Finite Group Databasebeta

← #46846843

Subgroups — #46846843

Order 34688.

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

normal subgroups only

This group has 564 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×#22576188C211yesmaximal, characteristic, complemented
2C2×#22576180C211yesmaximal, characteristic, complemented
3C22×#16864641C211yesmaximal, characteristic, complemented
4#31774471C211yesmaximal, characteristic, complemented
5#31774461C211yesmaximal, characteristic, complemented
6#31774290C212yesmaximal, complemented
7#31774283C212yesmaximal, complemented
8#31774282C212yesmaximal, complemented
9#31774268C212yesmaximal, complemented
10#31773855C211yesmaximal, characteristic, complemented
11C2×#137×C271C211yesmaximal, characteristic, complemented
12C2×#16864654C2211yescharacteristic
13C22×D1084C2211yescharacteristic
14#22576188—24no—
15#22576180—24no—
16C2×#16864644C2211yescharacteristic, complemented
17C2×#16864643C2212yes—
18C2×#16864642C2211yescharacteristic, complemented
19C2×#16864641—24no—
20C2×#16864641C2212yes—
21C2×#16864641C2212yes—
22C2×#16864641C2211yescharacteristic, complemented
23C2×D2168C2212yes—
24C2×#16864639C2211yescharacteristic, complemented
25C22×C271⋊C8C2211yescharacteristic
26#22576158C2212yescomplemented
27#22576156C2212yes—
28#22576149C2212yes—
29#22576149C2212yescomplemented
30#22576143—22no—
31#22576136—22no—
32#22576134—22no—
33#22576128—22no—
34#22576119C2211yescharacteristic
35#22576070C2212yescomplemented
36C8×Dic271C2212yescomplemented
37C2×#41×C271C2211yescharacteristic
38C22×Q8×C271C2211yescharacteristic
39#137×C271—24no—
40C2×Q16×C271C2212yescomplemented
41C2×SD16×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×D1084C2312yes—
51C2×D1084C2311yescharacteristic
52C2×C22⋊D271—24no—
53C2×C22⋊D271—21no—
54C2×C4×D271—21no—
55#16864644—22no—
56#16864643—24no—
57#16864642—22no—
58#16864641—24no—
59#16864641—24no—
60#16864641—22no—
61D2168—24no—
62#16864639—22no—
63C2×C271⋊C8D414yescomplemented
64C2×C271⋊C8C2311yescharacteristic
65C2×C271⋊C8C2311yescharacteristic
66C4⋊Dic271D412yes—
67Dic271⋊C4—24no—
68C22⋊Dic271D412yes—
69(C2×D271)⋊C4—24no—
70(C2×D271)⋊C4—24no—
71C4×Dic271C2312yes—
72#41×C271—22no—
73#41×C271—22no—
74C2×Q8×C271—21no—
75C2×Q8×C271—22no—
76C2×Q8×C271—22no—
77C2×Q8×C271C2311yescharacteristic
78C2×Q8×C271C2312yes—
79C2×D4×C271—21no—
80C2×D4×C271C2311yescharacteristic
81C22×C4×C271—21no—
82C22×C4×C271—21no—
83C22×C4×C271C2311yescharacteristic
84Q16×C271—24no—
85SD16×C271—24no—
86M16×C271D414yescomplemented
87C2×C8×C271C2312yes—
88C22×D271—21no—
89C22×D271—22no—
90C22×D271—42no—
91C22×D271—82no—
92Dic542—21no—
93Dic542—21no—
94D1084—21no—
95D1084—21no—
96D1084—22no—
97D1084—22no—
98D1084—42no—
99C22⋊D271—42no—
100C22⋊D271—84no—
101C2×Dic271—21no—
102C2×Dic271C2×D414yes—
103C4×D271—42no—
104C271⋊C8C2×D412yes—
105C271⋊C8C2×D412yes—
106C23×C271—21no—
107C23×C271C2×D411yescharacteristic
108Q8×C271—22no—
109Q8×C271—21no—
110Q8×C271—22no—
111Q8×C271—21no—
112Q8×C271—42no—
113D4×C271—21no—
114D4×C271—21no—
115D4×C271—42no—
116C2×C4×C271—21no—
117C2×C4×C271—22no—
118C2×C4×C271—42no—
119C2×C4×C271—42no—
120C2×C4×C271C2×D411yescharacteristic
121C2×C4×C271C2×D412yes—
122C2×C4×C271C2×D412yes—
123C2×C4×C271C2411yescharacteristic
124C8×C271C2×D414yes—
125C22×C271—22no—
126C22×C271—41no—
127C22×C271—42no—
128C22×C271#12712yes—
129C22×C271C22×D411yescharacteristic
130C2×D271—41no—
131C2×D271—42no—
132C2×D271—44no—
133C2×D271—42no—
134C2×D271—84no—
135C4×C271—41no—
136C4×C271—42no—
137C4×C271#12712yes—
138C4×C271C22×D411yescharacteristic
139C4×C271C22×D411yescharacteristic
140Dic271—24no—
141Dic271—41no—
142C2×C271—22no—
143C2×C271—81no—
144C2×C271C2×#12711yescharacteristic
145C2×C271#57712yes—
146D271—81no—
147D271—82no—
148C271#315411yescharacteristic, complemented, Sylow
149#3154—2711nomaximal, Sylow
150C2×#137D27111yescharacteristic, complemented
151C2×AGL(1,ℤ/8)—2711no—
152C2×#135—2711no—
153C22×SD16—2711no—
154#536—2711no—
155#535—2711no—
156#497—2712no—
157#496—2712no—
158#495—2712no—
159#492—2712no—
160#405—2711no—
161C2×#41—2711no—
162C2×#41C2×D27111yescharacteristic
163C22×Q8C2×D27111yescharacteristic
164C22×D4—2711no—
165#137—24no—
166AGL(1,ℤ/8)—5424no—
167#135—5424no—
168C2×Q16—2711no—
169C2×Q16C2×D27112yescomplemented
170C2×SD16—2712no—
171C2×SD16—2712no—
172C2×SD16—2711no—
173C2×SD16—2711no—
174C2×SD16—2712no—
175C2×SD16—5424no—
176C2×SD16C2×D27112yescomplemented
177C2×D8—2711no—
178C2×D8—2712no—
179C2×M16C2×D27111yescharacteristic
180C22×C8—2711no—
181C4⋊Q8—2712no—
182#127—2712no—
183#124—2712no—
184#124—2712no—
185#123—5422no—
186C22⋊Q8—5422no—
187#121—5422no—
188#120—5422no—
189#117—2711no—
190#97—2712no—
191C4×C8—2712no—
192C24—5421no—
193#41—22no—
194#41—22no—
195#41—5422no—
196#41—5422no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—2711no—
201C2×Q8C22×D27111yescharacteristic
202C2×Q8C22×D27112yes—
203C2×D4—21no—
204C2×D4—2711no—
205C2×D4—2712no—
206C2×D4—5424no—
207C2×D4—5421no—
208C2×D4—5422no—
209C2×D4—5421no—
210C2×D4—5422no—
211C2×D4C22×D27111yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—5421no—
215C22×C4C22×D27111yescharacteristic
216Q16—24no—
217Q16—5422no—
218SD16—24no—
219SD16—5422no—
220SD16—5424no—
221SD16—5424no—
222SD16—5422no—
223SD16—5424no—
224D8—5422no—
225D8—5424no—
226M16C22⋊D27114yescomplemented
227C2×C8—2714no—
228C2×C8—2711no—
229C2×C8—2711no—
230C2×C8C22×D27112yes—
231C4⋊C4—2712no—
232C4⋊C4—5424no—
233C22⋊C4—2712no—
234C22⋊C4—5424no—
235C22⋊C4—5424no—
236C42—2712no—
237C23—21no—
238C23—5421no—
239C23—5422no—
240C23—10842no—
241C23—21682no—
242C23D4×D27111yescharacteristic
243Q8—22no—
244Q8—21no—
245Q8—22no—
246Q8—21no—
247Q8—42no—
248Q8—5421no—
249Q8—5421no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—5421no—
254D4—5422no—
255D4—5421no—
256D4—5422no—
257D4—10842no—
258D4—10842no—
259D4—21684no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—2714no—
265C2×C4—5421no—
266C2×C4—10842no—
267C2×C4D4×D27111yescharacteristic
268C2×C4D4×D27112yes—
269C2×C4C2×C22⋊D27112yes—
270C2×C4C23×D27111yescharacteristic
271C8—2712no—
272C8—2712no—
273C8C2×C22⋊D27114yes—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—10841no—
278C22—10842no—
279C22—10844no—
280C22—10842no—
281C22—21684no—
282C22#2257615612yes—
283C22C2×D4×D27111yescharacteristic
284C4—41no—
285C4—42no—
286C4—5424no—
287C4—10841no—
288C4#2257615612yes—
289C4C2×D4×D27111yescharacteristic
290C4C22×C22⋊D27111yescharacteristic
291C2—22no—
292C2—81no—
293C2—21682no—
294C2—21681no—
295C2C2×#2257615611yescharacteristic
296C2#3177470612yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×#137×C271C2
socleC22×C271C22×D4
hypercenterC2×#137D271
hypocenterC271#3154
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#22576188C2semidirect
2. C2×#22576180C2semidirect
3. C22×#16864641C2semidirect
4. #31774471C2semidirect
5. #31774461C2semidirect
6. #31774290C2semidirect
7. #31774283C2semidirect
8. #31774282C2semidirect
9. #31774268C2semidirect
10. #31773855C2semidirect
11. C2×#137×C271C2semidirect
16. C2×#16864644C22semidirect
18. C2×#16864642C22semidirect
22. C2×#16864641C22semidirect
24. C2×#16864639C22semidirect
26. #22576158C22semidirect
29. #22576149C22semidirect
35. #22576070C22semidirect
36. C8×Dic271C22semidirect
40. C2×Q16×C271C22semidirect
41. C2×SD16×C271C22semidirect
63. C2×C271⋊C8D4semidirect
86. M16×C271D4semidirect
148. C271#3154semidirect
150. C2×#137D271semidirect
169. C2×Q16C2×D271semidirect
176. C2×SD16C2×D271semidirect
226. M16C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761882
C2×#225761802
C22×#168646412
#317744712
#317744612
#317742902
#317742832
#317742822
#317742682
#317738552
C2×#137×C2712
#3154271