Finite Group Databasebeta

← #46840003

Subgroups — #46840003

Order 34688.

800 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 181 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×Q8×D271C211yesmaximal, characteristic
2C2×#22576159C212yesmaximal, complemented
3C2×#22576138C211yesmaximal, characteristic
4#31773818C211yesmaximal, characteristic, complemented
5#31773817C211yesmaximal, characteristic, complemented
6#31773816C211yesmaximal, characteristic, complemented
7#31773801C212yesmaximal, complemented
8#31773771C212yesmaximal
9#31773768C212yesmaximal
10#31773744C211yesmaximal, characteristic
11#395×C271C211yesmaximal, characteristic, complemented
12C22×Dic542C2211yescharacteristic
13C22×Dic542C2211yescharacteristic
14C22×Dic542C2212yes—
15C2×Q8×D271—28no—
16C2×Q8×D271—24no—
17C22×C4×D271C2212yes—
18C22×C4×D271C2211yescharacteristic
19#22576159—28no—
20#22576138—24no—
21C2×C4⋊Dic271C2212yes—
22C2×Dic271⋊C4C2212yes—
23C2×Dic271⋊C4C2212yes—
24C2×Dic271⋊C4C2211yescharacteristic
25C2×Dic271⋊C4C2211yescharacteristic
26C2×(C2×D271)⋊C4C2211yescharacteristic
27C2×(C2×D271)⋊C4C2211yescharacteristic
28C2×(C2×D271)⋊C4C2212yes—
29C2×C4×Dic271C2212yes—
30C271⋊#95C2211yescharacteristic
31C271⋊#95C2212yes—
32C271⋊#95C2212yes—
33C271⋊#95C2212yes—
34C22×Q8×C271C2211yescharacteristic
35C2×C4⋊C4×C271C2212yes—
36C2×C4⋊C4×C271C2211yescharacteristic
37#95×C271C2211yescharacteristic
38#95×C271C2212yes—
39C23×D271C2311yescharacteristic
40C2×Dic542—24no—
41C2×Dic542—24no—
42C2×Dic542—24no—
43C2×Dic542—22no—
44C2×Dic542—24no—
45C2×Dic542—22no—
46C2×Dic542D418yes—
47Q8×D271—232no—
48C22×Dic271C2311yescharacteristic
49C22×Dic271C2311yescharacteristic
50C22×Dic271C2312yes—
51C22×Dic271C2312yes—
52C22×Dic271C2311yescharacteristic
53C2×C4×D271—24no—
54C2×C4×D271—24no—
55C2×C4×D271—24no—
56C2×C4×D271—24no—
57C2×C4×D271D414yes—
58C4⋊Dic271—24no—
59Dic271⋊C4—22no—
60Dic271⋊C4—24no—
61Dic271⋊C4—22no—
62Dic271⋊C4—24no—
63(C2×D271)⋊C4—22no—
64(C2×D271)⋊C4—22no—
65(C2×D271)⋊C4Q818yescomplemented
66C4×Dic271—24no—
67C2×Q8×C271—24no—
68C2×Q8×C271—22no—
69C22×C4×C271C2311yescharacteristic
70C22×C4×C271C2311yescharacteristic
71C22×C4×C271C2312yes—
72C22×C4×C271C2311yescharacteristic
73C22×C4×C271C2312yes—
74C4⋊C4×C271—22no—
75C4⋊C4×C271—24no—
76C22×D271—22no—
77C22×D271—21no—
78C22×D271—21no—
79C22×D271C2×D412yes—
80C22×D271C2×Q814yes—
81Dic542—28no—
82Dic542—216no—
83Dic542—44no—
84C2×Dic271—22no—
85C2×Dic271—22no—
86C2×Dic271—22no—
87C2×Dic271—22no—
88C2×Dic271—21no—
89C2×Dic271—21no—
90C2×Dic271—22no—
91C2×Dic271—21no—
92C2×Dic271—21no—
93C2×Dic271C2×D414yes—
94C2×Dic271C2×D414yes—
95C2×Dic271C2×D412yes—
96C2×Dic271C2×Q814yescomplemented
97C4×D271—216no—
98C4×D271—28no—
99C23×C271C2411yescharacteristic
100Q8×C271—28no—
101C2×C4×C271—22no—
102C2×C4×C271—22no—
103C2×C4×C271—22no—
104C2×C4×C271—22no—
105C2×C4×C271—22no—
106C2×C4×C271—21no—
107C2×C4×C271—21no—
108C2×C4×C271—22no—
109C2×C4×C271—21no—
110C2×C4×C271—21no—
111C2×C4×C271C2×D414yes—
112C2×C4×C271C2×D412yes—
113C2×C4×C271C2×Q814yescomplemented
114C22×C271C22×D411yescharacteristic
115C22×C271C22×D412yes—
116C22×C271C22×Q812yes—
117C22×C2712−1+411yescharacteristic
118C22×C2712−1+411yescharacteristic
119C2×D271—22no—
120C2×D271—22no—
121C2×D271—24no—
122C2×D271—24no—
123C2×D271C4⋊Q814yescomplemented
124C4×C271—24no—
125C4×C271—22no—
126C4×C271—24no—
127C4×C271—41no—
128C4×C271—41no—
129Dic271—24no—
130Dic271—24no—
131Dic271—41no—
132Dic271—41no—
133Dic271#12014yes—
134C2×C271C2×#12011yescharacteristic
135C2×C271C2×C4⋊Q811yescharacteristic
136C2×C271#53611yescharacteristic
137C2×C271D4×Q812yes—
138C2×C271D4×Q812yes—
139D271—24no—
140C271#243411yescharacteristic, complemented, Sylow
141#2434—2711nomaximal, Sylow
142C23×Q8—2711no—
143C2×C4⋊Q8—2712no—
144C2×C22⋊Q8—2711no—
145#395—2711no—
146#395—2711no—
147#395—2711no—
148#395D27111yescharacteristic, complemented
149#391—2712no—
150#385—2712no—
151#385—2712no—
152C4×C22⋊C4—2711no—
153C22×Q8—2712no—
154C22×Q8—2711no—
155C22×Q8—2711no—
156C22×Q8—5424no—
157C22×Q8—5428no—
158C22×Q8C2×D27111yescharacteristic
159C23×C4—2711no—
160C23×C4—2712no—
161C4⋊Q8—5428no—
162C22⋊Q8—5424no—
163C2×C4⋊C4—2711no—
164C2×C4⋊C4—2711no—
165C2×C4⋊C4—2712no—
166C2×C4⋊C4—2712no—
167C2×C4⋊C4—2712no—
168C2×C4⋊C4C2×D27111yescharacteristic
169C2×C4⋊C4C2×D27112yes—
170C2×C22⋊C4—2712no—
171C2×C22⋊C4—2711no—
172C2×C22⋊C4—2711no—
173C2×C42—2712no—
174#95—2711no—
175#95—2712no—
176#95—2712no—
177#95—2712no—
178#95C2×D27111yescharacteristic
179#95C2×D27112yes—
180C24—2711no—
181C2×Q8—22no—
182C2×Q8—24no—
183C2×Q8—2718no—
184C2×Q8—5424no—
185C2×Q8—5424no—
186C2×Q8—5424no—
187C2×Q8—5422no—
188C2×Q8—5424no—
189C2×Q8—54232no—
190C2×Q8—5422no—
191C22×C4—2711no—
192C22×C4—2712no—
193C22×C4—2712no—
194C22×C4—2714no—
195C22×C4—2711no—
196C22×C4—2711no—
197C22×C4—5424no—
198C22×C4—5424no—
199C22×C4—5424no—
200C22×C4—5424no—
201C22×C4C22×D27111yescharacteristic
202C22×C4C22×D27111yescharacteristic
203C22×C4C22×D27111yescharacteristic
204C22×C4C22×D27112yes—
205C22×C4C22×D27112yes—
206C4⋊C4—22no—
207C4⋊C4—24no—
208C4⋊C4—5422no—
209C4⋊C4—5424no—
210C4⋊C4—5424no—
211C4⋊C4—5422no—
212C4⋊C4—5424no—
213C22⋊C4—2718no—
214C22⋊C4—5422no—
215C22⋊C4—5422no—
216C42—5424no—
217C23—2712no—
218C23—2714no—
219C23—5422no—
220C23—5421no—
221C23—5421no—
222C23C23×D27111yescharacteristic
223Q8—28no—
224Q8—54216no—
225Q8—5428no—
226Q8—10844no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—2714no—
238C2×C4—2712no—
239C2×C4—2714no—
240C2×C4—2714no—
241C2×C4—5428no—
242C2×C4—54216no—
243C2×C4—5422no—
244C2×C4—5422no—
245C2×C4—5422no—
246C2×C4—5422no—
247C2×C4—5421no—
248C2×C4—5421no—
249C2×C4—5422no—
250C2×C4—5421no—
251C2×C4—5421no—
252C2×C4D4×D27112yes—
253C2×C4D4×D27114yes—
254C2×C4Q8×D27114yescomplemented
255C22—2714no—
256C22—5422no—
257C22—5424no—
258C22—5424no—
259C22—5422no—
260C22C2×D4×D27111yescharacteristic
261C22C2×D4×D27112yes—
262C22C2×Q8×D27112yes—
263C22#2257620711yescharacteristic
264C22#2257620811yescharacteristic
265C4—24no—
266C4—22no—
267C4—24no—
268C4—41no—
269C4—41no—
270C4—2714no—
271C4—5424no—
272C4—5424no—
273C4—10841no—
274C4—10841no—
275C2—5424no—
276C2#120×D27111yescharacteristic
277C2C4⋊Q8×D27111yescharacteristic
278C2#3177447011yescharacteristic
279C2#3177456012yes—
280C2#3177456212yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#395×C271C2
prime radicalC23×D271C23
hypercenter#395D271
hypocenterC271#2434
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#22576159C2semidirect
4. #31773818C2semidirect
5. #31773817C2semidirect
6. #31773816C2semidirect
7. #31773801C2semidirect
11. #395×C271C2semidirect
65. (C2×D271)⋊C4Q8semidirect
96. C2×Dic271C2×Q8semidirect
113. C2×C4×C271C2×Q8semidirect
123. C2×D271C4⋊Q8semidirect
140. C271#2434semidirect
148. #395D271semidirect
254. C2×C4Q8×D271semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D2712
C2×#225761592
C2×#225761382
#317738182
#317738172
#317738162
#317738012
#317737712
#317737682
#317737442
#395×C2712
#2434271