Finite Group Databasebeta

← #25069016

Subgroups — #25069016

Order 11008.

1054 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1054 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C2×#14583525C211yesmaximal, characteristic
2#19144544C212yesmaximal, complemented
3#19138869C211yesmaximal, characteristic
4#19133622C214yesmaximal, complemented
5#19133588C214yesmaximal, complemented
6#19133129C212yesmaximal, complemented
7C43×#1462C211yesmaximal, characteristic
8#14583525—24no—
9#14583525—22no—
10#14583525C2214yes—
11C2×#13063304—21no—
12C2×#13063304C2212yes—
13C2×#13063302—22no—
14C2×#13063302—22no—
15C2×#13063284C2211yescharacteristic
16C22×C22⋊Dic43—21no—
17#14583241—24no—
18#14583235C2214yescomplemented
19#14583230C2214yescomplemented
20#14582941—22no—
21#14582909—22no—
22#14582835C2212yes—
23#14582824C2212yes—
24#14582787—22no—
25#14582772—21no—
26#14582762C2211yescharacteristic
27#14582755—21no—
28#14582637C2218yescomplemented
29C22×C22⋊C4×C43C2211yescharacteristic
30C43×#407C2212yes—
31C43×#323C2214yes—
32C22×Dic86—21no—
33C22×Dic86D412yes—
34C23×Dic43—21no—
35C23×Dic43—21no—
36#13063304—24no—
37#13063304—24no—
38#13063304—22no—
39#13063304—22no—
40#13063304D414yes—
41#13063304C2314yes—
42#13063302—28no—
43#13063302—28no—
44#13063302—28no—
45#13063302—28no—
46#13063284—22no—
47#13063284—22no—
48C2×C4⋊Dic43—21no—
49C2×C4⋊Dic43D412yes—
50C2×C4⋊Dic43C2311yescharacteristic
51C2×C4⋊Dic43C2311yescharacteristic
52C2×Dic43⋊C4—22no—
53C2×Dic43⋊C4—22no—
54C2×Dic43⋊C4—22no—
55C2×C22⋊Dic43—22no—
56C2×C22⋊Dic43—24no—
57C2×C22⋊Dic43—22no—
58C2×C22⋊Dic43—24no—
59C2×C22⋊Dic43—22no—
60C2×C22⋊Dic43—21no—
61C2×C22⋊Dic43—21no—
62C2×C22⋊Dic43—22no—
63C2×C4×Dic43—21no—
64#13063259—24no—
65#13063257—24no—
66#13063250—24no—
67#13063250—24no—
68C43⋊#95—21no—
69C43⋊#95—22no—
70C43⋊#95C2312yes—
71C25×C43D411yescharacteristic
72C23×C4×C43D411yescharacteristic
73C23×C4×C43C2311yescharacteristic
74C2×M16×C43—22no—
75C2×C22⋊C4×C43—22no—
76C2×C22⋊C4×C43—22no—
77C2×C22⋊C4×C43D412yes—
78C2×C22⋊C4×C43C2312yes—
79C22⋊C8×C43—22no—
80C22⋊C8×C43C2314yes—
81C24×C43—21no—
82C24×C43—21no—
83C24×C43—24no—
84C24×C43—42no—
85C24×C43—81no—
86C24×C43C2×D411yescharacteristic
87C24×C43C2×D412yes—
88C22×C4×C43—21no—
89C22×C4×C43—21no—
90C22×C4×C43—42no—
91C22×C4×C43—81no—
92C22×C4×C43C2×D412yes—
93C22×C4×C43C2×D412yes—
94C22×C4×C43C2×D412yes—
95C22×C4×C43C2×D411yescharacteristic
96C22×C4×C43C2411yescharacteristic
97C22×Dic43—24no—
98C22×Dic43—24no—
99C22×Dic43—22no—
100C22×Dic43—22no—
101C22×Dic43—22no—
102C22×Dic43—21no—
103C22×Dic43—21no—
104C22×Dic43—82no—
105C22×Dic43—82no—
106C2×Dic86—24no—
107C2×Dic86—22no—
108C2×Dic86—44no—
109C2×Dic86—82no—
110C2×Dic86C2×D414yes—
111M16×C43—44no—
112C2×C8×C43—24no—
113C22⋊C4×C43—24no—
114C22⋊C4×C43—24no—
115C22⋊Dic43—28no—
116C22⋊Dic43—24no—
117C22⋊Dic43—24no—
118C22⋊Dic43—42no—
119C22⋊Dic43—41no—
120C22⋊Dic43—41no—
121C22⋊Dic43—44no—
122C22⋊Dic43—82no—
123C4⋊Dic43—24no—
124C4⋊Dic43—22no—
125C4⋊Dic43—22no—
126C4⋊Dic43—22no—
127C4⋊Dic43C2×D414yes—
128Dic43⋊C4—28no—
129Dic43⋊C4—24no—
130Dic43⋊C4—24no—
131Dic43⋊C4—82no—
132C4×Dic43—42no—
133C23×C43—24no—
134C23×C43—24no—
135C23×C43—22no—
136C23×C43—22no—
137C23×C43—22no—
138C23×C43—22no—
139C23×C43—41no—
140C23×C43—42no—
141C23×C43—42no—
142C23×C43—42no—
143C23×C43—42no—
144C23×C43—81no—
145C23×C43—81no—
146C23×C43—88no—
147C23×C43#12011yescharacteristic
148C23×C43#12011yescharacteristic
149C23×C43#13714yes—
150C23×C43C22×D411yescharacteristic
151C2×C4×C43—24no—
152C2×C4×C43—22no—
153C2×C4×C43—41no—
154C2×C4×C43—42no—
155C2×C4×C43—41no—
156C2×C4×C43—42no—
157C2×C4×C43—42no—
158C2×C4×C43—81no—
159C2×C4×C43#12012yes—
160C2×C4×C43C22×D412yes—
161C2×Dic43—24no—
162C2×Dic43—24no—
163C2×Dic43—44no—
164C2×Dic43—44no—
165C2×Dic43—42no—
166C2×Dic43—42no—
167C2×Dic43—42no—
168C2×Dic43—41no—
169C2×Dic43—41no—
170C2×Dic43—42no—
171C2×Dic43—42no—
172C2×Dic43—84no—
173C2×Dic43—84no—
174Dic86—44no—
175Dic86—82no—
176Dic86—84no—
177C8×C43—44no—
178#18138—431nomaximal, Sylow
179C22×C43—21no—
180C22×C43—21no—
181C22×C43—21no—
182C22×C43—21no—
183C22×C43—21no—
184C22×C43—24no—
185C22×C43—21no—
186C22×C43—21no—
187C22×C43—44no—
188C22×C43—44no—
189C22×C43—48no—
190C22×C43—42no—
191C22×C43—42no—
192C22×C43—82no—
193C22×C43—84no—
194C22×C43#45812yes—
195C22×C43C2×#12011yescharacteristic
196C22×C43C2×#13712yes—
197C4×C43—22no—
198C4×C43—22no—
199C4×C43—42no—
200Dic43—44no—
201Dic43—44no—
202Dic43—82no—
203Dic43—81no—
204Dic43—81no—
205C2×#543—431no—
206#3095—432no—
207#2482—431no—
208#1614—434no—
209#1608—434no—
210#1511—432no—
211#1462D4311yescharacteristic
212C2×C43—22no—
213C2×C43—22no—
214C2×C43—41no—
215C2×C43—44no—
216C2×C43#300411yescharacteristic
217C2×C43#301911yescharacteristic
218C2×C43C2×#45811yescharacteristic
219#543—434no—
220#543—864no—
221#543—862no—
222C2×C22⋊Q8—432no—
223C2×C22⋊Q8—862no—
224C2×C22⋊Q8—862no—
225C2×C22⋊Q8—861no—
226C2×#117—431no—
227C22×C22⋊C4—861no—
228C22×C22⋊C4C2×D4311yescharacteristic
229#485—864no—
230#484—434no—
231#483—434no—
232#428—862no—
233#419—862no—
234#407C2×D4312yes—
235#399—432no—
236#397—432no—
237#388—862no—
238#385—861no—
239#381—431no—
240#379—861no—
241#328—438no—
242#323C2×D4314yes—
243C43#1813811yescharacteristic, complemented, Sylow
244C25D17211yescharacteristic
245C22×Q8—432no—
246C22×Q8—861no—
247C23×C4—861no—
248C23×C4—861no—
249C23×C4D17211yescharacteristic
250C23×C4C22×D4311yescharacteristic
251C2×M16—22no—
252C22⋊Q8—434no—
253C22⋊Q8—434no—
254C22⋊Q8—868no—
255C22⋊Q8—868no—
256C22⋊Q8—868no—
257C22⋊Q8—868no—
258C22⋊Q8—864no—
259C22⋊Q8—862no—
260C22⋊Q8—864no—
261C22⋊Q8—862no—
262#117—862no—
263#117—862no—
264C2×C4⋊C4—432no—
265C2×C4⋊C4—431no—
266C2×C4⋊C4—431no—
267C2×C4⋊C4—862no—
268C2×C4⋊C4—861no—
269C2×C4⋊C4—862no—
270C2×C4⋊C4—862no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—862no—
274C2×C22⋊C4—864no—
275C2×C22⋊C4—862no—
276C2×C22⋊C4—864no—
277C2×C22⋊C4—862no—
278C2×C22⋊C4—861no—
279C2×C22⋊C4—861no—
280C2×C22⋊C4—862no—
281C2×C22⋊C4D17212yes—
282C2×C22⋊C4C22×D4312yes—
283C2×C42—861no—
284#107—864no—
285#106—864no—
286#103—864no—
287#103—864no—
288C22⋊C8—22no—
289C22⋊C8C22×D4314yes—
290#95—432no—
291#95—861no—
292#95—862no—
293C24—21no—
294C24—21no—
295C24—24no—
296C24—42no—
297C24—81no—
298C24C2×D17211yescharacteristic
299C24C2×D17212yes—
300C2×Q8—434no—
301C2×Q8—864no—
302C2×Q8—862no—
303C2×Q8—3442no—
304C2×Q8—1724no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—81no—
309C22×C4—864no—
310C22×C4—864no—
311C22×C4—862no—
312C22×C4—862no—
313C22×C4—862no—
314C22×C4—861no—
315C22×C4—861no—
316C22×C4—3442no—
317C22×C4—3442no—
318C22×C4C2×D17212yes—
319C22×C4C2×D17211yescharacteristic
320C22×C4D4×D4312yes—
321C22×C4D4×D4312yes—
322C22×C4C23×D4311yescharacteristic
323M16—44no—
324C2×C8—24no—
325C4⋊C4—434no—
326C4⋊C4—868no—
327C4⋊C4—864no—
328C4⋊C4—862no—
329C4⋊C4—864no—
330C4⋊C4—864no—
331C4⋊C4—862no—
332C4⋊C4—862no—
333C4⋊C4—3442no—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—868no—
337C22⋊C4—864no—
338C22⋊C4—864no—
339C22⋊C4—3442no—
340C22⋊C4—1722no—
341C22⋊C4—1721no—
342C22⋊C4—1721no—
343C22⋊C4—1724no—
344C42—1722no—
345C23—24no—
346C23—24no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—81no—
357C23—81no—
358C23—88no—
359C23#1306329511yescharacteristic
360C23#1306329511yescharacteristic
361C23#1306335814yes—
362C23C22×D17211yescharacteristic
363Q8—3442no—
364Q8—3444no—
365Q8—1724no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—81no—
374C2×C4—864no—
375C2×C4—864no—
376C2×C4—3444no—
377C2×C4—3444no—
378C2×C4—1722no—
379C2×C4—1722no—
380C2×C4—1722no—
381C2×C4—1721no—
382C2×C4—1721no—
383C2×C4—1722no—
384C2×C4—1722no—
385C2×C4—1724no—
386C2×C4—1724no—
387C2×C4#1306329512yes—
388C2×C4C2×D4×D4312yes—
389C8—44no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—24no—
398C22—44no—
399C22—44no—
400C22—48no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—84no—
405C22#1458309012yes—
406C22C2×#1306329511yescharacteristic
407C22C2×#1306335812yes—
408C4—22no—
409C4—22no—
410C4—42no—
411C4—3442no—
412C4—3441no—
413C4—3441no—
414C4—1724no—
415C4—1724no—
416C2—22no—
417C2—22no—
418C2—41no—
419C2—44no—
420C2#1914383211yescharacteristic
421C2#1914395511yescharacteristic
422C2C2×#1458309011yescharacteristic
subgroupG/H
centerC22C2×#13063295
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1462C2
socleC22×C43C2×#120
prime radicalC25×C43D4
hypercenter#1462D43
hypocenterC43#18138
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
2. #19144544C2semidirect
4. #19133622C2semidirect
5. #19133588C2semidirect
6. #19133129C2semidirect
18. #14583235C22semidirect
19. #14583230C22semidirect
28. #14582637C22semidirect
243. C43#18138semidirect

Maximal subgroups

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

subgroupindex
C2×#145835252
#191445442
#191388692
#191336222
#191335882
#191331292
C43×#14622
#1813843