Finite Group Databasebeta

← #46839982

Subgroups — #46839982

Order 34688.

952 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C22×#16864653C211yesmaximal, characteristic, complemented
2C2×#22576156C211yesmaximal, characteristic, complemented
3C2×#22576148C211yesmaximal, characteristic, complemented
4C2×#22576135C211yesmaximal, characteristic, complemented
5#31773806C214yesmaximal, complemented
6#31773802C211yesmaximal, characteristic, complemented
7#31773774C214yesmaximal, complemented
8C4×(C2×D271)⋊C4C211yesmaximal, characteristic, complemented
9#390×C271C211yesmaximal, characteristic, complemented
10C2×#16864653—24no—
11C2×#16864653—24no—
12C2×#16864653—24no—
13C22×Dic542C2211yescharacteristic
14C22×D1084C2211yescharacteristic
15C22×D1084C2212yes—
16C22×C22⋊D271C2212yes—
17C22×C22⋊D271C2214yes—
18C23×Dic271C2212yes—
19C22×C4×D271C2211yescharacteristic
20#22576156—24no—
21#22576148—24no—
22#22576135—24no—
23C2×(C2×D271)⋊C4C2212yes—
24C2×(C2×D271)⋊C4C2212yes—
25C2×(C2×D271)⋊C4C2214yes—
26C2×C4×Dic271C2211yescharacteristic
27C271⋊#95C2212yes—
28C271⋊#95C2214yes—
29C22×D4×C271C2211yescharacteristic
30C2×C4⋊C4×C271C2211yescharacteristic
31C2×C22⋊C4×C271C2214yes—
32C2×C42×C271C2211yescharacteristic
33C23×D271C2312yes—
34C23×D271C2311yescharacteristic
35#16864653—232no—
36C2×Dic542—22no—
37C2×Dic542—22no—
38C2×Dic542D414yescomplemented
39C2×D1084—24no—
40C2×D1084—22no—
41C2×D1084—22no—
42C2×D1084—22no—
43C2×D1084—44no—
44C2×C22⋊D271—24no—
45C2×C22⋊D271—24no—
46C2×C22⋊D271—28no—
47C2×C22⋊D271—24no—
48C2×C22⋊D271—28no—
49C2×C22⋊D271—44no—
50C22×Dic271—24no—
51C22×Dic271—24no—
52C22×Dic271—24no—
53C22×Dic271C2311yescharacteristic
54C22×Dic271C2312yes—
55C22×Dic271C2312yes—
56C2×C4×D271—22no—
57C2×C4×D271—22no—
58C2×C4×D271D414yescomplemented
59(C2×D271)⋊C4—24no—
60(C2×D271)⋊C4—28no—
61(C2×D271)⋊C4—24no—
62C4×Dic271—22no—
63C24×C271C2312yes—
64C2×D4×C271—22no—
65C2×D4×C271—22no—
66C2×D4×C271D414yescomplemented
67C22×C4×C271C2312yes—
68C22×C4×C271C2312yes—
69C22×C4×C271C2311yescharacteristic
70C4⋊C4×C271—22no—
71C22⋊C4×C271—44no—
72C42×C271—22no—
73C22×D271—21no—
74C22×D271—21no—
75C22×D271—22no—
76C22×D271—22no—
77C22×D271—22no—
78C22×D271—22no—
79C22×D271—22no—
80C22×D271—24no—
81C22×D271—24no—
82C22×D271C2×D412yes—
83Dic542—28no—
84D1084—48no—
85D1084—44no—
86C22⋊D271—216no—
87C22⋊D271—416no—
88C2×Dic271—216no—
89C2×Dic271—22no—
90C2×Dic271—21no—
91C2×Dic271—21no—
92C2×Dic271—22no—
93C2×Dic271—22no—
94C2×Dic271—22no—
95C2×Dic271C2×D412yes—
96C2×Dic271C2×D414yes—
97C2×Dic271#4114yescomplemented
98C4×D271—28no—
99C23×C271—22no—
100C23×C271—22no—
101C23×C271—24no—
102C23×C271—24no—
103C23×C271C2×D414yes—
104C23×C271C2411yescharacteristic
105D4×C271—28no—
106C2×C4×C271—22no—
107C2×C4×C271—22no—
108C2×C4×C271—22no—
109C2×C4×C271—22no—
110C2×C4×C271—24no—
111C2×C4×C271C2×D412yes—
112C2×C4×C271C2×D412yes—
113C2×C4×C271C2×D412yes—
114C22×C271—24no—
115C22×C271—28no—
116C22×C271—42no—
117C22×C271—42no—
118C22×C271—44no—
119C22×C271C22×D411yescharacteristic
120C22×C271C22×D411yescharacteristic
121C22×C271C22×D411yescharacteristic
122C22×C271C2×#4112yes—
123C22×C271H(5,𝔽2)12yes—
124C2×D271—22no—
125C2×D271—22no—
126C2×D271—22no—
127C2×D271—42no—
128C2×D271—42no—
129C2×D271—42no—
130C2×D271—42no—
131C2×D271—42no—
132C2×D271—44no—
133C2×D271—44no—
134C4×C271—42no—
135C4×C271—42no—
136C4×C271#12014yes—
137Dic271—22no—
138Dic271—24no—
139Dic271—24no—
140C2×C271—44no—
141C2×C271C2×#12011yescharacteristic
142C2×C271#53211yescharacteristic
143C2×C271#53511yescharacteristic
144C2×C271#54612yes—
145C2×C271#55012yes—
146D271—42no—
147D271—82no—
148C271#243111yescharacteristic, complemented, Sylow
149#2431—2711nomaximal, Sylow
150C22×#41—2711no—
151C2×#127—2711no—
152C2×#124—2711no—
153C2×#121—2711no—
154#392—2714no—
155#391—2711no—
156#390D27111yescharacteristic, complemented
157#386—2714no—
158C4×C22⋊C4—2711no—
159C2×#41—5424no—
160C2×#41—5424no—
161C2×#41—5424no—
162C22×Q8—2711no—
163C22×D4—2712no—
164C22×D4—2714no—
165C22×D4—2711no—
166C22×D4—2712no—
167C22×D4C2×D27111yescharacteristic
168C23×C4—2711no—
169C23×C4—2712no—
170#127—5424no—
171#124—5424no—
172#121—5424no—
173C2×C4⋊C4C2×D27111yescharacteristic
174C2×C22⋊C4—2712no—
175C2×C22⋊C4—2714no—
176C2×C22⋊C4—2712no—
177C2×C22⋊C4C2×D27114yes—
178C2×C42—2711no—
179C2×C42C2×D27111yescharacteristic
180#95—2712no—
181#95—2714no—
182C24—2711no—
183C24—2712no—
184C24C22×D27112yes—
185#41—54232no—
186C2×Q8—2714no—
187C2×Q8—5422no—
188C2×Q8—5422no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—5424no—
192C2×D4—5424no—
193C2×D4—5428no—
194C2×D4—5424no—
195C2×D4—5424no—
196C2×D4—5428no—
197C2×D4—5422no—
198C2×D4—5422no—
199C2×D4—5422no—
200C2×D4—10844no—
201C2×D4—10844no—
202C2×D4D108414yescomplemented
203C22×C4—2712no—
204C22×C4—2711no—
205C22×C4—2712no—
206C22×C4—2714no—
207C22×C4—5424no—
208C22×C4—5424no—
209C22×C4—5424no—
210C22×C4—5422no—
211C22×C4—5422no—
212C22×C4C22×D27112yes—
213C22×C4C22×D27112yes—
214C22×C4C22×D27111yescharacteristic
215C4⋊C4—22no—
216C22⋊C4—44no—
217C22⋊C4—5424no—
218C22⋊C4—5424no—
219C22⋊C4—5428no—
220C42—22no—
221C42—5422no—
222C23—22no—
223C23—22no—
224C23—24no—
225C23—24no—
226C23—2712no—
227C23—5421no—
228C23—5421no—
229C23—5422no—
230C23—5422no—
231C23—5422no—
232C23—5422no—
233C23—5422no—
234C23—5424no—
235C23—5424no—
236C23C2×D108414yes—
237C23C23×D27111yescharacteristic
238Q8—5428no—
239D4—28no—
240D4—54216no—
241D4—10848no—
242D4—108416no—
243D4—10844no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—2712no—
250C2×C4—2714no—
251C2×C4—2714no—
252C2×C4—54216no—
253C2×C4—5421no—
254C2×C4—5422no—
255C2×C4—5422no—
256C2×C4—5421no—
257C2×C4—5422no—
258C2×C4—5422no—
259C2×C4—5428no—
260C2×C4D4×D27112yes—
261C2×C4D4×D27112yes—
262C2×C4C2×D108412yes—
263C22—28no—
264C22—24no—
265C22—44no—
266C22—42no—
267C22—42no—
268C22—5422no—
269C22—5422no—
270C22—5422no—
271C22—10842no—
272C22—10842no—
273C22—10844no—
274C22—10844no—
275C22—10842no—
276C22—10842no—
277C22—10842no—
278C22C2×D4×D27111yescharacteristic
279C22C2×D4×D27111yescharacteristic
280C22C22×D108411yescharacteristic
281C22#41×D27112yes—
282C22#2257620512yes—
283C4—42no—
284C4—42no—
285C4—5424no—
286C4—5422no—
287C4—5424no—
288C4#2257613014yes—
289C2—44no—
290C2—10842no—
291C2—21682no—
292C2C2×#2257613011yescharacteristic
293C2#3177444611yescharacteristic
294C2#3177446011yescharacteristic
295C2#3177454212yes—
296C2#3177456512yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#390×C271C2
hypercenter#390D271
hypocenterC271#2431
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C22×#16864653C2semidirect
2. C2×#22576156C2semidirect
3. C2×#22576148C2semidirect
4. C2×#22576135C2semidirect
5. #31773806C2semidirect
6. #31773802C2semidirect
7. #31773774C2semidirect
8. C4×(C2×D271)⋊C4C2semidirect
9. #390×C271C2semidirect
38. C2×Dic542D4semidirect
58. C2×C4×D271D4semidirect
66. C2×D4×C271D4semidirect
97. C2×Dic271#41semidirect
148. C271#2431semidirect
156. #390D271semidirect
202. C2×D4D1084semidirect

Maximal subgroups

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

subgroupindex
C22×#168646532
C2×#225761562
C2×#225761482
C2×#225761352
#317738062
#317738022
#317737742
C4×(C2×D271)⋊C42
#390×C2712
#2431271