Finite Group Databasebeta

← #25960174

Subgroups — #25960174

Order 11008.

4810 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583666C211yesmaximal, characteristic, complemented
2#19147798C211yesmaximal, characteristic, complemented
3#19143128C2116yesmaximal, complemented
4#19143114C216yesmaximal, complemented
5C2×#14582908C216yesmaximal, complemented
6C22×2−1+4×C43C211yesmaximal, characteristic
7#14583666—28no—
8C2×#13063332C2218yescomplemented
9C22×#10621877C2216yes—
10#14583382C2218yescomplemented
11#14583377—21no—
12#14583374—23no—
13#41×Dic43—24no—
14C2×#13063284C2216yes—
15#14582908C22112yes—
16#14582908C22124yescomplemented
17#14582904C2214yes—
18#14582900C22124yescomplemented
19#14582895C22124yescomplemented
20C2×#13063249C22124yescomplemented
21C2×2−1+4×C43—28no—
22C2×2−1+4×C43—24no—
23C2×2−1+4×C43C4116yescomplemented
24C2×2−1+4×C43C2218yes—
25C22×#41×C43—23no—
26C22×#41×C43C2213yes—
27C22×#41×C43C2211yescharacteristic
28C23×Q8×C43—21no—
29C23×Q8×C43C2213yes—
30#13063332—232no—
31C2×#10621877—224no—
32C2×#10621877C2314yes—
33C2×#10621877C23112yes—
34C22×C43⋊C8C23112yes—
35Q8×Dic43—28no—
36D4×Dic43—224no—
37#13063284—26no—
38#13063284—26no—
39#13063284C2314yes—
40#13063284C23112yes—
41C2×C4⋊Dic43C23112yes—
42C2×C22⋊Dic43—26no—
43C2×C4×Dic43—26no—
44#13063249C23148yes—
45#13063246C23116yes—
462−1+4×C43—264no—
472−1+4×C43—232no—
482−1+4×C43C2×C4164yescomplemented
49C2×#41×C43—248no—
50C2×#41×C43—224no—
51C2×#41×C43—212no—
52C2×#41×C43—26no—
53C2×#41×C43C2×C418yes—
54C2×#41×C43C2×C4124yescomplemented
55C2×#41×C43D418yes—
56C2×#41×C43D414yes—
57C2×#41×C43C23112yes—
58C2×#41×C43C2314yes—
59C22×Q8×C43—216no—
60C22×Q8×C43—28no—
61C22×Q8×C43—23no—
62C22×Q8×C43—212no—
63C22×Q8×C43—26no—
64C22×Q8×C43C2×C4124yescomplemented
65C22×Q8×C43D411yescharacteristic
66C22×Q8×C43C2313yes—
67C22×Q8×C43C23112yes—
68C22×D4×C43—23no—
69C22×D4×C43D413yes—
70C22×D4×C43C2311yescharacteristic
71C23×C4×C43—23no—
72C23×C4×C43—23no—
73C23×C4×C43C2313yes—
74C24×C43—21no—
75C24×C43C2×D413yes—
76#41×C43—2192no—
77#41×C43—248no—
78#41×C43—224no—
79#41×C43C22⋊C4132yes—
80#41×C43C22×C4148yes—
81#41×C43C22×C4116yes—
82#41×C43C2×D4116yes—
83C2×Q8×C43—264no—
84C2×Q8×C43—224no—
85C2×Q8×C43—248no—
86C2×Q8×C43—26no—
87C2×Q8×C43—23no—
88C2×Q8×C43—212no—
89C2×Q8×C43—224no—
90C2×Q8×C43C22⋊C418yes—
91C2×Q8×C43C22×C4112yes—
92C2×Q8×C43C22×C4148yes—
93C2×Q8×C43C2×D414yes—
94C2×Q8×C43C2416yes—
95C2×D4×C43—22no—
96C2×D4×C43—21no—
97C2×D4×C43—224no—
98C2×D4×C43—212no—
99C2×D4×C43C22⋊C4124yes—
100C2×D4×C43C22×C414yes—
101C2×D4×C43C2×D4112yes—
102C2×D4×C43C2412yes—
103C22×C4×C43—224no—
104C22×C4×C43—212no—
105C22×C4×C43—224no—
106C22×C4×C43—23no—
107C22×C4×C43—212no—
108C22×C4×C43C22×C4112yes—
109C22×C4×C43C2×D4112yes—
110C22×C4×C43C2×D413yes—
111C22×C4×C43C2×D416yes—
112C22×C4×C43C2411yescharacteristic
113C22×C4×C43C2416yes—
114C22×Dic43—212no—
115C22⋊Dic43—24no—
116C22⋊Dic43—212no—
117C4⋊Dic43—212no—
118C4⋊Dic43C2418yes—
119C4×Dic43—24no—
120C4×Dic43—212no—
121#10621877—28no—
122#10621877—224no—
123C2×C43⋊C8—224no—
124C2×C43⋊C8C2418yes—
125C23×C43—24no—
126C23×C43—22no—
127C23×C43—46no—
128C23×C43—44no—
129C23×C43C2×C22⋊C4112yes—
130C23×C43C22×D411yescharacteristic
131C23×C43C22×D416yes—
132Q8×C43—248no—
133Q8×C43—212no—
134Q8×C43—26no—
135Q8×C43C2×C22⋊C4116yes—
136Q8×C43C23×C4112yes—
137D4×C43—248no—
138D4×C43—24no—
139D4×C43—22no—
140D4×C43C2×C22⋊C4148yes—
141D4×C43C23×C414yes—
142C2×C4×C43—212no—
143C2×C4×C43—248no—
144C2×C4×C43—26no—
145C2×C4×C43—248no—
146C2×C4×C43C2×C22⋊C4112yes—
147C2×C4×C43C2×C22⋊C4124yes—
148C2×C4×C43C23×C412yes—
149C2×C4×C43C23×C4112yes—
150C2×C4×C43C22×D411yescharacteristic
151C2×C4×C43C22×D416yes—
152C2×C4×C43C22×D4112yes—
153C2×C4×C43C22×D412yes—
154C2×C4×C43C2511yescharacteristic
155C2×Dic43—28no—
156C2×Dic43—812no—
157C43⋊C8—28no—
158#60088—431nomaximal, Sylow
159C22×C43—22no—
160C22×C43—212no—
161C22×C43—24no—
162C22×C43—412no—
163C22×C43—48no—
164C22×C43—44no—
165C22×C43C22×C22⋊C412yes—
166C22×C43C22×C22⋊C4112yes—
167C22×C43C23×D411yescharacteristic
168C4×C43—212no—
169C4×C43C22×C22⋊C4112yes—
170C4×C43C22×C22⋊C412yes—
171C4×C43C24×C411yescharacteristic
172C4×C43C23×D411yescharacteristic
173Dic43—48no—
174C22×2−1+4D4311yescharacteristic
175C2×#568—431no—
176#3428—431no—
177#2898—4316no—
178#2896—436no—
179C2×#419—436no—
180C2×C43—22no—
181C2×C43—212no—
182C2×C43—44no—
183C2×C43C23×C22⋊C411yescharacteristic
184C2×C43#358612yes—
185C2×2−1+4—28no—
186C2×2−1+4—24no—
187C2×2−1+4Dic43116yescomplemented
188C2×2−1+4C2×D4318yes—
189C22×#41—23no—
190C22×#41C2×D4313yes—
191C22×#41C2×D4311yescharacteristic
192C23×Q8—21no—
193C23×Q8C2×D4313yes—
194#568—868no—
195C2×#131—438no—
196C22×M16—436no—
197#520—438no—
198#519—861no—
199#518—863no—
200C4×#41—864no—
201C2×#117—436no—
202#419—4312no—
203#419—4324no—
204#418—434no—
205#417—4324no—
206#416—4324no—
207C2×#103—4324no—
208C43#6008811yescharacteristic, complemented, Sylow
2092−1+4—264no—
2102−1+4—232no—
2112−1+4C2×Dic43164yescomplemented
212C2×#41—248no—
213C2×#41—224no—
214C2×#41—212no—
215C2×#41—26no—
216C2×#41C2×Dic4318yes—
217C2×#41C2×Dic43124yescomplemented
218C2×#41C22⋊D4318yes—
219C2×#41C22⋊D4314yes—
220C2×#41C22×D43112yes—
221C2×#41C22×D4314yes—
222C22×Q8—216no—
223C22×Q8—28no—
224C22×Q8—23no—
225C22×Q8—212no—
226C22×Q8—26no—
227C22×Q8C2×Dic43124yescomplemented
228C22×Q8C22⋊D4311yescharacteristic
229C22×Q8C22×D4313yes—
230C22×Q8C22×D43112yes—
231C22×D4—23no—
232C22×D4C22⋊D4313yes—
233C22×D4C22×D4311yescharacteristic
234C23×C4—23no—
235C23×C4—23no—
236C23×C4C22×D4313yes—
237#131—8632no—
238C2×M16—434no—
239C2×M16—4312no—
240C2×M16—8624no—
241C22×C8—4312no—
242C4×Q8—868no—
243C4×D4—8624no—
244#117—434no—
245#117—4312no—
246#117—866no—
247#117—866no—
248C2×C4⋊C4—4312no—
249C2×C22⋊C4—866no—
250C2×C42—866no—
251#103—4348no—
252#102—4316no—
253C24—21no—
254C24C2×C22⋊D4313yes—
255#41—2192no—
256#41—248no—
257#41—224no—
258#41C22⋊Dic43132yes—
259#41C22×Dic43148yes—
260#41C22×Dic43116yes—
261#41C2×C22⋊D43116yes—
262C2×Q8—264no—
263C2×Q8—224no—
264C2×Q8—248no—
265C2×Q8—26no—
266C2×Q8—23no—
267C2×Q8—212no—
268C2×Q8—224no—
269C2×Q8C22⋊Dic4318yes—
270C2×Q8C22×Dic43112yes—
271C2×Q8C22×Dic43148yes—
272C2×Q8C2×C22⋊D4314yes—
273C2×Q8C23×D4316yes—
274C2×D4—224no—
275C2×D4—212no—
276C2×D4—21no—
277C2×D4—22no—
278C2×D4C22⋊Dic43124yes—
279C2×D4C22×Dic4314yes—
280C2×D4C2×C22⋊D43112yes—
281C2×D4C23×D4312yes—
282C22×C4—224no—
283C22×C4—23no—
284C22×C4—212no—
285C22×C4—224no—
286C22×C4—212no—
287C22×C4—8612no—
288C22×C4C22×Dic43112yes—
289C22×C4C2×C22⋊D4316yes—
290C22×C4C2×C22⋊D4313yes—
291C22×C4C2×C22⋊D43112yes—
292C22×C4C23×D4316yes—
293C22×C4C23×D4311yescharacteristic
294M16—868no—
295M16—8624no—
296C2×C8—438no—
297C2×C8—8624no—
298C4⋊C4—438no—
299C4⋊C4—8612no—
300C22⋊C4—864no—
301C22⋊C4—8612no—
302C42—864no—
303C42—8612no—
304C23—24no—
305C23—22no—
306C23—46no—
307C23—44no—
308C23C2×C22⋊Dic43112yes—
309C23C22×C22⋊D4311yescharacteristic
310C23C22×C22⋊D4316yes—
311Q8—248no—
312Q8—212no—
313Q8—26no—
314Q8C2×C22⋊Dic43116yes—
315Q8C23×Dic43112yes—
316D4—248no—
317D4—24no—
318D4—22no—
319D4C2×C22⋊Dic43148yes—
320D4C23×Dic4314yes—
321C2×C4—212no—
322C2×C4—248no—
323C2×C4—26no—
324C2×C4—248no—
325C2×C4—868no—
326C2×C4—34412no—
327C2×C4C2×C22⋊Dic43112yes—
328C2×C4C2×C22⋊Dic43124yes—
329C2×C4C23×Dic4312yes—
330C2×C4C23×Dic43112yes—
331C2×C4C22×C22⋊D4311yescharacteristic
332C2×C4C22×C22⋊D4316yes—
333C2×C4C22×C22⋊D43112yes—
334C2×C4C22×C22⋊D4312yes—
335C2×C4C24×D4311yescharacteristic
336C8—868no—
337C22—212no—
338C22—22no—
339C22—24no—
340C22—412no—
341C22—48no—
342C22—44no—
343C22C22×C22⋊Dic4312yes—
344C22C22×C22⋊Dic43112yes—
345C22C23×C22⋊D4311yescharacteristic
346C4—212no—
347C4—1728no—
348C4C22×C22⋊Dic43112yes—
349C4C22×C22⋊Dic4312yes—
350C4C24×Dic4311yescharacteristic
351C4C23×C22⋊D4311yescharacteristic
352C2—22no—
353C2—212no—
354C2—44no—
355C2C23×C22⋊Dic4311yescharacteristic
356C2#1914938212yes—
subgroupG/H
centerC22C23×C22⋊D43
commutatorC4×C43C24×C4
FrattiniC2×C4C24×D43
Fitting, prime radicalC22×2−1+4×C43C2
socleC22×C43C23×D4
cosocleC2×C4×C43C25
hypercenterC22×2−1+4D43
hypocenterC43#60088
LCSG ▷ C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C22×#41 ◁ C22×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#14583666C2semidirect
2. #19147798C2semidirect
3. #19143128C2semidirect
4. #19143114C2semidirect
5. C2×#14582908C2semidirect
8. C2×#13063332C22semidirect
10. #14583382C22semidirect
16. #14582908C22semidirect
18. #14582900C22semidirect
19. #14582895C22semidirect
20. C2×#13063249C22semidirect
23. C2×2−1+4×C43C4semidirect
48. 2−1+4×C43C2×C4semidirect
54. C2×#41×C43C2×C4semidirect
64. C22×Q8×C43C2×C4semidirect
187. C2×2−1+4Dic43semidirect
208. C43#60088semidirect
211. 2−1+4C2×Dic43semidirect
217. C2×#41C2×Dic43semidirect
227. C22×Q8C2×Dic43semidirect

Maximal subgroups

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

subgroupindex
C2×#145836662
#191477982
#191431282
#191431142
C2×#145829082
C22×2−1+4×C432
#6008843