Finite Group Databasebeta

← #46846749

Subgroups — #46846749

Order 34688.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#31774452C211yesmaximal, characteristic, complemented
2#124×D271C211yesmaximal, characteristic
3#31774259C211yesmaximal, characteristic, complemented
4#31774257C211yesmaximal, characteristic, complemented
5#31774256C211yesmaximal, characteristic, complemented
6#31774253C212yesmaximal, complemented
7#31774250C212yesmaximal, complemented
8#31773904C212yesmaximal, complemented
9#31773900C212yesmaximal
10#31773849C211yesmaximal, characteristic, complemented
11#489×C271C211yesmaximal, characteristic, complemented
12C2×Q8×D271C2211yescharacteristic
13C2×D4×D271C2211yescharacteristic
14C2×#16864637C2212yes—
15#22576160C2211yescharacteristic, complemented
16#22576151C2212yescomplemented
17#22576149C2211yescharacteristic
18#22576148C2211yescharacteristic
19#22576147C2211yescharacteristic
20#22576146—21no—
21#22576146—21no—
22#22576140—21no—
23#22576136—21no—
24Q8×Dic271—21no—
25#22576125—21no—
26#22576118C2212yes—
27C22⋊C4×D271—21no—
28C22⋊C4×D271—21no—
29C42×D271C2211yescharacteristic
30#22576085C2212yescomplemented
31#22576084C2212yescomplemented
32#22576083C2212yescomplemented
33#22576082C2212yes—
34#22576081C2212yes—
35#22576080C2212yes—
36#22576071C2211yescharacteristic, complemented
37C8×Dic271C2212yescomplemented
38C4⋊Q8×C271C2211yescharacteristic, complemented
39#124×C271C2211yescharacteristic
40#103×C271C2212yescomplemented
41#102×C271C2212yes—
42#97×C271C2211yescharacteristic, complemented
43C23×D271—21no—
44C2×Dic542—21no—
45C2×Dic542—21no—
46C2×Dic542C2311yescharacteristic
47Q8×D271—24no—
48C2×D1084C2311yescharacteristic
49C2×C22⋊D271—21no—
50D4×D271—24no—
51C2×C4×D271—21no—
52C2×C4×D271D411yescharacteristic
53C2×C4×D271D411yescharacteristic
54C2×C4×D271C2311yescharacteristic
55#16864637—24no—
56C2×C271⋊C8C2312yes—
57C4⋊Dic271—22no—
58C4⋊Dic271—21no—
59C4⋊Dic271C2312yes—
60Dic271⋊C4—22no—
61C22⋊Dic271—21no—
62C22⋊Dic271—21no—
63(C2×D271)⋊C4—22no—
64(C2×D271)⋊C4—21no—
65(C2×D271)⋊C4—21no—
66(C2×D271)⋊C4—21no—
67(C2×D271)⋊C4—21no—
68C4×Dic271—22no—
69C4×Dic271D411yescharacteristic
70C4×Dic271D411yescharacteristic
71C4×Dic271C2311yescharacteristic
72C2×Q8×C271—21no—
73C2×Q8×C271C2311yescharacteristic
74C2×D4×C271C2311yescharacteristic
75C2×C8×C271C2312yes—
76C4⋊C4×C271—21no—
77C4⋊C4×C271C2312yes—
78C22⋊C4×C271—21no—
79C22⋊C4×C271—21no—
80C42×C271C2311yescharacteristic
81C22×D271—21no—
82C22×D271—42no—
83C22×D271—42no—
84C22×D271—42no—
85C22×D271C2×D411yescharacteristic
86Dic542—22no—
87Dic542—42no—
88Dic542—42no—
89D1084—22no—
90C22⋊D271—42no—
91C2×Dic271—22no—
92C2×Dic271—21no—
93C2×Dic271C2×D411yescharacteristic
94C2×Dic271C2×D411yescharacteristic
95C2×Dic271C2×D411yescharacteristic
96C4×D271—41no—
97C4×D271—41no—
98C4×D271—42no—
99C4×D271#4114yes—
100C271⋊C8—22no—
101C23×C271—21no—
102Q8×C271—22no—
103Q8×C271—42no—
104D4×C271—22no—
105C2×C4×C271—21no—
106C2×C4×C271—22no—
107C2×C4×C271C2×D411yescharacteristic
108C2×C4×C271C2×D411yescharacteristic
109C2×C4×C271C2411yescharacteristic
110C8×C271—22no—
111C22×C271—41no—
112C22×C271—41no—
113C22×C271—41no—
114C22×C271C22×D411yescharacteristic
115C2×D271—21no—
116C2×D271—21no—
117C2×D271—41no—
118C2×D271—41no—
119C2×D271—41no—
120C2×D271—84no—
121C2×D271#12412yes—
122C4×C271—21no—
123C4×C271—21no—
124C4×C271—41no—
125C4×C271—42no—
126C4×C271C2×#4112yes—
127Dic271—21no—
128Dic271—21no—
129Dic271—41no—
130Dic271—42no—
131Dic271#12412yes—
132C2×C271—81no—
133C2×C271C2×#12411yescharacteristic
134C2×C271#57511yescharacteristic
135C2×C271#57511yescharacteristic
136D271—22no—
137D271—81no—
138C271#314011yescharacteristic, complemented, Sylow
139#3140—2711nomaximal, Sylow
140#533—2711no—
141C2×#124—2711no—
142#489—2711no—
143#489—2711no—
144#489—2711no—
145#489D27111yescharacteristic, complemented
146#488—2712no—
147#488—2712no—
148#419—2712no—
149#418—2712no—
150C4×M16—2711no—
151C22×Q8—2711no—
152C22×D4—2711no—
153C2×M16—2712no—
154C4⋊Q8—2711no—
155C4⋊Q8C2×D27111yescharacteristic, complemented
156#125—2712no—
157#124—2711no—
158#124—2711no—
159#124—2711no—
160#124—5421no—
161#124—5421no—
162#124C2×D27111yescharacteristic
163C22⋊Q8—5421no—
164C22⋊Q8—5421no—
165C4×Q8—5421no—
166C4×Q8—5421no—
167#117—2712no—
168C2×C22⋊C4—5421no—
169C2×C22⋊C4—5421no—
170C2×C42—2711no—
171#103—2712no—
172#103—2712no—
173#103—2712no—
174#103C2×D27112yescomplemented
175#102—2712no—
176#102—2712no—
177#102—2712no—
178#102C2×D27112yes—
179#97—2711no—
180#97C2×D27111yescharacteristic, complemented
181C4×C8—2712no—
182C24—5421no—
183C2×Q8—21no—
184C2×Q8—2711no—
185C2×Q8—5421no—
186C2×Q8—5424no—
187C2×Q8—5421no—
188C2×Q8C22×D27111yescharacteristic
189C2×D4—2711no—
190C2×D4—5424no—
191C2×D4—5421no—
192C2×D4C22×D27111yescharacteristic
193C22×C4—2711no—
194C22×C4—2711no—
195C22×C4—2711no—
196C22×C4—5421no—
197M16—5424no—
198C2×C8—2712no—
199C2×C8C22×D27112yes—
200C4⋊C4—21no—
201C4⋊C4—2712no—
202C4⋊C4—5422no—
203C4⋊C4—5422no—
204C4⋊C4—5421no—
205C4⋊C4C22×D27112yes—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—5421no—
209C22⋊C4—5421no—
210C22⋊C4—5422no—
211C22⋊C4—5421no—
212C22⋊C4—5421no—
213C22⋊C4—5421no—
214C22⋊C4—5421no—
215C42—2711no—
216C42—2711no—
217C42—2711no—
218C42—5422no—
219C42C22×D27111yescharacteristic
220C23—21no—
221C23—2711no—
222C23—5421no—
223C23—10842no—
224C23—10842no—
225C23—10842no—
226Q8—22no—
227Q8—42no—
228Q8—5422no—
229Q8—10842no—
230Q8—10842no—
231D4—22no—
232D4—5422no—
233D4—10842no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—2711no—
237C2×C4—2714no—
238C2×C4—2711no—
239C2×C4—2711no—
240C2×C4—5422no—
241C2×C4—5421no—
242C2×C4—10841no—
243C2×C4—10841no—
244C2×C4—10842no—
245C2×C4D4×D27111yescharacteristic
246C2×C4D4×D27111yescharacteristic
247C2×C4C23×D27111yescharacteristic
248C8—22no—
249C8—5422no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—2712no—
254C22—5421no—
255C22—5421no—
256C22—10841no—
257C22—10841no—
258C22—10841no—
259C22—21684no—
260C22C2×D4×D27111yescharacteristic
261C4—21no—
262C4—21no—
263C4—41no—
264C4—42no—
265C4—2712no—
266C4—5421no—
267C4—5421no—
268C4—10841no—
269C4—10842no—
270C4#41×D27112yes—
271C2—81no—
272C2—5422no—
273C2—21681no—
274C2#124×D27111yescharacteristic
275C2#3177469311yescharacteristic
276C2#3177469611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#489×C271C2
socleC22×C271C22×D4
prime radicalC2×D4×D271C22
hypercenter#489D271
hypocenterC271#3140
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #31774452C2semidirect
3. #31774259C2semidirect
4. #31774257C2semidirect
5. #31774256C2semidirect
6. #31774253C2semidirect
7. #31774250C2semidirect
8. #31773904C2semidirect
10. #31773849C2semidirect
11. #489×C271C2semidirect
15. #22576160C22semidirect
16. #22576151C22semidirect
30. #22576085C22semidirect
31. #22576084C22semidirect
32. #22576083C22semidirect
36. #22576071C22semidirect
37. C8×Dic271C22semidirect
38. C4⋊Q8×C271C22semidirect
40. #103×C271C22semidirect
42. #97×C271C22semidirect
138. C271#3140semidirect
145. #489D271semidirect
155. C4⋊Q8C2×D271semidirect
174. #103C2×D271semidirect
180. #97C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317744522
#124×D2712
#317742592
#317742572
#317742562
#317742532
#317742502
#317739042
#317739002
#317738492
#489×C2712
#3140271