Finite Group Databasebeta

← #46846812

Subgroups — #46846812

Order 34688.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#22576188C211yesmaximal, characteristic, complemented
2C2×#22576186C211yesmaximal, characteristic, complemented
3C2×#22576185C211yesmaximal, characteristic, complemented
4#31774443C211yesmaximal, characteristic, complemented
5C2×#22576156C211yesmaximal, characteristic, complemented
6#31774285C212yesmaximal, complemented
7#31774283C212yesmaximal, complemented
8#31774272C212yesmaximal, complemented
9#31774268C212yesmaximal, complemented
10M16×Dic271C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C271C211yesmaximal, characteristic, complemented
12C2×#16864654C2211yescharacteristic
13C22×D1084C2211yescharacteristic
14C22×C22⋊D271—22no—
15#22576188—24no—
16#22576186—24no—
17#22576185—24no—
18C2×#16864643C2212yescomplemented
19C2×#16864642C2212yescomplemented
20C2×#16864641C2212yescomplemented
21C2×D2168C2212yescomplemented
22C2×#16864639C2212yescomplemented
23C2×#16864639C2212yescomplemented
24C2×#16864638C2211yescharacteristic
25#22576158C2211yescharacteristic, complemented
26#22576156—22no—
27#22576156C2212yes—
28#22576156C2212yes—
29#22576156C2211yescharacteristic, complemented
30#22576149C2211yescharacteristic, complemented
31#22576147C2211yescharacteristic, complemented
32#22576132—22no—
33C4×C22⋊D271—22no—
34C2×C4×Dic271C2211yescharacteristic
35#22576070C2212yescomplemented
36C8×Dic271C2212yescomplemented
37C2×#41×C271C2211yescharacteristic
38C22×D4×C271C2211yescharacteristic
39AGL(1,ℤ/8)×C271—24no—
40C2×SD16×C271C2212yescomplemented
41C2×D8×C271C2212yescomplemented
42C2×M16×C271C2211yescharacteristic
43C23×D271—21no—
44#16864654—22no—
45#16864654—22no—
46C2×Dic542C2311yescharacteristic
47C2×D1084—22no—
48C2×D1084—22no—
49C2×D1084—21no—
50C2×D1084C2311yescharacteristic
51C2×D1084C2312yes—
52C2×C22⋊D271—22no—
53C2×C22⋊D271—24no—
54C2×C22⋊D271—24no—
55C2×C22⋊D271—21no—
56C2×C22⋊D271—48no—
57C22×Dic271D412yes—
58C2×C4×D271—21no—
59#16864643—24no—
60#16864642—24no—
61#16864641—24no—
62D2168—24no—
63#16864639—24no—
64#16864639—24no—
65#16864638D414yescomplemented
66C2×C271⋊C8C2312yes—
67Dic271⋊C4—22no—
68C22⋊Dic271—22no—
69(C2×D271)⋊C4—22no—
70C4×Dic271D412yes—
71C4×Dic271C2311yescharacteristic
72C4×Dic271C2311yescharacteristic
73C24×C271—21no—
74#41×C271—22no—
75#41×C271—22no—
76C2×Q8×C271C2311yescharacteristic
77C2×D4×C271—21no—
78C2×D4×C271—22no—
79C2×D4×C271—22no—
80C2×D4×C271—21no—
81C2×D4×C271C2311yescharacteristic
82C2×D4×C271C2312yes—
83C22×C4×C271—21no—
84C22×C4×C271C2311yescharacteristic
85SD16×C271—24no—
86D8×C271—24no—
87M16×C271D414yescomplemented
88C2×C8×C271C2312yes—
89C22×D271—21no—
90C22×D271—22no—
91C22×D271—44no—
92C22×D271—42no—
93Dic542—21no—
94Dic542—21no—
95D1084—21no—
96D1084—22no—
97D1084—21no—
98D1084—22no—
99D1084—42no—
100C22⋊D271—48no—
101C22⋊D271—42no—
102C22⋊D271—44no—
103C22⋊D271—48no—
104C2×Dic271—24no—
105C2×Dic271—21no—
106C2×Dic271C2×D412yes—
107C2×Dic271C2×D412yes—
108C4×D271—42no—
109C271⋊C8C2×D414yescomplemented
110C23×C271—21no—
111C23×C271—22no—
112C23×C271—44no—
113C23×C271—42no—
114C23×C271C2×D411yescharacteristic
115Q8×C271—21no—
116Q8×C271—21no—
117D4×C271—21no—
118D4×C271—22no—
119D4×C271—22no—
120D4×C271—21no—
121D4×C271—42no—
122D4×C271—42no—
123C2×C4×C271—21no—
124C2×C4×C271—42no—
125C2×C4×C271C2×D411yescharacteristic
126C2×C4×C271C2×D412yes—
127C2×C4×C271C2×D412yes—
128C2×C4×C271C2411yescharacteristic
129C8×C271C2×D414yescomplemented
130C22×C271—22no—
131C22×C271—41no—
132C22×C271—42no—
133C22×C271—44no—
134C22×C271—42no—
135C22×C271—84no—
136C22×C271#12712yes—
137C22×C271C22×D411yescharacteristic
138C2×D271—41no—
139C2×D271—42no—
140C2×D271—44no—
141C2×D271—42no—
142C2×D271—84no—
143C4×C271—41no—
144C4×C271#12712yes—
145C4×C271C22×D411yescharacteristic
146C4×C271C22×D411yescharacteristic
147Dic271—22no—
148Dic271—41no—
149Dic271AGL(1,ℤ/8)14yescomplemented
150C2×C271—22no—
151C2×C271—81no—
152C2×C271—82no—
153C2×C271C2×#12711yescharacteristic
154C2×C271C2×AGL(1,ℤ/8)12yes—
155D271—81no—
156D271—82no—
157C271#315111yescharacteristic, complemented, Sylow
158#3151—2711nomaximal, Sylow
159C2×AGL(1,ℤ/8)—2711no—
160C2×AGL(1,ℤ/8)—2711no—
161C2×AGL(1,ℤ/8)—2711no—
162C2×AGL(1,ℤ/8)D27111yescharacteristic, complemented
163#532—2711no—
164C2×#127—2711no—
165#496—2712no—
166#496—2712no—
167#493—2712no—
168#492—2712no—
169C4×M16—2711no—
170C2×#41—2711no—
171C2×#41C2×D27111yescharacteristic
172C22×D4—2711no—
173C22×D4—5422no—
174C22×D4C2×D27111yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—5424no—
177AGL(1,ℤ/8)—5424no—
178AGL(1,ℤ/8)—5424no—
179C2×SD16—2712no—
180C2×SD16—2712no—
181C2×SD16—2712no—
182C2×SD16C2×D27112yescomplemented
183C2×D8—2712no—
184C2×D8—2712no—
185C2×D8—2712no—
186C2×D8C2×D27112yescomplemented
187C2×M16—2711no—
188C2×M16C2×D27111yescharacteristic
189C4⋊Q8—2711no—
190#127—2712no—
191#127—2712no—
192#127—2711no—
193#127—5422no—
194#124—2711no—
195#124—2711no—
196#121—5422no—
197C4×D4—5422no—
198C2×C42—2711no—
199#97—2712no—
200C4×C8—2712no—
201C24—21no—
202C24—5421no—
203#41—22no—
204#41—22no—
205#41—5422no—
206#41—5422no—
207C2×Q8—2711no—
208C2×Q8C22×D27111yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—2711no—
214C2×D4—2712no—
215C2×D4—5422no—
216C2×D4—5424no—
217C2×D4—5424no—
218C2×D4—5421no—
219C2×D4—5422no—
220C2×D4—5421no—
221C2×D4—5422no—
222C2×D4—10848no—
223C2×D4C22×D27111yescharacteristic
224C2×D4C22×D27112yes—
225C22×C4—21no—
226C22×C4—2712no—
227C22×C4—5421no—
228C22×C4C22×D27111yescharacteristic
229SD16—24no—
230SD16—5424no—
231SD16—5424no—
232SD16—5424no—
233D8—24no—
234D8—5424no—
235D8—5424no—
236D8—5424no—
237M16—2714no—
238M16C22⋊D27114yescomplemented
239C2×C8—2712no—
240C2×C8C22×D27112yes—
241C4⋊C4—5422no—
242C22⋊C4—5422no—
243C22⋊C4—5422no—
244C42—2712no—
245C42—2711no—
246C42—2711no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—5421no—
252C23—5422no—
253C23—10844no—
254C23—10842no—
255C23D4×D27111yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—5421no—
259Q8—5421no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—5421no—
267D4—5422no—
268D4—5421no—
269D4—5422no—
270D4—10848no—
271D4—10844no—
272D4—10848no—
273D4—10842no—
274D4—10842no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—2712no—
278C2×C4—2712no—
279C2×C4—5424no—
280C2×C4—5421no—
281C2×C4—10842no—
282C2×C4D4×D27111yescharacteristic
283C2×C4D4×D27112yes—
284C2×C4C2×C22⋊D27112yes—
285C2×C4C23×D27111yescharacteristic
286C8—2714no—
287C8C2×C22⋊D27114yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—10844no—
295C22—10842no—
296C22—10841no—
297C22—10842no—
298C22—21684no—
299C22#2257615612yes—
300C22C2×D4×D27111yescharacteristic
301C4—41no—
302C4—2714no—
303C4—5422no—
304C4—10841no—
305C4#2257615612yes—
306C4C2×D4×D27111yescharacteristic
307C4C22×C22⋊D27111yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—21682no—
312C2—21681no—
313C2C2×#2257615611yescharacteristic
314C2AGL(1,ℤ/8)×D27112yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×AGL(1,ℤ/8)×C271C2
socleC22×C271C22×D4
hypercenterC2×AGL(1,ℤ/8)D271
hypocenterC271#3151
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22576188C2semidirect
2. C2×#22576186C2semidirect
3. C2×#22576185C2semidirect
4. #31774443C2semidirect
5. C2×#22576156C2semidirect
6. #31774285C2semidirect
7. #31774283C2semidirect
8. #31774272C2semidirect
9. #31774268C2semidirect
10. M16×Dic271C2semidirect
11. C2×AGL(1,ℤ/8)×C271C2semidirect
18. C2×#16864643C22semidirect
19. C2×#16864642C22semidirect
20. C2×#16864641C22semidirect
21. C2×D2168C22semidirect
22. C2×#16864639C22semidirect
23. C2×#16864639C22semidirect
25. #22576158C22semidirect
29. #22576156C22semidirect
30. #22576149C22semidirect
31. #22576147C22semidirect
35. #22576070C22semidirect
36. C8×Dic271C22semidirect
40. C2×SD16×C271C22semidirect
41. C2×D8×C271C22semidirect
65. #16864638D4semidirect
87. M16×C271D4semidirect
109. C271⋊C8C2×D4semidirect
129. C8×C271C2×D4semidirect
149. Dic271AGL(1,ℤ/8)semidirect
157. C271#3151semidirect
162. C2×AGL(1,ℤ/8)D271semidirect
182. C2×SD16C2×D271semidirect
186. C2×D8C2×D271semidirect
238. M16C22⋊D271semidirect
287. C8C2×C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761882
C2×#225761862
C2×#225761852
#317744432
C2×#225761562
#317742852
#317742832
#317742722
#317742682
M16×Dic2712
C2×AGL(1,ℤ/8)×C2712
#3151271