Finite Group Databasebeta

← #46846840

Subgroups — #46846840

Order 34688.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#22576188C211yesmaximal, characteristic, complemented
3C2×#22576182C211yesmaximal, characteristic, complemented
4C22×#16864643C211yesmaximal, characteristic, complemented
5#31774473C211yesmaximal, characteristic, complemented
6#31774466C211yesmaximal, characteristic, complemented
7#31774291C212yesmaximal, complemented
8#31774286C212yesmaximal, complemented
9#31774281C212yesmaximal, complemented
10#31774270C212yesmaximal, complemented
11#405×C271C211yesmaximal, characteristic, complemented
12C2×#16864654C2211yescharacteristic
13C2×#16864654C2211yescharacteristic
14C22×Dic542C2211yescharacteristic
15C22×D1084C2211yescharacteristic
16#22576193—24no—
17#22576188—24no—
18#22576182—24no—
19C2×Dic1084C2212yescomplemented
20C2×Dic1084C2211yescharacteristic, complemented
21C2×#16864643—24no—
22C2×#16864643C2212yes—
23C2×#16864643C2212yes—
24C2×#16864643C2212yes—
25C2×#16864643C2211yescharacteristic, complemented
26C2×#16864643C2212yescomplemented
27C2×#16864643C2211yescharacteristic, complemented
28C2×D2168C2212yes—
29C2×D2168C2211yescharacteristic, complemented
30#22576160C2212yescomplemented
31#22576157C2212yes—
32#22576148C2212yes—
33#22576148C2212yescomplemented
34#22576144—22no—
35#22576140—22no—
36#22576135—22no—
37#22576130—22no—
38C2×M16×C271C2211yescharacteristic
39C22×C8×C271C2211yescharacteristic
40#117×C271C2211yescharacteristic
41#97×C271C2212yescomplemented
42C4×C8×C271C2212yescomplemented
43C23×D271—21no—
44#16864654—22no—
45#16864654—22no—
46#16864654—22no—
47#16864654—22no—
48C2×Dic542—21no—
49C2×Dic542—22no—
50C2×Dic542—22no—
51C2×Dic542C2311yescharacteristic
52C2×Dic542C2312yes—
53C2×Dic542C2311yescharacteristic
54C2×D1084—22no—
55C2×D1084—22no—
56C2×D1084—21no—
57C2×D1084—24no—
58C2×D1084C2312yes—
59C2×D1084C2311yescharacteristic
60C2×D1084C2311yescharacteristic
61C2×C22⋊D271—21no—
62C2×C22⋊D271—21no—
63C22×Dic271—21no—
64C2×C4×D271—21no—
65C2×C4×D271—21no—
66Dic1084—22no—
67Dic1084—24no—
68#16864643—24no—
69#16864643—24no—
70#16864643—24no—
71#16864643—22no—
72#16864643—22no—
73#16864643—24no—
74D2168—24no—
75D2168—22no—
76C4⋊Dic271—24no—
77(C2×D271)⋊C4—24no—
78(C2×D271)⋊C4—24no—
79C22×C4×C271C2311yescharacteristic
80M16×C271D414yescomplemented
81C2×C8×C271D414yescomplemented
82C2×C8×C271C2312yes—
83C2×C8×C271C2311yescharacteristic
84C2×C8×C271C2311yescharacteristic
85C4⋊C4×C271D412yes—
86C22⋊C4×C271D412yes—
87C42×C271C2312yes—
88C22×D271—21no—
89C22×D271—22no—
90C22×D271—21no—
91C22×D271—42no—
92C22×D271—82no—
93Dic542—22no—
94Dic542—21no—
95Dic542—22no—
96Dic542—21no—
97Dic542—21no—
98Dic542—21no—
99Dic542—42no—
100D1084—21no—
101D1084—22no—
102D1084—21no—
103D1084—21no—
104D1084—22no—
105D1084—21no—
106D1084—42no—
107D1084—84no—
108C22⋊D271—42no—
109C22⋊D271—42no—
110C2×Dic271—21no—
111C2×Dic271—22no—
112C2×Dic271—21no—
113C2×Dic271—42no—
114C4×D271—42no—
115C4×D271—42no—
116C23×C271C2×D411yescharacteristic
117C2×C4×C271C2×D414yes—
118C2×C4×C271C2×D411yescharacteristic
119C2×C4×C271C2×D412yes—
120C2×C4×C271C2×D412yes—
121C2×C4×C271C2411yescharacteristic
122C8×C271C2×D412yes—
123C8×C271C2×D412yes—
124C8×C271C2×D414yes—
125C22×C271—22no—
126C22×C271#12712yes—
127C22×C271C22×D411yescharacteristic
128C2×D271—41no—
129C2×D271—41no—
130C2×D271—42no—
131C2×D271—42no—
132C2×D271—44no—
133C2×D271—42no—
134C2×D271—84no—
135C4×C271—24no—
136C4×C271#12712yes—
137C4×C271C22×D411yescharacteristic
138C4×C271C22×D411yescharacteristic
139Dic271—41no—
140Dic271—41no—
141Dic271—42no—
142C2×C271—22no—
143C2×C271C2×#12711yescharacteristic
144C2×C271#57712yes—
145D271—81no—
146D271—81no—
147D271—82no—
148C271#315411yescharacteristic, complemented, Sylow
149#3154—2711nomaximal, Sylow
150C2×#137—2711no—
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#405D27111yescharacteristic, complemented
161C2×#41—2711no—
162C2×#41—2711no—
163C22×Q8—2711no—
164C22×D4—2711no—
165#137—5424no—
166AGL(1,ℤ/8)—5424no—
167#135—5424no—
168C2×Q16—2711no—
169C2×Q16—2712no—
170C2×SD16—2712no—
171C2×SD16—2712no—
172C2×SD16—2712no—
173C2×SD16—2711no—
174C2×SD16—2711no—
175C2×SD16—2712no—
176C2×SD16—5424no—
177C2×D8—2712no—
178C2×D8—2711no—
179C2×M16C2×D27111yescharacteristic
180C22×C8C2×D27111yescharacteristic
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#117C2×D27111yescharacteristic
190#97C2×D27112yescomplemented
191C4×C8C2×D27112yescomplemented
192C24—5421no—
193#41—5422no—
194#41—5422no—
195#41—5422no—
196#41—5422no—
197C2×Q8—2711no—
198C2×Q8—2712no—
199C2×Q8—2711no—
200C2×Q8—5421no—
201C2×Q8—5422no—
202C2×Q8—5422no—
203C2×D4—2711no—
204C2×D4—2712no—
205C2×D4—2711no—
206C2×D4—5421no—
207C2×D4—5421no—
208C2×D4—5422no—
209C2×D4—5422no—
210C2×D4—5421no—
211C2×D4—5424no—
212C22×C4—5421no—
213C22×C4—5421no—
214C22×C4—5421no—
215C22×C4C22×D27111yescharacteristic
216Q16—5422no—
217Q16—5424no—
218SD16—5424no—
219SD16—5424no—
220SD16—5422no—
221SD16—5422no—
222SD16—5424no—
223SD16—5424no—
224D8—5422no—
225D8—5424no—
226M16D108414yescomplemented
227C2×C8D108414yescomplemented
228C2×C8C22×D27111yescharacteristic
229C2×C8C22×D27112yes—
230C2×C8C22×D27111yescharacteristic
231C4⋊C4—5424no—
232C4⋊C4D108412yes—
233C22⋊C4—5424no—
234C22⋊C4—5424no—
235C22⋊C4D108412yes—
236C42C22×D27112yes—
237C23—5421no—
238C23—5422no—
239C23—5421no—
240C23—10842no—
241C23—21682no—
242C23C2×D108411yescharacteristic
243Q8—5422no—
244Q8—5421no—
245Q8—5422no—
246Q8—5421no—
247Q8—5421no—
248Q8—5421no—
249Q8—10842no—
250D4—5421no—
251D4—5422no—
252D4—5421no—
253D4—5421no—
254D4—5422no—
255D4—5421no—
256D4—10842no—
257D4—10842no—
258D4—10842no—
259D4—21684no—
260C2×C4—5421no—
261C2×C4—5422no—
262C2×C4—5421no—
263C2×C4—10842no—
264C2×C4—10842no—
265C2×C4—10842no—
266C2×C4C2×D108414yes—
267C2×C4C2×D108411yescharacteristic
268C2×C4C2×D108412yes—
269C2×C4C2×D108412yes—
270C2×C4C23×D27111yescharacteristic
271C8C2×D108412yes—
272C8C2×D108412yes—
273C8C2×D108414yes—
274C22—22no—
275C22—10841no—
276C22—10841no—
277C22—10842no—
278C22—10842no—
279C22—10844no—
280C22—10842no—
281C22—21684no—
282C22#2257615712yes—
283C22C22×D108411yescharacteristic
284C4—24no—
285C4—10841no—
286C4—10841no—
287C4—10842no—
288C4#2257615712yes—
289C4C22×D108411yescharacteristic
290C4C22×D108411yescharacteristic
291C2—22no—
292C2—21682no—
293C2—21681no—
294C2—21681no—
295C2C2×#2257615711yescharacteristic
296C2#3177470912yes—
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#405×C271C2
socleC22×C271C22×D4
hypercenter#405D271
hypocenterC271#3154
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. C2×#22576193C2semidirect
2. C2×#22576188C2semidirect
3. C2×#22576182C2semidirect
4. C22×#16864643C2semidirect
5. #31774473C2semidirect
6. #31774466C2semidirect
7. #31774291C2semidirect
8. #31774286C2semidirect
9. #31774281C2semidirect
10. #31774270C2semidirect
11. #405×C271C2semidirect
19. C2×Dic1084C22semidirect
20. C2×Dic1084C22semidirect
25. C2×#16864643C22semidirect
26. C2×#16864643C22semidirect
27. C2×#16864643C22semidirect
29. C2×D2168C22semidirect
30. #22576160C22semidirect
33. #22576148C22semidirect
41. #97×C271C22semidirect
42. C4×C8×C271C22semidirect
80. M16×C271D4semidirect
81. C2×C8×C271D4semidirect
148. C271#3154semidirect
160. #405D271semidirect
190. #97C2×D271semidirect
191. C4×C8C2×D271semidirect
226. M16D1084semidirect
227. C2×C8D1084semidirect

Maximal subgroups

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

subgroupindex
C2×#225761932
C2×#225761882
C2×#225761822
C22×#168646432
#317744732
#317744662
#317742912
#317742862
#317742812
#317742702
#405×C2712
#3154271