Finite Group Databasebeta

← #46846826

Subgroups — #46846826

Order 34688.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22576193C211yesmaximal, characteristic, complemented
2C2×#22576190C211yesmaximal, characteristic, complemented
3C2×#22576186C211yesmaximal, characteristic, complemented
4#31774443C211yesmaximal, characteristic, complemented
5C2×#22576147C211yesmaximal, characteristic, complemented
6#31774288C212yesmaximal, complemented
7#31774285C212yesmaximal, complemented
8#31774280C212yesmaximal, complemented
9#31774278C212yesmaximal, complemented
10M16×Dic271C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C271C211yesmaximal, characteristic, complemented
12C2×#16864654C2211yescharacteristic
13C22×Dic542C2211yescharacteristic
14#22576193—24no—
15#22576190—24no—
16#22576186—24no—
17C2×Dic1084C2212yescomplemented
18C2×#16864644C2212yescomplemented
19C2×#16864643C2212yescomplemented
20C2×#16864642C2212yes—
21C2×#16864642C2212yescomplemented
22C2×#16864639C2212yes—
23C2×#16864638C2211yescharacteristic
24#22576158C2211yescharacteristic, complemented
25#22576156C2211yescharacteristic, complemented
26#22576149C2211yescharacteristic, complemented
27#22576147—22no—
28#22576147C2212yes—
29#22576147C2212yes—
30#22576147C2211yescharacteristic, complemented
31#22576132—22no—
32C4×C22⋊D271—22no—
33C2×C22⋊Dic271—22no—
34C2×C4×Dic271C2211yescharacteristic
35#22576070C2212yescomplemented
36C8×Dic271C2212yescomplemented
37C2×#41×C271C2211yescharacteristic
38C22×D4×C271C2211yescharacteristic
39AGL(1,ℤ/8)×C271—24no—
40C2×SD16×C271C2212yes—
41C2×D8×C271C2212yes—
42C2×M16×C271C2211yescharacteristic
43#16864654—22no—
44#16864654—22no—
45C2×Dic542—21no—
46C2×Dic542—22no—
47C2×Dic542—22no—
48C2×Dic542C2312yes—
49C2×Dic542C2311yescharacteristic
50C2×D1084C2311yescharacteristic
51C2×C22⋊D271—22no—
52C2×C22⋊D271—21no—
53C22×Dic271—21no—
54C22×Dic271D412yes—
55C2×C4×D271—21no—
56Dic1084—24no—
57#16864644—24no—
58#16864643—24no—
59#16864642—24no—
60#16864642—24no—
61#16864639—24no—
62#16864638D414yescomplemented
63C2×C271⋊C8C2312yes—
64Dic271⋊C4—22no—
65C22⋊Dic271—22no—
66C22⋊Dic271—24no—
67C22⋊Dic271—24no—
68(C2×D271)⋊C4—22no—
69C4×Dic271D412yes—
70C4×Dic271C2311yescharacteristic
71C4×Dic271C2311yescharacteristic
72C24×C271—21no—
73#41×C271—22no—
74#41×C271—22no—
75C2×Q8×C271C2311yescharacteristic
76C2×D4×C271—21no—
77C2×D4×C271—22no—
78C2×D4×C271—22no—
79C2×D4×C271—21no—
80C2×D4×C271C2312yes—
81C2×D4×C271C2311yescharacteristic
82C22×C4×C271—21no—
83C22×C4×C271C2311yescharacteristic
84SD16×C271—24no—
85D8×C271—24no—
86M16×C271D414yescomplemented
87C2×C8×C271C2312yes—
88C22×D271—21no—
89Dic542—21no—
90Dic542—22no—
91Dic542—22no—
92Dic542—21no—
93Dic542—42no—
94D1084—21no—
95D1084—21no—
96C22⋊D271—44no—
97C22⋊D271—42no—
98C2×Dic271—22no—
99C2×Dic271—21no—
100C2×Dic271—42no—
101C2×Dic271—42no—
102C2×Dic271C2×D412yes—
103C2×Dic271C2×D412yes—
104C4×D271—42no—
105C271⋊C8C2×D414yes—
106C23×C271—21no—
107C23×C271—22no—
108C23×C271—44no—
109C23×C271—42no—
110C23×C271C2×D411yescharacteristic
111Q8×C271—21no—
112Q8×C271—21no—
113D4×C271—21no—
114D4×C271—22no—
115D4×C271—22no—
116D4×C271—21no—
117D4×C271—42no—
118D4×C271—42no—
119C2×C4×C271—21no—
120C2×C4×C271—42no—
121C2×C4×C271C2×D411yescharacteristic
122C2×C4×C271C2×D412yes—
123C2×C4×C271C2×D412yes—
124C2×C4×C271C2411yescharacteristic
125C8×C271C2×D414yes—
126C22×C271—22no—
127C22×C271—41no—
128C22×C271—42no—
129C22×C271—44no—
130C22×C271—42no—
131C22×C271—84no—
132C22×C271#12712yes—
133C22×C271C22×D411yescharacteristic
134C2×D271—41no—
135C2×D271—42no—
136C4×C271—41no—
137C4×C271#12712yes—
138C4×C271C22×D411yescharacteristic
139C4×C271C22×D411yescharacteristic
140Dic271—22no—
141Dic271—22no—
142Dic271—41no—
143Dic271—42no—
144C2×C271—22no—
145C2×C271—81no—
146C2×C271—82no—
147C2×C271C2×#12711yescharacteristic
148C2×C271#57512yes—
149D271—81no—
150C271#315311yescharacteristic, complemented, Sylow
151#3153—2711nomaximal, Sylow
152C2×#137—2711no—
153C2×#137—2711no—
154C2×AGL(1,ℤ/8)—2711no—
155C2×AGL(1,ℤ/8)D27111yescharacteristic, complemented
156#532—2711no—
157C2×#124—2711no—
158#497—2712no—
159#496—2712no—
160#495—2712no—
161#495—2712no—
162C4×M16—2711no—
163C2×#41—2711no—
164C2×#41C2×D27111yescharacteristic
165C22×Q8—2711no—
166C22×D4C2×D27111yescharacteristic
167#137—5424no—
168#137—5424no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—5424no—
171C2×Q16—2712no—
172C2×Q16—2712no—
173C2×SD16—2712no—
174C2×SD16—2712no—
175C2×SD16—2712no—
176C2×SD16C2×D27112yes—
177C2×D8—2712no—
178C2×D8C2×D27112yes—
179C2×M16—2711no—
180C2×M16C2×D27111yescharacteristic
181C4⋊Q8—2711no—
182#127—2711no—
183#124—2711no—
184#124—2712no—
185#124—2712no—
186#124—2711no—
187#124—5422no—
188#121—5422no—
189C4×D4—5422no—
190C2×C22⋊C4—5422no—
191C2×C42—2711no—
192#97—2712no—
193C4×C8—2712no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—5422no—
198#41—5422no—
199C2×Q8—2712no—
200C2×Q8—2711no—
201C2×Q8—5421no—
202C2×Q8—5422no—
203C2×Q8—5422no—
204C2×Q8C22×D27111yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—2711no—
210C2×D4—5422no—
211C2×D4—5421no—
212C2×D4C22×D27112yes—
213C2×D4C22×D27111yescharacteristic
214C22×C4—21no—
215C22×C4—2712no—
216C22×C4—5421no—
217C22×C4—5421no—
218C22×C4C22×D27111yescharacteristic
219Q16—5424no—
220Q16—5424no—
221SD16—24no—
222SD16—5424no—
223SD16—5424no—
224SD16—5424no—
225D8—24no—
226D8—5424no—
227M16—2714no—
228M16C22⋊D27114yescomplemented
229C2×C8—2712no—
230C2×C8C22×D27112yes—
231C4⋊C4—5422no—
232C22⋊C4—5422no—
233C22⋊C4—5424no—
234C22⋊C4—5424no—
235C22⋊C4—5422no—
236C42—2712no—
237C42—2711no—
238C42—2711no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—5421no—
244C23D4×D27111yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—5421no—
248Q8—5422no—
249Q8—5422no—
250Q8—5421no—
251Q8—10842no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—5421no—
259D4—5421no—
260D4—10844no—
261D4—10842no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—2712no—
265C2×C4—2712no—
266C2×C4—5422no—
267C2×C4—5421no—
268C2×C4—10842no—
269C2×C4—10842no—
270C2×C4—10842no—
271C2×C4D4×D27111yescharacteristic
272C2×C4D4×D27112yes—
273C2×C4C2×C22⋊D27112yes—
274C2×C4C23×D27111yescharacteristic
275C8—2714no—
276C8C2×C22⋊D27114yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—10841no—
284C22—10842no—
285C22#2257615612yes—
286C22C2×D4×D27111yescharacteristic
287C4—41no—
288C4—5422no—
289C4—5422no—
290C4—10841no—
291C4—10842no—
292C4#2257615612yes—
293C4C2×D4×D27111yescharacteristic
294C4C22×C22⋊D27111yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—21681no—
299C2C2×#2257615611yescharacteristic
300C2#3177469312yes—
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#3153
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×#22576193C2semidirect
2. C2×#22576190C2semidirect
3. C2×#22576186C2semidirect
4. #31774443C2semidirect
5. C2×#22576147C2semidirect
6. #31774288C2semidirect
7. #31774285C2semidirect
8. #31774280C2semidirect
9. #31774278C2semidirect
10. M16×Dic271C2semidirect
11. C2×AGL(1,ℤ/8)×C271C2semidirect
17. C2×Dic1084C22semidirect
18. C2×#16864644C22semidirect
19. C2×#16864643C22semidirect
21. C2×#16864642C22semidirect
24. #22576158C22semidirect
25. #22576156C22semidirect
26. #22576149C22semidirect
30. #22576147C22semidirect
35. #22576070C22semidirect
36. C8×Dic271C22semidirect
62. #16864638D4semidirect
86. M16×C271D4semidirect
150. C271#3153semidirect
155. C2×AGL(1,ℤ/8)D271semidirect
228. M16C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761932
C2×#225761902
C2×#225761862
#317744432
C2×#225761472
#317742882
#317742852
#317742802
#317742782
M16×Dic2712
C2×AGL(1,ℤ/8)×C2712
#3153271