Finite Group Databasebeta

← #46839995

Subgroups — #46839995

Order 34688.

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

normal subgroups only

This group has 836 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
2C2×#22576158C211yesmaximal, characteristic, complemented
3C2×#22576136C211yesmaximal, characteristic, complemented
4#31773812C214yesmaximal, complemented
5#31773803C211yesmaximal, characteristic, complemented
6#31773800C211yesmaximal, characteristic, complemented
7#31773769C214yesmaximal, complemented
8C4×(C2×D271)⋊C4C211yesmaximal, characteristic, complemented
9C2×#124×C271C211yesmaximal, characteristic, complemented
10C2×#16864653—24no—
11C2×#16864653—24no—
12C2×#16864653—24no—
13C22×Dic542C2211yescharacteristic
14C22×C22⋊D271C2212yes—
15C23×Dic271C2212yes—
16C22×C4×D271C2211yescharacteristic
17#22576158—24no—
18#22576136—24no—
19C2×C4⋊Dic271C2211yescharacteristic
20C2×Dic271⋊C4C2212yes—
21C2×Dic271⋊C4C2214yes—
22C2×C22⋊Dic271C2214yes—
23C2×(C2×D271)⋊C4C2212yes—
24C2×(C2×D271)⋊C4C2212yes—
25C2×C4×Dic271C2211yescharacteristic
26C271⋊#95C2212yes—
27C271⋊#95C2214yes—
28C22×Q8×C271C2211yescharacteristic
29C22×D4×C271C2211yescharacteristic
30#124×C271—24no—
31C2×C22⋊C4×C271C2214yes—
32C2×C42×C271C2211yescharacteristic
33C23×D271C2311yescharacteristic
34#16864653—232no—
35C2×Dic542—22no—
36C2×Dic542—22no—
37C2×Dic542D414yes—
38C2×C22⋊D271—24no—
39C2×C22⋊D271—24no—
40C2×C22⋊D271—24no—
41C22×Dic271—24no—
42C22×Dic271—24no—
43C22×Dic271—24no—
44C22×Dic271C2312yes—
45C22×Dic271C2311yescharacteristic
46C22×Dic271C2312yes—
47C22×Dic271C2312yes—
48C2×C4×D271—22no—
49C2×C4×D271—22no—
50C2×C4×D271D414yes—
51C4⋊Dic271—22no—
52Dic271⋊C4—24no—
53Dic271⋊C4—28no—
54C22⋊Dic271—28no—
55(C2×D271)⋊C4—24no—
56(C2×D271)⋊C4—24no—
57C4×Dic271—22no—
58C24×C271C2312yes—
59C2×Q8×C271—22no—
60C2×Q8×C271—22no—
61C2×Q8×C271—22no—
62C2×D4×C271—22no—
63C2×D4×C271—22no—
64C2×D4×C271D414yes—
65C22×C4×C271C2312yes—
66C22×C4×C271C2312yes—
67C22×C4×C271C2311yescharacteristic
68C22⋊C4×C271—28no—
69C42×C271—22no—
70C22×D271—21no—
71C22×D271—21no—
72C22×D271—22no—
73C22×D271—22no—
74C22×D271C2×D412yes—
75Dic542—28no—
76C22⋊D271—216no—
77C2×Dic271—216no—
78C2×Dic271—22no—
79C2×Dic271—21no—
80C2×Dic271—21no—
81C2×Dic271—22no—
82C2×Dic271—22no—
83C2×Dic271—22no—
84C2×Dic271—22no—
85C2×Dic271—24no—
86C2×Dic271C2×D412yes—
87C2×Dic271C2×D414yes—
88C2×Dic271#4114yescomplemented
89C4×D271—28no—
90C23×C271—22no—
91C23×C271—22no—
92C23×C271—24no—
93C23×C271—24no—
94C23×C271C2×D414yes—
95C23×C271C2411yescharacteristic
96Q8×C271—44no—
97D4×C271—28no—
98C2×C4×C271—22no—
99C2×C4×C271—22no—
100C2×C4×C271—22no—
101C2×C4×C271—22no—
102C2×C4×C271—22no—
103C2×C4×C271—22no—
104C2×C4×C271C2×D412yes—
105C2×C4×C271C2×D412yes—
106C2×C4×C271C2×D412yes—
107C22×C271—24no—
108C22×C271—28no—
109C22×C271—42no—
110C22×C271—42no—
111C22×C271—44no—
112C22×C271C22×D411yescharacteristic
113C22×C271C22×D411yescharacteristic
114C22×C271C22×D411yescharacteristic
115C22×C271C2×#4112yes—
116C22×C2712−1+412yes—
117C2×D271—22no—
118C2×D271—22no—
119C2×D271—22no—
120C2×D271—42no—
121C2×D271—42no—
122C4×C271—42no—
123C4×C271—42no—
124C4×C271#12014yes—
125Dic271—22no—
126Dic271—24no—
127Dic271—24no—
128Dic271—42no—
129C2×C271—44no—
130C2×C271C2×#12011yescharacteristic
131C2×C271#53211yescharacteristic
132C2×C271#53611yescharacteristic
133C2×C271#54712yes—
134C2×C271#54812yes—
135D271—42no—
136C271#243211yescharacteristic, complemented, Sylow
137#2432—2711nomaximal, Sylow
138C22×#41—2711no—
139C2×C4⋊Q8—2711no—
140C2×#124D27111yescharacteristic, complemented
141C2×C22⋊Q8—2711no—
142#394—2714no—
143#391—2711no—
144#390—2711no—
145#385—2714no—
146C4×C22⋊C4—2711no—
147C2×#41—5424no—
148C2×#41—5424no—
149C2×#41—5424no—
150C22×Q8—2711no—
151C22×Q8C2×D27111yescharacteristic
152C22×D4—2712no—
153C22×D4C2×D27111yescharacteristic
154C23×C4—2711no—
155C23×C4—2712no—
156C4⋊Q8—5424no—
157#124—24no—
158C22⋊Q8—5424no—
159C2×C4⋊C4—2712no—
160C2×C4⋊C4—2711no—
161C2×C4⋊C4—2714no—
162C2×C22⋊C4—2714no—
163C2×C22⋊C4—2712no—
164C2×C22⋊C4—2712no—
165C2×C22⋊C4C2×D27114yes—
166C2×C42—2711no—
167C2×C42C2×D27111yescharacteristic
168#95—2712no—
169#95—2714no—
170C24—2711no—
171C24C22×D27112yes—
172#41—54232no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—2714no—
177C2×Q8—5422no—
178C2×Q8—5422no—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—5424no—
182C2×D4—5424no—
183C2×D4—5424no—
184C2×D4C22⋊D27114yes—
185C22×C4—2711no—
186C22×C4—2712no—
187C22×C4—2712no—
188C22×C4—2714no—
189C22×C4—2712no—
190C22×C4—5424no—
191C22×C4—5424no—
192C22×C4—5424no—
193C22×C4—5422no—
194C22×C4—5422no—
195C22×C4C22×D27112yes—
196C22×C4C22×D27112yes—
197C22×C4C22×D27111yescharacteristic
198C4⋊C4—5424no—
199C4⋊C4—5428no—
200C4⋊C4—5422no—
201C22⋊C4—28no—
202C22⋊C4—5424no—
203C22⋊C4—5424no—
204C22⋊C4—5428no—
205C42—22no—
206C42—5422no—
207C23—22no—
208C23—22no—
209C23—24no—
210C23—24no—
211C23—2712no—
212C23—5421no—
213C23—5421no—
214C23—5422no—
215C23—5422no—
216C23C2×C22⋊D27114yes—
217C23C23×D27111yescharacteristic
218Q8—44no—
219Q8—5428no—
220D4—28no—
221D4—54216no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—2712no—
229C2×C4—2714no—
230C2×C4—2714no—
231C2×C4—54216no—
232C2×C4—5422no—
233C2×C4—5421no—
234C2×C4—5422no—
235C2×C4—5421no—
236C2×C4—5422no—
237C2×C4—5422no—
238C2×C4—5428no—
239C2×C4—5422no—
240C2×C4—5424no—
241C2×C4D4×D27112yes—
242C2×C4D4×D27112yes—
243C2×C4C2×C22⋊D27112yes—
244C22—28no—
245C22—24no—
246C22—42no—
247C22—44no—
248C22—42no—
249C22—5422no—
250C22—5422no—
251C22—5422no—
252C22—10842no—
253C22—10842no—
254C22C2×D4×D27111yescharacteristic
255C22C2×D4×D27111yescharacteristic
256C22C22×C22⋊D27111yescharacteristic
257C22#41×D27112yes—
258C22#2257620712yes—
259C4—42no—
260C4—42no—
261C4—5424no—
262C4—5422no—
263C4—5424no—
264C4—10842no—
265C4#2257612814yes—
266C2—44no—
267C2—10842no—
268C2C2×#2257612811yescharacteristic
269C2#3177444611yescharacteristic
270C2#3177446711yescharacteristic
271C2#3177454612yes—
272C2#3177455512yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC2×#124×C271C2
prime radicalC22×#16864653C2
hypercenterC2×#124D271
hypocenterC271#2432
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#22576158C2semidirect
3. C2×#22576136C2semidirect
4. #31773812C2semidirect
5. #31773803C2semidirect
6. #31773800C2semidirect
7. #31773769C2semidirect
8. C4×(C2×D271)⋊C4C2semidirect
9. C2×#124×C271C2semidirect
88. C2×Dic271#41semidirect
136. C271#2432semidirect
140. C2×#124D271semidirect

Maximal subgroups

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

subgroupindex
C22×#168646532
C2×#225761582
C2×#225761362
#317738122
#317738032
#317738002
#317737692
C4×(C2×D271)⋊C42
C2×#124×C2712
#2432271