complemented
10#31773849C211yesmaximal, characteristic, complemented
11#496×C271C211yesmaximal, characteristic, complemented
12C2×#16864655C2211yescharacteristic
13C2×#16864653C2211yescharacteristic
14C2×D4×D271—22no—
15C2×D4×D271C2212yes—
16#22576187—28no—
17#22576184—28no—
18C2×#16864643C2212yescomplemented
19C2×#16864642C2212yescomplemented
20C2×#16864641C2212yescomplemented
21C2×D2168C2212yescomplemented
22C2×#16864639C2212yescomplemented
23C2×#16864639C2212yescomplemented
24C2×#16864637C2212yes—
25#22576158C2211yescharacteristic, complemented
26#22576157C2211yescharacteristic
27#22576156—22no—
28#22576156C2212yes—
29#22576156C2211yescharacteristic, complemented
30#22576148C2211yescharacteristic, complemented
31#22576133—22no—
32(C22⋊D271)⋊C4—22no—
33C42×D271C2211yescharacteristic
34#22576071C2211yescharacteristic, complemented
35C8×Dic271C2212yescomplemented
36C2×SD16×C271C2212yescomplemented
37C2×D8×C271C2212yescomplemented
38#127×C271C2211yescharacteristic
39#124×C271C2211yescharacteristic, complemented
40#97×C271C2211yescharacteristic, complemented
41C23×D271—22no—
42#16864655—24no—
43#16864653—24no—
44C2×Dic542C2311yescharacteristic
45C2×D1084—21no—
46C2×D1084—22no—
47C2×D1084C2311yescharacteristic
48C2×D1084C2312yes—
49C2×C22⋊D271—21no—
50C2×C22⋊D271—22no—
51C2×C22⋊D271—22no—
52C2×C22⋊D271—24no—
53D4×D271—24no—
54D4×D271—24no—
55D4×D271—48no—
56C22×Dic271—21no—
57C2×C4×D271—21no—
58C2×C4×D271D412yes—
59C2×C4×D271C2311yescharacteristic
60#16864643—24no—
61#16864642—24no—
62#16864641—24no—
63D2168—24no—
64#16864639—24no—
65#16864639—24no—
66#16864637D418yescomplemented
67C2×C271⋊C8C2312yes—
68Dic271⋊C4—22no—
69(C2×D271)⋊C4—22no—
70C4×Dic271D412yes—
71C4×Dic271C2311yescharacteristic
72C2×Q8×C271C2311yescharacteristic
73C2×D4×C271—22no—
74C2×D4×C271C2312yes—
75C2×D4×C271C2311yescharacteristic
76SD16×C271—24no—
77D8×C271—24no—
78C2×C8×C271C2312yes—
79C22⋊C4×C271—22no—
80C42×C271C2311yescharacteristic
81C22×D271—21no—
82C22×D271—22no—
83C22×D271—48no—
84C22×D271—44no—
85C22×D271C2×D411yescharacteristic
86Dic542—22no—
87D1084—22no—
88D1084—22no—
89D1084—22no—
90D1084—44no—
91D1084—42no—
92C22⋊D271—42no—
93C22⋊D271—44no—
94C22⋊D271—48no—
95C22⋊D271—44no—
96C2×Dic271—21no—
97C2×Dic271—42no—
98C2×Dic271C2×D411yescharacteristic
99C2×Dic271C2×D412yes—
100C4×D271—24no—
101C4×D271—42no—
102C4×D271C2×D414yes—
103C271⋊C8C2×D414yescomplemented
104C23×C271—21no—
105C23×C271—22no—
106Q8×C271—22no—
107D4×C271—22no—
108D4×C271—22no—
109D4×C271—22no—
110D4×C271—44no—
111C2×C4×C271—21no—
112C2×C4×C271C2×D412yes—
113C2×C4×C271C2411yescharacteristic
114C8×C271C2×D414yescomplemented
115C22×C271—41no—
116C22×C271—42no—
117C22×C271—44no—
118C22×C271—42no—
119C22×C271C22×D411yescharacteristic
120C2×D271—22no—
121C2×D271—41no—
122C2×D271—42no—
123C2×D271—44no—
124C2×D271—42no—
125C2×D271—88no—
126C2×D271#12712yes—
127C4×C271—22no—
128C4×C271—41no—
129C4×C271C22×D412yes—
130Dic271—41no—
131Dic271#12712yes—
132Dic271AGL(1,ℤ/8)14yescomplemented
133C2×C271—82no—
134C2×C271—81no—
135C2×C271C2×#12711yescharacteristic
136C2×C271C2×AGL(1,ℤ/8)12yes—
137D271—22no—
138D271—81no—
139D271—82no—
140C271#315111yescharacteristic, complemented, Sylow
141#3151—2711nomaximal, Sylow
142C2×AGL(1,ℤ/8)—2712no—
143C2×AGL(1,ℤ/8)—2712no—
144#532—2711no—
145C2×#127—2711no—
146#496—2711no—
147#496—2711no—
148#496—2711no—
149#496D27111yescharacteristic, complemented
150#493—2712no—
151#492—2712no—
152C4×M16—2711no—
153C2×#41—2711no—
154C2×#41—2711no—
155C22×D4—2712no—
156C22×D4—5422no—
157AGL(1,ℤ/8)—5428no—
158AGL(1,ℤ/8)—5428no—
159C2×SD16—2712no—
160C2×SD16—2712no—
161C2×SD16—2712no—
162C2×SD16C2×D27112yescomplemented
163C2×D8—2712no—
164C2×D8—2712no—
165C2×D8—2712no—
166C2×D8C2×D27112yescomplemented
167C2×M16—2712no—
168C4⋊Q8—2711no—
169#127—2712no—
170#127—2711no—
171#127—2711no—
172#127—5422no—
173#127C2×D27111yescharacteristic
174#124—2711no—
175#124C2×D27111yescharacteristic, complemented
176#121—5422no—
177C4×D4—5422no—
178C2×C42—2711no—
179#97—2711no—
180#97C2×D27111yescharacteristic, complemented
181C4×C8—2712no—
182C24—5422no—
183#41—5424no—
184#41—5424no—
185C2×Q8—2711no—
186C2×Q8C22×D27111yescharacteristic
187C2×D4—22no—
188C2×D4—2711no—
189C2×D4—2712no—
190C2×D4—5421no—
191C2×D4—5424no—
192C2×D4—5424no—
193C2×D4—5422no—
194C2×D4—5424no—
195C2×D4—5422no—
196C2×D4—5421no—
197C2×D4—5422no—
198C2×D4—10848no—
199C2×D4C22×D27111yescharacteristic
200C2×D4C22×D27112yes—
201C22×C4—2711no—
202C22×C4—2712no—
203C22×C4—5421no—
204C22×C4—5421no—
205SD16—24no—
206SD16—5424no—
207SD16—5424no—
208SD16—5424no—
209D8—24no—
210D8—5424no—
211D8—5424no—
212D8—5424no—
213M16—2718no—
214C2×C8—2712no—
215C2×C8C22×D27112yes—
216C4⋊C4—5422no—
217C22⋊C4—22no—
218C22⋊C4—5422no—
219C42—2712no—
220C42—2711no—
221C42C22×D27111yescharacteristic
222C23—21no—
223C23—22no—
224C23—2711no—
225C23—5421no—
226C23—5422no—
227C23—10844no—
228C23—10848no—
229Q8—22no—
230Q8—5422no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—5422no—
236D4—5422no—
237D4—5422no—
238D4—10844no—
239D4—10842no—
240D4—10844no—
241D4—10848no—
242D4—10844no—
243D4—10842no—
244C2×C4—21no—
245C2×C4—2712no—
246C2×C4—2711no—
247C2×C4—2714no—
248C2×C4—5424no—
249C2×C4—5421no—
250C2×C4—10842no—
251C2×C4—10842no—
252C2×C4D4×D27112yes—
253C2×C4C23×D27111yescharacteristic
254C8—2714no—
255C8D4×D27114yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—2712no—
261C22—5422no—
262C22—10844no—
263C22—10842no—
264C22—10841no—
265C22—10842no—
266C22—21688no—
267C22C2×D4×D27111yescharacteristic
268C4—22no—
269C4—41no—
270C4—2714no—
271C4—2712no—
272C4—10841no—
273C4C2×D4×D27112yes—
274C2—82no—
275C2—81no—
276C2—5422no—
277C2—21682no—
278C2—21681no—
279C2#127×D27111yescharacteristic
280C2AGL(1,ℤ/8)×D27112yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#496×C271C2
socleC22×C271C22×D4
hypercenter#496D271
hypocenterC271#3151
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#22576187C2semidirect
2. C2×#22576184C2semidirect
3. #31774446C2semidirect
4. #127×D271C2semidirect
5. #31774287C2semidirect
6. #31774286C2semidirect
7. #31774284C2semidirect
8. #31774272C2semidirect
9. #31774268C2semidirect
10. #31773849C2semidirect
11. #496×C271C2semidirect
18. C2×#16864643C22semidirect
19. C2×#16864642C22semidirect
20. C2×#16864641C22semidirect
21. C2×D2168C22semidirect
22. C2×#16864639C22semidirect
23. C2×#16864639C22semidirect
25. #22576158C22semidirect
29. #22576156C22semidirect
30. #22576148C22semidirect
34. #22576071C22semidirect
35. C8×Dic271C22semidirect
36. C2×SD16×C271C22semidirect
37. C2×D8×C271C22semidirect
39. #124×C271C22semidirect
40. #97×C271C22semidirect
66. #16864637D4semidirect
103. C271⋊C8C2×D4semidirect
114. C8×C271C2×D4semidirect
132. Dic271AGL(1,ℤ/8)semidirect
140. C271#3151semidirect
149. #496D271semidirect
162. C2×SD16C2×D271semidirect
166. C2×D8C2×D271semidirect
175. #124C2×D271semidirect
180. #97C2×D271semidirect
255. C8D4×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761872
C2×#225761842
#317744462
#127×D2712
#317742872
#317742862
#317742842
#317742722
#317742682
#317738492
#496×C2712
#3151271