Finite Group Databasebeta

← #25070505

Subgroups — #25070505

Order 11008.

1220 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1220 conjugacy classes of subgroups — too many to draw. Only its 187 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×#14583480C211yesmaximal, characteristic
2#19145348C212yesmaximal, complemented
3#19133631C214yesmaximal, complemented
4#19133563C214yesmaximal, complemented
5#19133162C212yesmaximal, complemented
6C2×#14582625C211yesmaximal, characteristic, complemented
7C43×#2627C211yesmaximal, characteristic, complemented
8C22×#10621898—21no—
9#14583480—24no—
10#14583480—22no—
11#14583480C2214yes—
12C2×#13063305—22no—
13C2×#13063300—22no—
14C2×#13063299—21no—
15C2×#13063299C2212yes—
16#14583227—22no—
17#14583217C2214yescomplemented
18#14583130—24no—
19#14583049—22no—
20#14583038C2214yescomplemented
21C2×C22⋊(C43⋊C8)C2212yes—
22#14582632C2218yescomplemented
23#14582625C2214yescomplemented
24C2×C22⋊Q8×C43C2211yescharacteristic
25C22×C4⋊C4×C43C2211yescharacteristic
26C43×#400C2212yescomplemented
27C43×#398—22no—
28C43×#395C2212yescomplemented
29C43×#385—21no—
30C43×#381—21no—
31C43×#380C2211yescharacteristic
32C2×#10621898—24no—
33C2×#10621898—24no—
34C2×#10621898—44no—
35C2×#10621898—44no—
36C22×Dic86—21no—
37C22×D172—21no—
38C22×D172—21no—
39C22×D172D412yes—
40C22×C22⋊D43—22no—
41C22×C4×D43—22no—
42C2×#10621884—24no—
43C22×C43⋊C8—22no—
44#13063305—28no—
45#13063305—28no—
46#13063300—28no—
47#13063300—28no—
48#13063299—24no—
49#13063299—24no—
50#13063299—22no—
51#13063299—22no—
52#13063299D414yes—
53#13063299C2314yes—
54C2×C4⋊Dic43—21no—
55C2×C4⋊Dic43D412yes—
56C2×(C2×D43)⋊C4—22no—
57C2×(C2×D43)⋊C4—22no—
58#13063256—24no—
59#13063249—24no—
60#13063245—24no—
61C22⋊(C43⋊C8)—22no—
62C22⋊(C43⋊C8)C2314yes—
63C22×Q8×C43C2311yescharacteristic
64C23×C4×C43D411yescharacteristic
65C23×C4×C43D411yescharacteristic
66C23×C4×C43C2311yescharacteristic
67C22⋊Q8×C43—22no—
68C22⋊Q8×C43—22no—
69C2×C4⋊C4×C43—22no—
70C2×C4⋊C4×C43—22no—
71C2×C4⋊C4×C43—21no—
72C2×C4⋊C4×C43—21no—
73C2×C4⋊C4×C43—22no—
74C2×C4⋊C4×C43D412yes—
75C2×C4⋊C4×C43C2311yescharacteristic
76C2×C4⋊C4×C43C2312yes—
77C2×C22⋊C4×C43—21no—
78C2×C22⋊C4×C43—21no—
79#95×C43—22no—
80#95×C43—22no—
81#95×C43—22no—
82#95×C43C2312yes—
83C24×C43C2×D411yescharacteristic
84C23×D43—22no—
85C2×Q8×C43—22no—
86C2×Q8×C43—81no—
87C22×C4×C43—21no—
88C22×C4×C43—21no—
89C22×C4×C43—21no—
90C22×C4×C43—21no—
91C22×C4×C43—22no—
92C22×C4×C43—21no—
93C22×C4×C43—22no—
94C22×C4×C43—21no—
95C22×C4×C43—81no—
96C22×C4×C43—81no—
97C22×C4×C43SD1614yescomplemented
98C22×C4×C43C2×D412yes—
99C22×C4×C43C2×D412yes—
100C22×C4×C43C2×D412yes—
101C22×C4×C43C2×D412yes—
102C22×C4×C43C2×D411yescharacteristic
103C22×C4×C43C2411yescharacteristic
104C2×C22⋊D43—24no—
105C2×C22⋊D43—24no—
106C2×C22⋊D43—44no—
107C2×C22⋊D43—44no—
108C22×Dic43—22no—
109#10621898—48no—
110#10621898—416no—
111#10621898—48no—
112C2×D172—24no—
113C2×D172—24no—
114C2×D172—22no—
115C2×D172—44no—
116C2×D172—82no—
117C2×D172—82no—
118C2×D172C2×D414yes—
119C2×C4×D43—28no—
120C2×C4×D43—48no—
121C2×Dic86—24no—
122C2×Dic86—44no—
123C4⋊C4×C43—24no—
124C4⋊C4×C43—24no—
125C4⋊C4×C43—22no—
126C4⋊C4×C43—42no—
127C4⋊C4×C43—81no—
128C4⋊C4×C43—82no—
129C22⋊C4×C43—41no—
130C22⋊C4×C43—41no—
131C22⋊C4×C43—42no—
132#10621884—84no—
133(C2×D43)⋊C4—28no—
134(C2×D43)⋊C4—24no—
135(C2×D43)⋊C4—24no—
136C4⋊Dic43—24no—
137C4⋊Dic43—22no—
138C4⋊Dic43C2×D414yes—
139C2×C43⋊C8—24no—
140C2×C43⋊C8—82no—
141C23×C43—41no—
142C23×C43—41no—
143C23×C43#12011yescharacteristic
144C23×C43#12011yescharacteristic
145C23×C43C2×SD1612yes—
146C23×C43AGL(1,ℤ/8)12yes—
147C23×C43C22×D411yescharacteristic
148C22×D43—24no—
149C22×D43—44no—
150C22×D43—44no—
151C22×D43—44no—
152Q8×C43—44no—
153C2×C4×C43—22no—
154C2×C4×C43—22no—
155C2×C4×C43—24no—
156C2×C4×C43—24no—
157C2×C4×C43—22no—
158C2×C4×C43—42no—
159C2×C4×C43—41no—
160C2×C4×C43—42no—
161C2×C4×C43—41no—
162C2×C4×C43—42no—
163C2×C4×C43—42no—
164C2×C4×C43—42no—
165C2×C4×C43—42no—
166C2×C4×C43—42no—
167C2×C4×C43—42no—
168C2×C4×C43—41no—
169C2×C4×C43—41no—
170C2×C4×C43—42no—
171C2×C4×C43—81no—
172C2×C4×C43#12012yes—
173C2×C4×C43C2×SD1614yes—
174C2×C4×C43C22×D412yes—
175C22⋊D43—44no—
176C22⋊D43—44no—
177C22⋊D43—48no—
178C2×Dic43—24no—
179C2×Dic43—44no—
180D172—44no—
181D172—48no—
182D172—82no—
183D172—84no—
184C4×D43—416no—
185Dic86—48no—
186C43⋊C8—44no—
187#18293—431nomaximal, Sylow
188C22×C43—21no—
189C22×C43—21no—
190C22×C43—21no—
191C22×C43—21no—
192C22×C43—21no—
193C22×C43—21no—
194C22×C43—42no—
195C22×C43—42no—
196C22×C43#45012yes—
197C22×C43#45012yes—
198C22×C43C2×#12011yescharacteristic
199C22×C43C22×SD1611yescharacteristic
200C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
201C2×D43—44no—
202C2×D43—44no—
203C2×D43—44no—
204C2×D43—88no—
205C4×C43—24no—
206C4×C43—22no—
207C4×C43—22no—
208C4×C43—42no—
209C4×C43—82no—
210C4×C43—81no—
211C4×C43—81no—
212Dic43—44no—
213C2×#537—431no—
214#3187—432no—
215#2627D4311yescharacteristic, complemented
216#1616—434no—
217#1604—434no—
218#1519—432no—
219C2×#324—431no—
220C2×C43—21no—
221C2×C43—22no—
222C2×C43—21no—
223C2×C43—22no—
224C2×C43C2×#45011yescharacteristic
225C2×C43#302011yescharacteristic
226C2×C43#302611yescharacteristic
227D43—84no—
228C22×#41—861no—
229#537—434no—
230#537—864no—
231#537—862no—
232C2×C22⋊Q8—862no—
233C2×C22⋊Q8C2×D4311yescharacteristic
234C2×#121—432no—
235C2×#121—862no—
236C2×#121—861no—
237C22×C4⋊C4C2×D4311yescharacteristic
238#483—862no—
239#481—434no—
240#465—864no—
241#450—862no—
242#448—434no—
243C2×C22⋊C8—432no—
244#400C2×D4312yescomplemented
245#398—22no—
246#395C2×D4312yescomplemented
247#385—21no—
248#381—21no—
249#380C2×D4311yescharacteristic
250#327—438no—
251#324—434no—
252C43#1829311yescharacteristic, complemented, Sylow
253C2×#41—864no—
254C2×#41—864no—
255C2×#41—1724no—
256C2×#41—1724no—
257C22×Q8—861no—
258C22×Q8C22×D4311yescharacteristic
259C22×D4—432no—
260C22×D4—861no—
261C22×D4—861no—
262C22×D4—862no—
263C23×C4—862no—
264C23×C4C22⋊D4311yescharacteristic
265C23×C4C22⋊D4311yescharacteristic
266C23×C4C22×D4311yescharacteristic
267C2×SD16—864no—
268C22×C8—862no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C22⋊Q8—868no—
272C22⋊Q8—868no—
273#121—434no—
274#121—434no—
275#121—868no—
276#121—868no—
277#121—864no—
278#121—862no—
279#121—862no—
280#121—864no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—432no—
287C2×C4⋊C4—861no—
288C2×C4⋊C4C22⋊D4312yes—
289C2×C4⋊C4C22×D4311yescharacteristic
290C2×C4⋊C4C22×D4312yes—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—862no—
294C2×C22⋊C4—862no—
295#106—864no—
296#103—864no—
297#102—864no—
298C22⋊C8—434no—
299C22⋊C8—862no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95C22×D4312yes—
304C24—862no—
305C24C2×C22⋊D4311yescharacteristic
306#41—17216no—
307#41—1728no—
308#41—1728no—
309C2×Q8—22no—
310C2×Q8—81no—
311C2×Q8—864no—
312C2×Q8—1724no—
313C2×D4—434no—
314C2×D4—864no—
315C2×D4—864no—
316C2×D4—864no—
317C2×D4—864no—
318C2×D4—862no—
319C2×D4—3442no—
320C2×D4—3442no—
321C2×D4—1724no—
322C2×D4—1724no—
323C2×D4—1724no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—81no—
333C22×C4—81no—
334C22×C4—868no—
335C22×C4—862no—
336C22×C4—1728no—
337C22×C4#1062188414yescomplemented
338C22×C4D4×D4312yes—
339C22×C4D4×D4312yes—
340C22×C4C2×C22⋊D4312yes—
341C22×C4C2×C22⋊D4312yes—
342C22×C4C2×C22⋊D4311yescharacteristic
343C22×C4C23×D4311yescharacteristic
344SD16—3444no—
345C2×C8—864no—
346C2×C8—3442no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—42no—
351C4⋊C4—81no—
352C4⋊C4—82no—
353C4⋊C4—434no—
354C4⋊C4—864no—
355C4⋊C4—862no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—42no—
359C22⋊C4—868no—
360C22⋊C4—864no—
361C22⋊C4—864no—
362C23—41no—
363C23—41no—
364C23—864no—
365C23—1724no—
366C23—1724no—
367C23—1724no—
368C23#1306329311yescharacteristic
369C23#1306329311yescharacteristic
370C23C2×#1062188412yes—
371C23#1306335012yes—
372C23C22×C22⋊D4311yescharacteristic
373Q8—44no—
374Q8—1728no—
375D4—3442no—
376D4—3444no—
377D4—1724no—
378D4—1728no—
379D4—1728no—
380D4—1724no—
381D4—1724no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—81no—
401C2×C4—864no—
402C2×C4—17216no—
403C2×C4—1724no—
404C2×C4#1306329312yes—
405C2×C4C2×#1062188414yes—
406C2×C4C2×D4×D4312yes—
407C8—1724no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—42no—
415C22—42no—
416C22—3448no—
417C22—1724no—
418C22—1724no—
419C22—1724no—
420C22#1458304912yes—
421C22#1458304912yes—
422C22C2×#1306329311yescharacteristic
423C22C22×#1062188411yescharacteristic
424C22C2×#1306335011yescharacteristic
425C4—22no—
426C4—22no—
427C4—24no—
428C4—42no—
429C4—82no—
430C4—81no—
431C4—81no—
432C4—1724no—
433C2—21no—
434C2—22no—
435C2—22no—
436C2—21no—
437C2—3444no—
438C2C2×#1458304911yescharacteristic
439C2#1914395811yescharacteristic
440C2#1914399111yescharacteristic
subgroupG/H
centerC22C2×#13063293
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2627C2
socleC22×C43C2×#120
prime radicalC2×#14583480C2
hypercenter#2627D43
hypocenterC43#18293
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
2. #19145348C2semidirect
3. #19133631C2semidirect
4. #19133563C2semidirect
5. #19133162C2semidirect
6. C2×#14582625C2semidirect
7. C43×#2627C2semidirect
17. #14583217C22semidirect
20. #14583038C22semidirect
22. #14582632C22semidirect
23. #14582625C22semidirect
26. C43×#400C22semidirect
28. C43×#395C22semidirect
97. C22×C4×C43SD16semidirect
215. #2627D43semidirect
244. #400C2×D43semidirect
246. #395C2×D43semidirect
252. C43#18293semidirect
337. C22×C4#10621884semidirect

Maximal subgroups

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

subgroupindex
C2×#145834802
#191453482
#191336312
#191335632
#191331622
C2×#145826252
C43×#26272
#1829343