Finite Group Databasebeta

← #24995837

Subgroups — #24995837

Order 11008.

518 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583278C211yesmaximal, characteristic
2#19135547C211yesmaximal, characteristic
3C4⋊C4×C43⋊C8C211yesmaximal, characteristic
4C2×#14582651C211yesmaximal, characteristic
5C2×#14582645C211yesmaximal, characteristic
6C2×#14582644C211yesmaximal, characteristic
7C43×#1988C211yesmaximal, characteristic
8C22×#10621885—21no—
9C22×#10621885—21no—
10C2×#13063325C2211yescharacteristic
11#14583278—22no—
12#14583278C414yescomplemented
13C2×C4⋊(C43⋊C8)C2211yescharacteristic
14C2×C4⋊(C43⋊C8)C2211yescharacteristic
15C2×C4×C43⋊C8—21no—
16C2×C4×C43⋊C8C2211yescharacteristic
17#14582661—22no—
18#14582654—21no—
19#14582651—22no—
20#14582645—22no—
21#14582644—22no—
22C2×C4⋊Q8×C43C2211yescharacteristic
23C2×C4⋊C8×C43C2211yescharacteristic
24C4×C4⋊C4×C43C2211yescharacteristic
25C43×#340—22no—
26C22×Dic86—21no—
27C2×#10621885—24no—
28C2×#10621885—24no—
29C2×#10621885—42no—
30C2×#10621885—42no—
31C2×#10621885—42no—
32C2×#10621885—42no—
33C22×C43⋊C8—21no—
34C22×C43⋊C8—21no—
35C22×C43⋊C8D411yescharacteristic
36C22×C43⋊C8D411yescharacteristic
37#13063325—21no—
38#13063325—21no—
39#13063325—21no—
40#13063325C2×C412yes—
41C2×C4⋊Dic43—21no—
42C2×C4⋊Dic43—21no—
43C4⋊(C43⋊C8)—21no—
44C4⋊(C43⋊C8)—21no—
45C4⋊(C43⋊C8)—21no—
46C4⋊(C43⋊C8)D412yes—
47C4×C43⋊C8—42no—
48C4×C43⋊C8C2×C412yes—
49C4×C43⋊C8D412yes—
50C22×Q8×C43—21no—
51C22×C8×C43—21no—
52C4⋊Q8×C43—21no—
53C4⋊Q8×C43—21no—
54C4⋊Q8×C43—21no—
55C4⋊Q8×C43C2×C412yes—
56C2×C4⋊C4×C43—21no—
57C2×C4⋊C4×C43—21no—
58C2×C4⋊C4×C43D412yes—
59C2×C42×C43—21no—
60C2×C42×C43C2311yescharacteristic
61C4⋊C8×C43—21no—
62C4⋊C8×C43—21no—
63#95×C43—21no—
64C2×Q8×C43—22no—
65C2×Q8×C43—41no—
66C2×Q8×C43—41no—
67C2×Q8×C43—41no—
68C2×Q8×C43—41no—
69C2×Q8×C43—41no—
70C22×C4×C43—22no—
71C22×C4×C43—41no—
72C22×C4×C43C2×D411yescharacteristic
73C22×C4×C43C2×D411yescharacteristic
74C22×C4×C43#4111yescharacteristic
75C22×Dic43—41no—
76C2×Dic86—22no—
77C2×Dic86—41no—
78C2×Dic86—41no—
79C2×Dic86—41no—
80C2×Dic86—41no—
81C2×Dic86—41no—
82C2×C8×C43—41no—
83C2×C8×C43—42no—
84C4⋊C4×C43—22no—
85C4⋊C4×C43—22no—
86C4⋊C4×C43—22no—
87C4⋊C4×C43—41no—
88C4⋊C4×C43—41no—
89C4⋊C4×C43Q1614yescomplemented
90C42×C43—41no—
91C42×C43—41no—
92C42×C43C22×C411yescharacteristic
93C42×C43C2×D411yescharacteristic
94C42×C43C2×D411yescharacteristic
95C42×C43#4111yescharacteristic
96#10621885—48no—
97#10621885—48no—
98C4⋊Dic43—22no—
99C4⋊Dic43—22no—
100C4⋊Dic43—41no—
101C4⋊Dic43—41no—
102C2×C43⋊C8—21no—
103C2×C43⋊C8—21no—
104C2×C43⋊C8—22no—
105C2×C43⋊C8—41no—
106C2×C43⋊C8—41no—
107C2×C43⋊C8—41no—
108C2×C43⋊C8—42no—
109C2×C43⋊C8C22⋊C412yes—
110C2×C43⋊C8C22⋊C412yes—
111C2×C43⋊C8D812yes—
112C2×C43⋊C8SD1612yes—
113C23×C43#12111yescharacteristic
114Q8×C43—42no—
115Q8×C43—42no—
116Q8×C43—81no—
117Q8×C43—81no—
118C2×C4×C43—22no—
119C2×C4×C43—24no—
120C2×C4×C43—42no—
121C2×C4×C43—42no—
122C2×C4×C43—42no—
123C2×C4×C43—81no—
124C2×C4×C43—81no—
125C2×C4×C43C2×C22⋊C411yescharacteristic
126C2×C4×C43C4×D411yescharacteristic
127C2×C4×C43C4×D411yescharacteristic
128C2×C4×C43#12111yescharacteristic
129C2×C4×C43#12411yescharacteristic
130C2×C4×C43#12711yescharacteristic
131C2×C4×C43C2×D811yescharacteristic
132C2×C4×C43C2×SD1611yescharacteristic
133C2×C4×C43C2×Q1612yes—
134C2×C4×C43#13512yes—
135C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
137C2×Dic43—42no—
138C2×Dic43—81no—
139C2×Dic43—81no—
140Dic86—42no—
141Dic86—42no—
142Dic86—81no—
143Dic86—81no—
144C8×C43—81no—
145C43⋊C8—22no—
146C43⋊C8—42no—
147C43⋊C8—81no—
148C43⋊C8#10214yes—
149#9733—431nomaximal, Sylow
150C22×C43#39011yescharacteristic
151C22×C43#45911yescharacteristic
152C22×C43#46011yescharacteristic
153C22×C43#46712yes—
154C22×C43#47312yes—
155C4×C43—21no—
156C4×C43—21no—
157C4×C43—44no—
158C4×C43—82no—
159C4×C43C2×#10211yescharacteristic
160C4×C43#41811yescharacteristic
161C4×C43C4×Q1612yes—
162C4×C43#45911yescharacteristic
163C4×C43#46011yescharacteristic
164C4×C43#49411yescharacteristic
165C4×C43#49511yescharacteristic
166Dic43—82no—
167C2×#494—431no—
168#1988—431no—
169#1988D4311yescharacteristic
170C8×C4⋊C4—431no—
171C2×#335—431no—
172C2×#332—431no—
173C2×#332—431no—
174C2×C43#166912yes—
175C2×C43#167012yes—
176C2×C43#196911yescharacteristic
177C2×C43#197911yescharacteristic
178C2×C43#198211yescharacteristic
179C22×Q16—861no—
180C22×Q16—861no—
181C2×C4⋊Q8—431no—
182C2×C4⋊Q8C2×D4311yescharacteristic
183#494—434no—
184#494—862no—
185C2×C4⋊C8—431no—
186C2×C4⋊C8—431no—
187C2×C4⋊C8C2×D4311yescharacteristic
188C2×C4×C8—431no—
189C2×C4×C8—861no—
190C4×C4⋊C4C2×D4311yescharacteristic
191#340—22no—
192#340—862no—
193#336—861no—
194#335—862no—
195#332—862no—
196#332—862no—
197C43#973311yescharacteristic, complemented, Sylow
198C22×Q8—21no—
199C22×Q8—861no—
200C2×Q16—864no—
201C2×Q16—864no—
202C2×Q16—1722no—
203C2×Q16—1722no—
204C2×Q16—1722no—
205C2×Q16—1722no—
206C22×C8—21no—
207C22×C8—431no—
208C22×C8—431no—
209C22×C8—861no—
210C22×C8—861no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—432no—
215C4⋊Q8—861no—
216C4⋊Q8—861no—
217C4⋊Q8—861no—
218C4⋊Q8C4×D4312yes—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—861no—
222C2×C4⋊C4—861no—
223C2×C4⋊C4C22⋊D4312yes—
224C2×C42—21no—
225C2×C42C22×D4311yescharacteristic
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—432no—
229C4⋊C8—861no—
230C4⋊C8—861no—
231C4⋊C8—861no—
232C4×C8—432no—
233C4×C8—432no—
234C4×C8—1722no—
235#95—21no—
236C2×Q8—22no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—862no—
243C2×Q8—1721no—
244C2×Q8—1721no—
245C2×Q8—1721no—
246C2×Q8—1721no—
247C2×Q8—1721no—
248C22×C4—22no—
249C22×C4—41no—
250C22×C4—1721no—
251C22×C4D4×D4311yescharacteristic
252C22×C4#1062190011yescharacteristic
253C22×C4C2×C22⋊D4311yescharacteristic
254Q16—1728no—
255Q16—1728no—
256C2×C8—41no—
257C2×C8—42no—
258C2×C8—432no—
259C2×C8—432no—
260C2×C8—432no—
261C2×C8—432no—
262C2×C8—861no—
263C2×C8—861no—
264C2×C8—862no—
265C2×C8—1721no—
266C2×C8—1722no—
267C2×C8—1721no—
268C2×C8—1721no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—862no—
275C4⋊C4—862no—
276C4⋊C4—1721no—
277C4⋊C4—1721no—
278C4⋊C4#1062188514yescomplemented
279C42—41no—
280C42—41no—
281C42C2×C4×D4311yescharacteristic
282C42#1062189811yescharacteristic
283C42D4×D4311yescharacteristic
284C42D4×D4311yescharacteristic
285C23#1306329711yescharacteristic
286Q8—42no—
287Q8—42no—
288Q8—81no—
289Q8—81no—
290Q8—3441no—
291Q8—3441no—
292Q8—1722no—
293Q8—1722no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—3441no—
302C2×C4—3441no—
303C2×C4—1722no—
304C2×C4C22⋊C4×D4311yescharacteristic
305C2×C4(C22⋊D43)⋊C411yescharacteristic
306C2×C4C4×C22⋊D4311yescharacteristic
307C2×C4#1306329811yescharacteristic
308C2×C4#1306331111yescharacteristic
309C2×C4#1306332111yescharacteristic
310C2×C4D8×D4311yescharacteristic
311C2×C4SD16×D4311yescharacteristic
312C2×C4C2×#1062188512yes—
313C2×C4#1306334512yes—
314C2×C4#1306334911yescharacteristic
315C2×C4#1306335211yescharacteristic
316C8—81no—
317C8—434no—
318C8—862no—
319C8—3441no—
320C8—1722no—
321C22#1458279911yescharacteristic
322C22#1458309211yescharacteristic
323C22#1458309711yescharacteristic
324C22#1458313812yes—
325C22#1458317212yes—
326C4—21no—
327C4—21no—
328C4—44no—
329C4—82no—
330C4—3442no—
331C4#102×D4311yescharacteristic
332C4#1458290211yescharacteristic
333C4C4×#1062188512yes—
334C4#1458309311yescharacteristic
335C4#1458309811yescharacteristic
336C4#1458327811yescharacteristic
337C4#1458328111yescharacteristic
338C2#1913395812yes—
339C2#1913396312yes—
340C2#1913545511yescharacteristic
341C2#1913550911yescharacteristic
342C2#1913552011yescharacteristic
subgroupG/H
centerC23#13063297
commutatorC42×C43C22×C4
FrattiniC2×C42C22×D43
FittingC43×#1988C2
socle, prime radicalC23×C43#121
cosocleC2×C42×C43C23
hypercenter#1988D43
hypocenterC43#9733
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
12. #14583278C4semidirect
89. C4⋊C4×C43Q16semidirect
197. C43#9733semidirect
278. C4⋊C4#10621885semidirect

Maximal subgroups

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

subgroupindex
C2×#145832782
#191355472
C4⋊C4×C43⋊C82
C2×#145826512
C2×#145826452
C2×#145826442
C43×#19882
#973343