Finite Group Databasebeta

← #46847964

Subgroups — #46847964

Order 34688.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#22576185C212yesmaximal, complemented
2D4×D1084C211yesmaximal, characteristic, complemented
3#31774271C211yesmaximal, characteristic, complemented
4#31774145C212yesmaximal, complemented
5#31774099C211yesmaximal, characteristic, complemented
6#31774092C211yesmaximal, characteristic, complemented
7#31774047C212yesmaximal, complemented
8#31774042C212yesmaximal, complemented
9C4×#16864639C211yesmaximal, characteristic, complemented
10#31773975C211yesmaximal, characteristic, complemented
11#547×C271C211yesmaximal, characteristic, complemented
12C22×D1084C2212yes—
13C2×D4×D271—21no—
14#22576185—28no—
15C2×#16864641C2212yes—
16C2×#16864641C2212yescomplemented
17C2×#16864641C2212yescomplemented
18C2×#16864639C2211yescharacteristic
19C2×#16864639C2212yescomplemented
20C2×#16864638C2212yescomplemented
21#22576157C2211yescharacteristic
22#22576135—21no—
23#22576134C2212yes—
24#22576130—22no—
25C4×D1084C2211yescharacteristic
26#22576093C2211yescharacteristic, complemented
27C4⋊(C271⋊C8)C2211yescharacteristic, complemented
28#22576084C2212yescomplemented
29#22576081C2211yescharacteristic, complemented
30#22576080C2211yescharacteristic
31#22576080C2212yescomplemented
32C22⋊(C271⋊C8)C2212yescomplemented
33C4×C271⋊C8C2211yescharacteristic, complemented
34C2×#41×C271C2212yescomplemented
35C4⋊Q8×C271C2211yescharacteristic, complemented
36#123×C271—22no—
37C22⋊Q8×C271C2212yescomplemented
38#121×C271—21no—
39C4×D4×C271C2211yescharacteristic
40C4×D4×C271C2211yescharacteristic, complemented
41C2×C4⋊C4×C271—21no—
42C23×D271—22no—
43C2×D1084—24no—
44C2×D1084—22no—
45C2×D1084—21no—
46C2×D1084—21no—
47C2×D1084—42no—
48C2×D1084C2312yes—
49C2×D1084C2311yescharacteristic
50C2×C22⋊D271—22no—
51D4×D271—44no—
52C2×C4×D271—21no—
53#16864641—24no—
54#16864641—24no—
55#16864641—42no—
56#16864639—24no—
57#16864639D414yescomplemented
58#16864638—24no—
59C2×C271⋊C8C2312yes—
60C2×C271⋊C8C2311yescharacteristic
61C2×C271⋊C8C2311yescharacteristic
62C4⋊Dic271C2311yescharacteristic
63(C2×D271)⋊C4—21no—
64(C2×D271)⋊C4—22no—
65#41×C271—24no—
66#41×C271—42no—
67C2×Q8×C271C2312yes—
68C2×D4×C271—22no—
69C2×D4×C271D411yescharacteristic
70C2×D4×C271C2311yescharacteristic
71C22×C4×C271—21no—
72C22×C4×C271—22no—
73C22×C4×C271C2312yes—
74C4⋊C4×C271—21no—
75C4⋊C4×C271—22no—
76C4⋊C4×C271—21no—
77C4⋊C4×C271D411yescharacteristic
78C4⋊C4×C271C2311yescharacteristic
79C22⋊C4×C271—22no—
80C22⋊C4×C271—21no—
81C22⋊C4×C271D412yes—
82C42×C271C2311yescharacteristic
83C22×D271—21no—
84C22×D271—22no—
85C22×D271—44no—
86C22×D271—44no—
87C22×D271—82no—
88D1084—22no—
89D1084—21no—
90D1084—22no—
91D1084—44no—
92D1084—42no—
93D1084—81no—
94D1084C2×D412yes—
95C22⋊D271—44no—
96C2×Dic271—21no—
97C4×D271—42no—
98C271⋊C8—22no—
99C271⋊C8—21no—
100C271⋊C8C2×D412yescomplemented
101C23×C271—21no—
102C23×C271C2×D412yes—
103Q8×C271—22no—
104Q8×C271—22no—
105D4×C271—22no—
106D4×C271—21no—
107D4×C271—44no—
108D4×C271C2×D412yescomplemented
109C2×C4×C271—22no—
110C2×C4×C271—22no—
111C2×C4×C271—21no—
112C2×C4×C271—22no—
113C2×C4×C271—44no—
114C2×C4×C271—42no—
115C2×C4×C271C2×D411yescharacteristic
116C2×C4×C271C2×D412yes—
117C2×C4×C271C2×D411yescharacteristic
118C2×C4×C271C2411yescharacteristic
119C22×C271—22no—
120C22×C271—22no—
121C22×C271—22no—
122C22×C271—22no—
123C22×C271—41no—
124C22×C271—41no—
125C22×C271C22×D411yescharacteristic
126C2×D271—22no—
127C2×D271—41no—
128C2×D271—41no—
129C2×D271—42no—
130C2×D271—42no—
131C2×D271—42no—
132C2×D271—88no—
133C4×C271—22no—
134C4×C271—21no—
135C4×C271—42no—
136C4×C271—41no—
137C4×C271AGL(1,ℤ/8)12yescomplemented
138C4×C271C22×D411yescharacteristic
139C4×C271H(5,𝔽2)11yescharacteristic
140Dic271—41no—
141C2×C271—42no—
142C2×C271—42no—
143C2×C271D4211yescharacteristic
144C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C271#57711yescharacteristic
146D271—42no—
147D271—82no—
148C271#327211yescharacteristic, complemented, Sylow
149#3272—2711nomaximal, Sylow
150C2×AGL(1,ℤ/8)—2712no—
151#547D27111yescharacteristic, complemented
152D42—2711no—
153#492—2711no—
154#468—2712no—
155#461—2711no—
156#459—2711no—
157#450—2712no—
158#449—2712no—
159C4×D8—2711no—
160#436—2711no—
161C2×#41C2×D27112yescomplemented
162C22×D4—2712no—
163C22×D4—5421no—
164AGL(1,ℤ/8)—5428no—
165C2×SD16—2712no—
166C2×SD16—2712no—
167C2×SD16—2712no—
168C2×D8—2711no—
169C2×D8—2712no—
170C2×M16—2712no—
171C4⋊Q8C2×D27111yescharacteristic, complemented
172#127—2711no—
173#123—22no—
174C22⋊Q8C2×D27112yescomplemented
175#121—21no—
176#121—2712no—
177#121—5421no—
178#120—5422no—
179C4×D4—2711no—
180C4×D4C2×D27111yescharacteristic
181C4×D4C2×D27111yescharacteristic, complemented
182C2×C4⋊C4—21no—
183#107—2711no—
184C4⋊C8—2711no—
185#103—2712no—
186#102—2711no—
187#102—2711no—
188#102—2712no—
189C22⋊C8—2712no—
190C4×C8—2711no—
191C24—5422no—
192#41—24no—
193#41—42no—
194C2×Q8C22×D27112yes—
195C2×D4—22no—
196C2×D4—2711no—
197C2×D4—2712no—
198C2×D4—5424no—
199C2×D4—5422no—
200C2×D4—5422no—
201C2×D4—5421no—
202C2×D4—5421no—
203C2×D4—10842no—
204C2×D4—10844no—
205C2×D4C22⋊D27111yescharacteristic
206C2×D4C22×D27111yescharacteristic
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—5421no—
210C22×C4C22×D27112yes—
211SD16—5424no—
212SD16—5424no—
213SD16—10842no—
214D8—2714no—
215D8—5424no—
216M16—5424no—
217C2×C8—2711no—
218C2×C8—2711no—
219C2×C8—2712no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—2711no—
224C4⋊C4C22⋊D27111yescharacteristic
225C4⋊C4C22×D27111yescharacteristic
226C22⋊C4—22no—
227C22⋊C4—21no—
228C22⋊C4—5421no—
229C22⋊C4—5422no—
230C22⋊C4C22⋊D27112yes—
231C42C22×D27111yescharacteristic
232C23—21no—
233C23—5421no—
234C23—5422no—
235C23—10844no—
236C23—10844no—
237C23—21682no—
238C23C2×C22⋊D27112yes—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—21no—
243D4—44no—
244D4—2712no—
245D4—5422no—
246D4—5421no—
247D4—5422no—
248D4—10844no—
249D4—10844no—
250D4—10842no—
251D4—21681no—
252D4D4×D27112yescomplemented
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—5421no—
260C2×C4—10842no—
261C2×C4C2×C22⋊D27111yescharacteristic
262C2×C4C2×C22⋊D27112yes—
263C2×C4C2×C22⋊D27111yescharacteristic
264C2×C4C23×D27111yescharacteristic
265C8—2712no—
266C8—5422no—
267C8—5421no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—5422no—
275C22—10841no—
276C22—10841no—
277C22—10842no—
278C22—10842no—
279C22—10842no—
280C22—21688no—
281C22C22×C22⋊D27111yescharacteristic
282C4—22no—
283C4—21no—
284C4—42no—
285C4—41no—
286C4—10841no—
287C4#2257618512yescomplemented
288C4C2×D4×D27111yescharacteristic
289C4#2257620511yescharacteristic
290C2—42no—
291C2—42no—
292C2—10842no—
293C2—21682no—
294C2D4×C22⋊D27111yescharacteristic
295C2C2×#2257618511yescharacteristic
296C2#3177470511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#547×C271C2
socleC22×C271C22×D4
hypercenter#547D271
hypocenterC271#3272
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#22576185C2semidirect
2. D4×D1084C2semidirect
3. #31774271C2semidirect
4. #31774145C2semidirect
5. #31774099C2semidirect
6. #31774092C2semidirect
7. #31774047C2semidirect
8. #31774042C2semidirect
9. C4×#16864639C2semidirect
10. #31773975C2semidirect
11. #547×C271C2semidirect
16. C2×#16864641C22semidirect
17. C2×#16864641C22semidirect
19. C2×#16864639C22semidirect
20. C2×#16864638C22semidirect
26. #22576093C22semidirect
27. C4⋊(C271⋊C8)C22semidirect
28. #22576084C22semidirect
29. #22576081C22semidirect
31. #22576080C22semidirect
32. C22⋊(C271⋊C8)C22semidirect
33. C4×C271⋊C8C22semidirect
34. C2×#41×C271C22semidirect
35. C4⋊Q8×C271C22semidirect
37. C22⋊Q8×C271C22semidirect
40. C4×D4×C271C22semidirect
57. #16864639D4semidirect
100. C271⋊C8C2×D4semidirect
108. D4×C271C2×D4semidirect
137. C4×C271AGL(1,ℤ/8)semidirect
148. C271#3272semidirect
151. #547D271semidirect
161. C2×#41C2×D271semidirect
171. C4⋊Q8C2×D271semidirect
174. C22⋊Q8C2×D271semidirect
181. C4×D4C2×D271semidirect
252. D4D4×D271semidirect
287. C4#22576185semidirect

Maximal subgroups

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

subgroupindex
C2×#225761852
D4×D10842
#317742712
#317741452
#317740992
#317740922
#317740472
#317740422
C4×#168646392
#317739752
#547×C2712
#3272271