Finite Group Databasebeta

← #46846283

Subgroups — #46846283

Order 34688.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#31774407C211yesmaximal, characteristic
2C2×C4⋊C4×D271C211yesmaximal, characteristic, complemented
3#31774234C212yesmaximal, complemented
4#31774228C211yesmaximal, characteristic, complemented
5#31774227C211yesmaximal, characteristic, complemented
6#31774214C212yesmaximal, complemented
7#31774208C211yesmaximal, characteristic, complemented
8#31773938C212yesmaximal, complemented
9#31773895C212yesmaximal, complemented
10#31773859C211yesmaximal, characteristic, complemented
11#480×C271C211yesmaximal, characteristic, complemented
12C2×#16864653C2211yescharacteristic
13C22×C4×D271—21no—
14C22×C4×D271C2211yescharacteristic
15C2×#16864637C2212yes—
16#22576142—21no—
17#22576140C2211yescharacteristic
18#22576139C2211yescharacteristic
19#22576131C2211yescharacteristic
20#22576128—21no—
21D4×Dic271—21no—
22(C22⋊D271)⋊C4—21no—
23#22576118C2211yescharacteristic
24C2×C4⋊Dic271C2211yescharacteristic
25C2×Dic271⋊C4—21no—
26C4⋊C4×D271—21no—
27C4⋊C4×D271—21no—
28C4⋊C4×D271C2212yes—
29C4⋊C4×D271C2212yes—
30#22576093C2212yescomplemented
31#22576092C2212yescomplemented
32#22576091C2212yescomplemented
33#22576085C2212yescomplemented
34#22576083C2212yescomplemented
35#22576081C2212yescomplemented
36C22⋊(C271⋊C8)C2211yescharacteristic
37(C2×D271)⋊C8C2212yes—
38#121×C271C2211yescharacteristic
39C2×C4⋊C4×C271C2211yescharacteristic
40#107×C271C2212yescomplemented
41#102×C271C2212yescomplemented
42C22⋊C8×C271C2211yescharacteristic
43C23×D271D411yescharacteristic
44#16864653—24no—
45C2×Dic542C2311yescharacteristic
46C2×C22⋊D271—21no—
47C2×C22⋊D271—21no—
48C22×Dic271—21no—
49C22×Dic271—21no—
50C22×Dic271D411yescharacteristic
51C2×C4×D271—24no—
52C2×C4×D271—22no—
53C2×C4×D271—22no—
54C2×C4×D271—44no—
55C2×C4×D271D411yescharacteristic
56C2×C4×D271D411yescharacteristic
57C2×C4×D271C2311yescharacteristic
58C2×C4×D271C2311yescharacteristic
59#16864637—24no—
60C2×C271⋊C8C2312yes—
61C4⋊Dic271—21no—
62C4⋊Dic271—21no—
63C4⋊Dic271C2311yescharacteristic
64C4⋊Dic271C2312yes—
65Dic271⋊C4—22no—
66Dic271⋊C4—22no—
67Dic271⋊C4—21no—
68C22⋊Dic271—21no—
69C22⋊Dic271—21no—
70(C2×D271)⋊C4—21no—
71(C2×D271)⋊C4—21no—
72C4×Dic271—21no—
73C2×D4×C271—21no—
74C2×D4×C271C2311yescharacteristic
75C22×C4×C271—21no—
76C22×C4×C271C2311yescharacteristic
77C2×C8×C271C2312yes—
78C4⋊C4×C271—21no—
79C4⋊C4×C271C2311yescharacteristic
80C4⋊C4×C271C2312yes—
81C22⋊C4×C271—21no—
82C22×D271—22no—
83C22×D271—22no—
84C22×D271—22no—
85C22×D271C2×D411yescharacteristic
86C22×D271C2×D411yescharacteristic
87Dic542—22no—
88C22⋊D271—42no—
89C22⋊D271—42no—
90C2×Dic271—22no—
91C2×Dic271—21no—
92C2×Dic271—21no—
93C2×Dic271—22no—
94C2×Dic271—42no—
95C2×Dic271—42no—
96C2×Dic271C2×D411yescharacteristic
97C2×Dic271C2×D411yescharacteristic
98C4×D271—22no—
99C4×D271—22no—
100C4×D271—22no—
101C4×D271—44no—
102C4×D271—44no—
103C4×D271#4114yes—
104C271⋊C8—22no—
105C23×C271—21no—
106C23×C271C2×D411yescharacteristic
107D4×C271—22no—
108D4×C271—42no—
109C2×C4×C271—22no—
110C2×C4×C271—21no—
111C2×C4×C271—22no—
112C2×C4×C271—42no—
113C2×C4×C271C2×D411yescharacteristic
114C2×C4×C271C2411yescharacteristic
115C8×C271—22no—
116C22×C271—21no—
117C22×C271—21no—
118C22×C271—41no—
119C22×C271—41no—
120C22×C271—41no—
121C22×C271AGL(1,ℤ/8)12yes—
122C22×C271C22×D411yescharacteristic
123C2×D271—28no—
124C2×D271—21no—
125C2×D271—21no—
126C2×D271—21no—
127C2×D271—21no—
128C2×D271#12312yes—
129C2×D271#13712yes—
130C4×C271—21no—
131C4×C271—42no—
132C4×C271—41no—
133C4×C271C2×#4112yes—
134Dic271—21no—
135Dic271—41no—
136Dic271—42no—
137Dic271—41no—
138Dic271#12312yes—
139C2×C271—22no—
140C2×C271—81no—
141C2×C271C2×#12311yescharacteristic
142C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C271C2×#13711yescharacteristic
144D271—22no—
145D271—22no—
146C271#309211yescharacteristic, complemented, Sylow
147#3092—2711nomaximal, Sylow
148#525—2711no—
149C22×C4⋊C4—2711no—
150#484—2712no—
151#483—2711no—
152#483—2711no—
153#481—2712no—
154#480—2711no—
155#480D27111yescharacteristic, complemented
156#428—2712no—
157#417—2712no—
158#407—2711no—
159C2×#41—2711no—
160C23×C4—2711no—
161C23×C4—5421no—
162C2×M16—2712no—
163#123—5421no—
164C22⋊Q8—2711no—
165C22⋊Q8—2711no—
166#121—2711no—
167#121C2×D27111yescharacteristic
168#120—5421no—
169C4×D4—5421no—
170C4×D4—5421no—
171#117—2711no—
172C2×C4⋊C4—2711no—
173C2×C4⋊C4—2712no—
174C2×C4⋊C4—2712no—
175C2×C4⋊C4—5421no—
176C2×C4⋊C4—5421no—
177C2×C4⋊C4—5421no—
178C2×C4⋊C4C2×D27111yescharacteristic
179#107—2712no—
180#107C2×D27112yescomplemented
181#106—2712no—
182#106—2712no—
183#103—2712no—
184#103—2712no—
185#102—2712no—
186#102C2×D27112yescomplemented
187C22⋊C8—2711no—
188C22⋊C8—2712no—
189C22⋊C8C2×D27111yescharacteristic
190C24—2711no—
191#41—5424no—
192C2×Q8—2711no—
193C2×D4—21no—
194C2×D4—5421no—
195C2×D4—5421no—
196C2×D4C22×D27111yescharacteristic
197C22×C4—21no—
198C22×C4—2711no—
199C22×C4—2711no—
200C22×C4—2711no—
201C22×C4—2711no—
202C22×C4—2711no—
203C22×C4—5424no—
204C22×C4—5422no—
205C22×C4—5422no—
206C22×C4—5421no—
207C22×C4—5421no—
208C22×C4—10844no—
209C22×C4C22×D27111yescharacteristic
210M16—5424no—
211C2×C8—2712no—
212C2×C8C22×D27112yes—
213C4⋊C4—21no—
214C4⋊C4—2712no—
215C4⋊C4—2711no—
216C4⋊C4—5422no—
217C4⋊C4—5422no—
218C4⋊C4—5421no—
219C4⋊C4—5421no—
220C4⋊C4—5421no—
221C4⋊C4C22×D27112yes—
222C4⋊C4C22×D27111yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—5421no—
225C22⋊C4—5421no—
226C22⋊C4—5421no—
227C22⋊C4—5421no—
228C42—5421no—
229C23—21no—
230C23—2711no—
231C23—2711no—
232C23—5422no—
233C23—5422no—
234C23—5422no—
235C23D4×D27111yescharacteristic
236Q8—5422no—
237D4—22no—
238D4—42no—
239D4—10842no—
240D4—10842no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—2711no—
246C2×C4—2714no—
247C2×C4—2711no—
248C2×C4—5422no—
249C2×C4—5422no—
250C2×C4—5422no—
251C2×C4—5422no—
252C2×C4—5421no—
253C2×C4—5421no—
254C2×C4—5422no—
255C2×C4—10844no—
256C2×C4—10844no—
257C2×C4—10842no—
258C2×C4—10842no—
259C2×C4D4×D27111yescharacteristic
260C2×C4C23×D27111yescharacteristic
261C8—22no—
262C8—5422no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—2712no—
269C22—2712no—
270C22—5428no—
271C22—5421no—
272C22—5421no—
273C22—5421no—
274C22—5421no—
275C22#2257618412yes—
276C22C2×D4×D27111yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—2712no—
281C4—5421no—
282C4—10841no—
283C4—10842no—
284C4—10841no—
285C4#41×D27112yes—
286C2—22no—
287C2—81no—
288C2—5422no—
289C2—5422no—
290C2#123×D27111yescharacteristic
291C2C2×#2257618411yescharacteristic
292C2#137×D27111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#480×C271C2
socleC22×C271C22×D4
prime radical#31774407C2
hypercenter#480D271
hypocenterC271#3092
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D271C2semidirect
3. #31774234C2semidirect
4. #31774228C2semidirect
5. #31774227C2semidirect
6. #31774214C2semidirect
7. #31774208C2semidirect
8. #31773938C2semidirect
9. #31773895C2semidirect
10. #31773859C2semidirect
11. #480×C271C2semidirect
30. #22576093C22semidirect
31. #22576092C22semidirect
32. #22576091C22semidirect
33. #22576085C22semidirect
34. #22576083C22semidirect
35. #22576081C22semidirect
40. #107×C271C22semidirect
41. #102×C271C22semidirect
146. C271#3092semidirect
155. #480D271semidirect
180. #107C2×D271semidirect
186. #102C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317744072
C2×C4⋊C4×D2712
#317742342
#317742282
#317742272
#317742142
#317742082
#317739382
#317738952
#317738592
#480×C2712
#3092271