Finite Group Databasebeta

← #46845643

Subgroups — #46845643

Order 34688.

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

normal subgroups only

This group has 752 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×#22576206C211yesmaximal, characteristic
2C2×#22576193C211yesmaximal, characteristic, complemented
3C22×#16864643C211yesmaximal, characteristic, complemented
4#31774468C211yesmaximal, characteristic, complemented
5#31774061C214yesmaximal, complemented
6#31774050C214yesmaximal, complemented
7#31773903C211yesmaximal, characteristic, complemented
8#31773894C211yesmaximal, characteristic, complemented
9#408×C271C211yesmaximal, characteristic, complemented
10#22576206—24no—
11#22576206—24no—
12C2×#16864655—21no—
13C2×#16864654C2212yes—
14C2×#16864654C2211yescharacteristic
15C22×Dic542C2211yescharacteristic, complemented
16C22×D1084C2212yes—
17C22×D1084C2211yescharacteristic
18C2×D4×D271—22no—
19C2×D4×D271—21no—
20#22576193—24no—
21C2×Dic1084C2212yescomplemented
22C2×#16864643—24no—
23C2×#16864643C2212yescomplemented
24C2×#16864643C2214yescomplemented
25#22576158—21no—
26#22576147—21no—
27#22576141—22no—
28#22576139C2214yescomplemented
29#22576119C2211yescharacteristic, complemented
30#22576085C2212yescomplemented
31#22576082C2212yescomplemented
32#22576082C2214yescomplemented
33C2×#41×C271C2211yescharacteristic
34C2×M16×C271C2211yescharacteristic, complemented
35C22×C8×C271C2211yescharacteristic, complemented
36C22⋊C8×C271C2214yescomplemented
37C23×D271—22no—
38C23×D271—21no—
39#16864655—28no—
40#16864654—24no—
41#16864654—24no—
42#16864654—22no—
43#16864654D414yes—
44C2×Dic542—21no—
45C2×Dic542—22no—
46C2×Dic542—22no—
47C2×Dic542C2312yes—
48C2×Dic542C2311yescharacteristic
49C2×D1084—24no—
50C2×D1084—24no—
51C2×D1084—22no—
52C2×D1084—21no—
53C2×D1084—22no—
54C2×D1084D414yes—
55C2×D1084C2311yescharacteristic
56C2×D1084C2312yes—
57C2×C22⋊D271—22no—
58C2×C22⋊D271—21no—
59D4×D271—216no—
60D4×D271—28no—
61C22×Dic271—21no—
62C2×C4×D271—22no—
63C2×C4×D271—21no—
64Dic1084—24no—
65#16864643—28no—
66#16864643—24no—
67C4⋊Dic271C2312yes—
68Dic271⋊C4—24no—
69C22⋊Dic271—24no—
70C22⋊Dic271—21no—
71C4×Dic271—21no—
72#41×C271—22no—
73#41×C271—22no—
74C2×Q8×C271D411yescharacteristic
75C2×D4×C271D412yes—
76C2×D4×C271D411yescharacteristic
77C22×C4×C271C2312yes—
78C22×C4×C271C2311yescharacteristic
79M16×C271—22no—
80C2×C8×C271—22no—
81C2×C8×C271C2312yes—
82C2×C8×C271C2312yes—
83C22×D271—22no—
84C22×D271—21no—
85C22×D271—28no—
86C22×D271—24no—
87C22×D271—82no—
88C22×D271—84no—
89Dic542—21no—
90Dic542—22no—
91Dic542—22no—
92Dic542—42no—
93Dic542C2×D412yes—
94D1084—28no—
95D1084—24no—
96D1084—22no—
97D1084—21no—
98D1084C2×D412yes—
99D1084C2×D414yes—
100C22⋊D271—28no—
101C22⋊D271—24no—
102C2×Dic271—21no—
103C2×Dic271—22no—
104C2×Dic271—22no—
105C2×Dic271—42no—
106C4×D271—28no—
107C4×D271—24no—
108C23×C271C2×D412yes—
109C23×C271C2×D411yescharacteristic
110Q8×C271—22no—
111D4×C271—24no—
112D4×C271—22no—
113C2×C4×C271—22no—
114C2×C4×C271—22no—
115C2×C4×C271C2×D412yes—
116C2×C4×C271C2×D412yes—
117C2×C4×C271C2×D412yes—
118C2×C4×C271C2×D411yescharacteristic
119C2×C4×C271C2411yescharacteristic
120C8×C271—22no—
121C8×C271—22no—
122C22×C271—22no—
123C22×C271—22no—
124C22×C271—24no—
125C22×C271#12012yes—
126C22×C271C22×D411yescharacteristic
127C2×D271—24no—
128C2×D271—22no—
129C2×D271—44no—
130C2×D271—42no—
131C2×D271—84no—
132C2×D271—88no—
133C4×C271—22no—
134C4×C271#12012yes—
135C4×C271C22×D411yescharacteristic
136C4×C271C22×D411yescharacteristic
137Dic271—22no—
138Dic271—42no—
139Dic271—42no—
140C2×C271—22no—
141C2×C271—42no—
142C2×C271C2×#12011yescharacteristic
143C2×C271#57712yes—
144D271—44no—
145D271—42no—
146C271#300911yescharacteristic, complemented, Sylow
147#3009—2711nomaximal, Sylow
148C2×H(5,𝔽2)—2711no—
149C2×#137—2711no—
150C22×SD16—2711no—
151#536—2711no—
152#452—2714no—
153#450—2714no—
154#418—2711no—
155#416—2711no—
156#408D27111yescharacteristic, complemented
157H(5,𝔽2)—5424no—
158H(5,𝔽2)—5424no—
159C2×#41—2711no—
160C2×#41—2712no—
161C2×#41—5421no—
162C2×#41C2×D27111yescharacteristic
163C22×Q8—2711no—
164C22×D4—2711no—
165C22×D4—2712no—
166C22×D4—5422no—
167C22×D4—5421no—
168#137—5424no—
169C2×Q16—2712no—
170C2×SD16—2714no—
171C2×SD16—2712no—
172C2×SD16—5424no—
173C2×M16C2×D27111yescharacteristic, complemented
174C22×C8C2×D27111yescharacteristic, complemented
175C4⋊Q8—5421no—
176#124—5421no—
177#123—5422no—
178C22⋊Q8—2714no—
179#117—2711no—
180#103—2712no—
181#102—2712no—
182#102—2714no—
183C22⋊C8C2×D27114yescomplemented
184C24—5422no—
185C24—5421no—
186#41—22no—
187#41—22no—
188#41—2714no—
189#41—5424no—
190#41—5424no—
191#41—5428no—
192#41—5422no—
193C2×Q8—2711no—
194C2×Q8—2712no—
195C2×Q8—5421no—
196C2×Q8—5422no—
197C2×Q8—5422no—
198C2×Q8D108411yescharacteristic
199C2×D4—2714no—
200C2×D4—2712no—
201C2×D4—2711no—
202C2×D4—5424no—
203C2×D4—5424no—
204C2×D4—54216no—
205C2×D4—5428no—
206C2×D4—5422no—
207C2×D4—5421no—
208C2×D4—5422no—
209C2×D4—5422no—
210C2×D4—5421no—
211C2×D4D108412yes—
212C2×D4D108411yescharacteristic
213C22×C4—5422no—
214C22×C4—5421no—
215C22×C4—5421no—
216C22×C4C22×D27112yes—
217C22×C4C22×D27111yescharacteristic
218Q16—5424no—
219SD16—5428no—
220SD16—5424no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D27112yes—
224C2×C8C22×D27112yes—
225C4⋊C4—2712no—
226C4⋊C4—5424no—
227C22⋊C4—5421no—
228C22⋊C4—5424no—
229C42—5421no—
230C23—5428no—
231C23—5422no—
232C23—5421no—
233C23—5424no—
234C23—21682no—
235C23—21684no—
236C23C2×D108412yes—
237C23C2×D108411yescharacteristic
238Q8—22no—
239Q8—2712no—
240Q8—5421no—
241Q8—5422no—
242Q8—5422no—
243Q8—10842no—
244D4—24no—
245D4—22no—
246D4—2714no—
247D4—2712no—
248D4—5428no—
249D4—5428no—
250D4—5424no—
251D4—5424no—
252D4—5421no—
253D4—5422no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—5428no—
257C2×C4—5421no—
258C2×C4—5424no—
259C2×C4—5422no—
260C2×C4—5422no—
261C2×C4—10842no—
262C2×C4D4×D27112yes—
263C2×C4D4×D27112yes—
264C2×C4C2×D108412yes—
265C2×C4C2×D108411yescharacteristic
266C2×C4C23×D27111yescharacteristic
267C8—22no—
268C8—22no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—5424no—
273C22—5422no—
274C22—10844no—
275C22—10842no—
276C22—21684no—
277C22—21688no—
278C22#2257613012yes—
279C22C22×D108411yescharacteristic
280C4—22no—
281C4—5422no—
282C4—10842no—
283C4—10842no—
284C4#2257613012yes—
285C4C2×D4×D27111yescharacteristic
286C4C2×D4×D27111yescharacteristic
287C2—22no—
288C2—42no—
289C2—10844no—
290C2—10842no—
291C2C2×#2257613011yescharacteristic
292C2#3177470912yes—
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#408×C271C2
socleC22×C271C22×D4
prime radicalC2×#22576206C2
hypercenter#408D271
hypocenterC271#3009
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#22576193C2semidirect
3. C22×#16864643C2semidirect
4. #31774468C2semidirect
5. #31774061C2semidirect
6. #31774050C2semidirect
7. #31773903C2semidirect
8. #31773894C2semidirect
9. #408×C271C2semidirect
15. C22×Dic542C22semidirect
21. C2×Dic1084C22semidirect
23. C2×#16864643C22semidirect
24. C2×#16864643C22semidirect
28. #22576139C22semidirect
29. #22576119C22semidirect
30. #22576085C22semidirect
31. #22576082C22semidirect
32. #22576082C22semidirect
34. C2×M16×C271C22semidirect
35. C22×C8×C271C22semidirect
36. C22⋊C8×C271C22semidirect
146. C271#3009semidirect
156. #408D271semidirect
173. C2×M16C2×D271semidirect
174. C22×C8C2×D271semidirect
183. C22⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225762062
C2×#225761932
C22×#168646432
#317744682
#317740612
#317740502
#317739032
#317738942
#408×C2712
#3009271