Finite Group Databasebeta

← #46846831

Subgroups — #46846831

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×#22576191C211yesmaximal, characteristic, complemented
2C2×#22576188C211yesmaximal, characteristic, complemented
3C2×#22576185C211yesmaximal, characteristic, complemented
4#31774443C211yesmaximal, characteristic, complemented
5C2×#22576149C211yesmaximal, characteristic, complemented
6#31774290C212yesmaximal, complemented
7#31774283C212yesmaximal, complemented
8#31774282C212yesmaximal, complemented
9#31774278C212yesmaximal, complemented
10M16×Dic271C211yesmaximal, characteristic, complemented
11C2×#137×C271C211yesmaximal, characteristic, complemented
12C2×#16864654C2211yescharacteristic
13C22×D1084C2211yescharacteristic
14#22576191—24no—
15#22576188—24no—
16#22576185—24no—
17C2×#16864644C2212yescomplemented
18C2×#16864643C2212yes—
19C2×#16864641C2212yes—
20C2×#16864641C2212yescomplemented
21C2×D2168C2212yes—
22C2×#16864639C2212yes—
23C2×#16864638C2211yescharacteristic
24#22576158C2211yescharacteristic, complemented
25#22576156C2211yescharacteristic, complemented
26#22576149—22no—
27#22576149C2212yes—
28#22576149C2212yes—
29#22576149C2211yescharacteristic, complemented
30#22576147C2211yescharacteristic, complemented
31#22576132—22no—
32C4×C22⋊D271—22no—
33C2×(C2×D271)⋊C4—22no—
34C2×C4×Dic271C2211yescharacteristic
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—22no—
53C2×C22⋊D271—21no—
54C22×Dic271D412yes—
55C2×C4×D271—21no—
56#16864644—24no—
57#16864643—24no—
58#16864641—24no—
59#16864641—24no—
60D2168—24no—
61#16864639—24no—
62#16864638D414yescomplemented
63C2×C271⋊C8C2312yes—
64Dic271⋊C4—22no—
65C22⋊Dic271—22no—
66(C2×D271)⋊C4—22no—
67(C2×D271)⋊C4—24no—
68(C2×D271)⋊C4—24no—
69C4×Dic271D412yes—
70C4×Dic271C2311yescharacteristic
71C4×Dic271C2311yescharacteristic
72#41×C271—22no—
73#41×C271—22no—
74C2×Q8×C271—21no—
75C2×Q8×C271—22no—
76C2×Q8×C271—22no—
77C2×Q8×C271C2312yes—
78C2×Q8×C271C2311yescharacteristic
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—22no—
89C22×D271—21no—
90C22×D271—44no—
91C22×D271—42no—
92Dic542—21no—
93Dic542—21no—
94D1084—22no—
95D1084—21no—
96D1084—22no—
97D1084—21no—
98D1084—42no—
99C22⋊D271—44no—
100C22⋊D271—42no—
101C2×Dic271—21no—
102C2×Dic271—42no—
103C2×Dic271C2×D412yes—
104C2×Dic271C2×D412yes—
105C4×D271—42no—
106C271⋊C8C2×D414yes—
107C23×C271—21no—
108C23×C271C2×D411yescharacteristic
109Q8×C271—22no—
110Q8×C271—21no—
111Q8×C271—22no—
112Q8×C271—21no—
113Q8×C271—42no—
114D4×C271—21no—
115D4×C271—21no—
116D4×C271—42no—
117C2×C4×C271—21no—
118C2×C4×C271—22no—
119C2×C4×C271—42no—
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#12712yes—
130C22×C271C22×D411yescharacteristic
131C2×D271—41no—
132C2×D271—42no—
133C2×D271—42no—
134C2×D271—44no—
135C2×D271—84no—
136C4×C271—41no—
137C4×C271—42no—
138C4×C271#12712yes—
139C4×C271C22×D411yescharacteristic
140C4×C271C22×D411yescharacteristic
141Dic271—22no—
142Dic271—22no—
143Dic271—41no—
144C2×C271—22no—
145C2×C271—81no—
146C2×C271C2×#12711yescharacteristic
147C2×C271#57512yes—
148D271—81no—
149D271—82no—
150C271#315311yescharacteristic, complemented, Sylow
151#3153—2711nomaximal, Sylow
152C2×#137—2711no—
153C2×#137D27111yescharacteristic, complemented
154C2×AGL(1,ℤ/8)—2711no—
155C2×AGL(1,ℤ/8)—2711no—
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×Q8C2×D27111yescharacteristic
166C22×D4—2711no—
167#137—24no—
168#137—5424no—
169AGL(1,ℤ/8)—5424no—
170AGL(1,ℤ/8)—5424no—
171C2×Q16—2712no—
172C2×Q16C2×D27112yescomplemented
173C2×SD16—2712no—
174C2×SD16—2712no—
175C2×SD16—2712no—
176C2×SD16C2×D27112yescomplemented
177C2×D8—2712no—
178C2×D8—2712no—
179C2×M16—2711no—
180C2×M16C2×D27111yescharacteristic
181C4⋊Q8—2711no—
182#127—2711no—
183#124—2712no—
184#124—2712no—
185#124—2711no—
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—5421no—
195#41—22no—
196#41—22no—
197#41—5422no—
198#41—5422no—
199C2×Q8—21no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—2711no—
203C2×Q8C22×D27112yes—
204C2×Q8C22×D27111yescharacteristic
205C2×D4—21no—
206C2×D4—2712no—
207C2×D4—2711no—
208C2×D4—5422no—
209C2×D4—5421no—
210C2×D4—5422no—
211C2×D4—5421no—
212C2×D4—5422no—
213C2×D4C22×D27111yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—2712no—
217C22×C4—5421no—
218C22×C4C22×D27111yescharacteristic
219Q16—24no—
220Q16—5424no—
221SD16—24no—
222SD16—5424no—
223SD16—5424no—
224SD16—5424no—
225D8—5424no—
226D8—5424no—
227M16—2714no—
228M16C22⋊D27114yescomplemented
229C2×C8—2712no—
230C2×C8C22×D27112yes—
231C4⋊C4—5422no—
232C22⋊C4—5424no—
233C22⋊C4—5424no—
234C22⋊C4—5422no—
235C22⋊C4—5422no—
236C42—2712no—
237C42—2711no—
238C42—2711no—
239C23—21no—
240C23—5422no—
241C23—5421no—
242C23—10844no—
243C23—10842no—
244C23D4×D27111yescharacteristic
245Q8—21no—
246Q8—22no—
247Q8—22no—
248Q8—21no—
249Q8—42no—
250Q8—5421no—
251Q8—5421no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—5422no—
256D4—5421no—
257D4—5421no—
258D4—5422no—
259D4—10844no—
260D4—10842no—
261D4—10842no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—2712no—
267C2×C4—2712no—
268C2×C4—5421no—
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—10841no—
281C22—10842no—
282C22—10842no—
283C22—10844no—
284C22—21684no—
285C22#2257615612yes—
286C22C2×D4×D27111yescharacteristic
287C4—41no—
288C4—42no—
289C4—5422no—
290C4—5422no—
291C4—10841no—
292C4#2257615612yes—
293C4C2×D4×D27111yescharacteristic
294C4C22×C22⋊D27111yescharacteristic
295C2—22no—
296C2—81no—
297C2—21682no—
298C2—21681no—
299C2C2×#2257615611yescharacteristic
300C2#3177469612yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×#137×C271C2
socleC22×C271C22×D4
hypercenterC2×#137D271
hypocenterC271#3153
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×#22576191C2semidirect
2. C2×#22576188C2semidirect
3. C2×#22576185C2semidirect
4. #31774443C2semidirect
5. C2×#22576149C2semidirect
6. #31774290C2semidirect
7. #31774283C2semidirect
8. #31774282C2semidirect
9. #31774278C2semidirect
10. M16×Dic271C2semidirect
11. C2×#137×C271C2semidirect
17. C2×#16864644C22semidirect
20. C2×#16864641C22semidirect
24. #22576158C22semidirect
25. #22576156C22semidirect
29. #22576149C22semidirect
30. #22576147C22semidirect
35. #22576070C22semidirect
36. C8×Dic271C22semidirect
40. C2×Q16×C271C22semidirect
41. C2×SD16×C271C22semidirect
62. #16864638D4semidirect
86. M16×C271D4semidirect
150. C271#3153semidirect
153. C2×#137D271semidirect
172. C2×Q16C2×D271semidirect
176. C2×SD16C2×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×#225761912
C2×#225761882
C2×#225761852
#317744432
C2×#225761492
#317742902
#317742832
#317742822
#317742782
M16×Dic2712
C2×#137×C2712
#3153271