Finite Group Databasebeta

← #11864171

Subgroups — #11864171

Order 1152.

2262 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3830C211yesmaximal, characteristic
2C2×#64124C212yesmaximal, complemented
3C3⋊D3×C22⋊Q8C211yesmaximal, characteristic, complemented
4C32×#450C211yesmaximal, characteristic, complemented
5#10592917C211yesmaximal, characteristic, complemented
6#10592872C212yesmaximal, complemented
7#10592749C211yesmaximal, characteristic, complemented
8#10592482C212yesmaximal, complemented
9#10592470C212yesmaximal, complemented
10#10592438C211yesmaximal, characteristic, complemented
11#10592425C211yesmaximal, characteristic, complemented
12#80147—34nomaximal
13C22×C32×D4C2211yescharacteristic
14C22×#638—21no—
15C23×C3⋊Dic3—21no—
16C2×Q8×C3⋊D3C2211yescharacteristic, complemented
17C2×#3830—28no—
18C2×#3830—21no—
19C2×#3830—21no—
20C2×#3830C2212yes—
21C2×#3830C2212yes—
22C22×C4×C3⋊D3C2211yescharacteristic
23C22×#634C2211yescharacteristic
24C2×C32×SD16C2212yescomplemented
25C32×C22⋊Q8C2211yescharacteristic
26#64155C2211yescharacteristic, complemented
27C2×#3756C2212yescomplemented
28C2×#3754C2212yescomplemented
29#64133C2211yescharacteristic
30#64124—28no—
31C2×#3746C2212yescomplemented
32C2×#3743C2212yescomplemented
33#64106C2211yescharacteristic, complemented
34#64105—21no—
35C4⋊C4×C3⋊D3—21no—
36C4⋊C4×C3⋊D3C2211yescharacteristic, complemented
37C22⋊C4×C3⋊D3—21no—
38#64087—21no—
39C32×#102C2212yescomplemented
40C32×C22⋊C8C2211yescharacteristic
41#63659C2212yescomplemented
42#63657C2211yescharacteristic
43#63647C2212yescomplemented
44#63643C2212yescomplemented
45#63637C2212yescomplemented
46C22×#237—34no—
47C2×#915—38no—
48D3×C22⋊Q8—34no—
49C3×#450D314yescomplemented
50#5218—34no—
51#5173—38no—
52#5050—34no—
53#4783—38no—
54#4771—38no—
55#4739—34no—
56#4726—34no—
57C24×C32—21no—
58C23×C3⋊D3D411yescharacteristic
59C2×C32×Q8C2311yescharacteristic
60C2×C32×D4—24no—
61C2×C32×D4—21no—
62C2×C32×D4C2312yes—
63C22×C32×C4C2311yescharacteristic
64C2×#638—28no—
65C2×#638—22no—
66C2×#638—22no—
67C22×C3⋊Dic3—28no—
68C22×C3⋊Dic3—21no—
69C22×C3⋊Dic3—22no—
70C22×C3⋊Dic3—22no—
71C22×C3⋊Dic3D411yescharacteristic
72Q8×C3⋊D3—24no—
73#3830—28no—
74#3830—28no—
75#3830—24no—
76#3830—24no—
77#3830—24no—
78#3830D418yes—
79C2×C4×C3⋊D3—24no—
80C2×C4×C3⋊D3—21no—
81C2×C4×C3⋊D3—21no—
82C2×C4×C3⋊D3D411yescharacteristic
83C2×C4×C3⋊D3D411yescharacteristic
84C2×C4×C3⋊D3C2311yescharacteristic
85C2×C4×C3⋊D3C2311yescharacteristic
86C2×#634—21no—
87C2×#634—24no—
88C2×#634—21no—
89C2×#634C2311yescharacteristic
90C2×#634C2312yes—
91C32×SD16—24no—
92C2×C32×C8C2312yes—
93C32×C4⋊C4—21no—
94C32×C4⋊C4C2311yescharacteristic
95C32×C22⋊C4—21no—
96#3757—21no—
97#3756—24no—
98#3754—24no—
99#3752—21no—
100#3752—21no—
101C4⋊(C3⋊Dic3)—21no—
102C4⋊(C3⋊Dic3)C2311yescharacteristic
103#3750—21no—
104#3750—21no—
105#3750—21no—
106C2×C3⋊(C3⋊C8)C2312yes—
107#3746—24no—
108#3743—24no—
109#3007—91noSylow
110C22×C3×D4C2×D314yes—
111C22×#67—64no—
112C23×Dic3—64no—
113C2×D3×Q8—34no—
114C2×#237—38no—
115C2×#237—38no—
116C2×#237—64no—
117C2×#237—64no—
118C2×#237—632no—
119C22×C4×D3—34no—
120C22×Dic6—34no—
121C2×C3×SD16C2×D318yescomplemented
122C3×C22⋊Q8C2×D314yes—
123#946—34no—
124C2×#216—38no—
125C2×#214—38no—
126#924—34no—
127#915—632no—
128C2×Dic12—38no—
129C2×ΓL(1,𝔽25)—38no—
130#897—34no—
131#896—64no—
132D3×C4⋊C4—34no—
133D3×C4⋊C4—64no—
134D3×C22⋊C4—64no—
135#878—64no—
136C3×#102C2×D318yescomplemented
137C3×C22⋊C8C2×D314yes—
138#832—38no—
139#830—34no—
140(C2×D3)⋊C8—38no—
141#816—38no—
142#810—38no—
143C23×C32—24no—
144C23×C32—22no—
145C23×C32—42no—
146C23×C32—42no—
147C23×C32C2×D411yescharacteristic
148C22×C3⋊D3—22no—
149C22×C3⋊D3—22no—
150C22×C3⋊D3—41no—
151C22×C3⋊D3C2×D411yescharacteristic
152C22×C3⋊D3C2×D411yescharacteristic
153C32×Q8—22no—
154C32×D4—24no—
155C32×D4—22no—
156C32×D4C2×D414yes—
157C2×C32×C4—22no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4C2×D411yescharacteristic
161C2×C32×C4C2411yescharacteristic
162#638—28no—
163#638—28no—
164C2×C3⋊Dic3—24no—
165C2×C3⋊Dic3—28no—
166C2×C3⋊Dic3—22no—
167C2×C3⋊Dic3—22no—
168C2×C3⋊Dic3—28no—
169C2×C3⋊Dic3—21no—
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3C2×D411yescharacteristic
172C2×C3⋊Dic3C2×D411yescharacteristic
173C4×C3⋊D3—22no—
174C4×C3⋊D3—22no—
175C4×C3⋊D3—22no—
176C4×C3⋊D3—42no—
177C4×C3⋊D3—42no—
178C4×C3⋊D3C2×D414yes—
179#634—24no—
180#634—22no—
181#634—22no—
182#634—42no—
183#634C2×D414yes—
184C32×C8—22no—
185C3⋊(C3⋊C8)—22no—
186C22×#41—91no—
187C2×#137—92no—
188C2×C22⋊Q8—91no—
189#452—92no—
190#452—92no—
191#451—91no—
192#451—91no—
193#450—91no—
194#450C3⋊D311yescharacteristic, complemented
195#417—92no—
196#407—91no—
197C24×C3—24no—
198C23×D3—34no—
199C2×C3×Q8C22×D314yes—
200C2×C3×D4—216no—
201C2×C3×D4—24no—
202C2×C3×D4C22×D318yes—
203C22×C3×C4C22×D314yes—
204C2×#67—68no—
205C2×#67—68no—
206C2×#67—632no—
207C22×Dic3—34no—
208C22×Dic3—68no—
209C22×Dic3—68no—
210C22×Dic3—632no—
211C22×Dic3—64no—
212D3×Q8—616no—
213#237—332no—
214#237—616no—
215#237—616no—
216#237—632no—
217#237—632no—
218#237—616no—
219C2×C4×D3—34no—
220C2×C4×D3—34no—
221C2×C4×D3—34no—
222C2×C4×D3—34no—
223C2×C4×D3—616no—
224C2×C4×D3—64no—
225C2×C4×D3—64no—
226C2×Dic6—38no—
227C2×Dic6—34no—
228C2×Dic6—616no—
229C2×Dic6—64no—
230C2×Dic6—64no—
231C3×SD16—216no—
232C2×C3×C8C22×D318yes—
233C3×C4⋊C4—24no—
234C3×C4⋊C4C22×D314yes—
235C3×C22⋊C4—24no—
236#217—64no—
237#216—616no—
238#214—616no—
239(C2×D3)⋊C4—64no—
240(C2×D3)⋊C4—64no—
241C4⋊Dic3—34no—
242C4⋊Dic3—64no—
243Dic3⋊C4—64no—
244Dic3⋊C4—64no—
245Dic3⋊C4—64no—
246C2×C3⋊C8—38no—
247Dic12—616no—
248ΓL(1,𝔽25)—616no—
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—24no—
252C22×C32—48no—
253C22×C32—42no—
254C22×C32—42no—
255C22×C32#13712yescomplemented
256C22×C32C22×D411yescharacteristic
257C2×C3⋊D3—21no—
258C2×C3⋊D3—21no—
259C2×C3⋊D3—21no—
260C2×C3⋊D3—21no—
261C2×C3⋊D3—21no—
262C2×C3⋊D3—44no—
263C2×C3⋊D3#12012yes—
264C32×C4—21no—
265C32×C4—41no—
266C32×C4—41no—
267C32×C4C22×D412yes—
268C3⋊Dic3—24no—
269C3⋊Dic3—21no—
270C3⋊Dic3—41no—
271C3⋊Dic3—41no—
272C3⋊Dic3#12012yes—
273C2×#41—92no—
274C2×#41—92no—
275C2×#41—181no—
276C2×#41—181no—
277C2×#41—188no—
278C22×Q8—91no—
279C22×Q8—91no—
280C22×D4—181no—
281C22×D4C2×C3⋊D311yescharacteristic
282C23×C4—91no—
283C23×C4—181no—
284#137—188no—
285C2×Q16—92no—
286C2×Q16—92no—
287C2×SD16—92no—
288C2×SD16C2×C3⋊D312yescomplemented
289C2×M16—92no—
290C22⋊Q8—91no—
291C22⋊Q8—91no—
292C22⋊Q8—91no—
293C22⋊Q8—181no—
294C22⋊Q8—181no—
295C22⋊Q8C2×C3⋊D311yescharacteristic
296C2×C4⋊C4—91no—
297C2×C4⋊C4—181no—
298C2×C22⋊C4—181no—
299#103—92no—
300#103—92no—
301#102—92no—
302#102C2×C3⋊D312yescomplemented
303C22⋊C8—91no—
304C22⋊C8—92no—
305C22⋊C8C2×C3⋊D311yescharacteristic
306C23×C3—216no—
307C23×C3—28no—
308C23×C3—48no—
309C23×C3—48no—
310C23×C3D3×D414yes—
311C22×D3—34no—
312C22×D3—34no—
313C22×D3—68no—
314C22×D3—68no—
315C22×D3—124no—
316C3×Q8—28no—
317C3×D4—216no—
318C3×D4—28no—
319C3×D4D3×D4116yes—
320C2×C3×C4—28no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4D3×D414yes—
324C2×C3×C4C23×D314yes—
325#67—632no—
326#67—632no—
327C2×Dic3—34no—
328C2×Dic3—34no—
329C2×Dic3—68no—
330C2×Dic3—632no—
331C2×Dic3—616no—
332C2×Dic3—632no—
333C2×Dic3—68no—
334C2×Dic3—64no—
335C2×Dic3—64no—
336C4×D3—316no—
337C4×D3—68no—
338C4×D3—68no—
339C4×D3—68no—
340C4×D3—128no—
341C4×D3—128no—
342Dic6—316no—
343Dic6—616no—
344Dic6—68no—
345Dic6—68no—
346Dic6—128no—
347C3×C8—28no—
348C3⋊C8—68no—
349C2×C32—22no—
350C2×C32—44no—
351C2×C32C2×#12011yescharacteristic
352C2×C32C2×#13711yescharacteristic
353C2×C32#57511yescharacteristic
354C3⋊D3—22no—
355C3⋊D3—41no—
356C24—21no—
357C24—91no—
358#41—98no—
359#41—184no—
360#41—184no—
361#41—188no—
362#41—188no—
363#41—184no—
364C2×Q8—92no—
365C2×Q8—91no—
366C2×Q8—181no—
367C2×Q8—184no—
368C2×Q8—184no—
369C2×Q8—181no—
370C2×Q8C22×C3⋊D311yescharacteristic
371C2×D4—24no—
372C2×D4—21no—
373C2×D4—182no—
374C2×D4—182no—
375C2×D4—188no—
376C2×D4C22×C3⋊D312yes—
377C22×C4—91no—
378C22×C4—91no—
379C22×C4—91no—
380C22×C4—91no—
381C22×C4—91no—
382C22×C4—184no—
383C22×C4—181no—
384C22×C4—181no—
385C22×C4—182no—
386C22×C4—182no—
387C22×C4—188no—
388C22×C4—181no—
389C22×C4C22×C3⋊D311yescharacteristic
390Q16—184no—
391Q16—184no—
392SD16—24no—
393SD16—184no—
394M16—184no—
395C2×C8—92no—
396C2×C8C22×C3⋊D312yes—
397C4⋊C4—21no—
398C4⋊C4—91no—
399C4⋊C4—181no—
400C4⋊C4—181no—
401C4⋊C4—181no—
402C4⋊C4—181no—
403C4⋊C4C22×C3⋊D311yescharacteristic
404C22⋊C4—21no—
405C22⋊C4—181no—
406C22⋊C4—181no—
407C22⋊C4—181no—
408C22×C3—24no—
409C22×C3—24no—
410C22×C3—216no—
411C22×C3—432no—
412C22×C3—48no—
413C22×C3—48no—
414C22×C3#91518yescomplemented
415C22×C3C2×D3×D414yes—
416C2×D3—38no—
417C2×D3—64no—
418C2×D3—64no—
419C2×D3—64no—
420C2×D3—64no—
421C2×D3—64no—
422C2×D3—1216no—
423C3×C4—24no—
424C3×C4—44no—
425C3×C4—44no—
426C3×C4C2×D3×D418yes—
427Dic3—38no—
428Dic3—616no—
429Dic3—64no—
430Dic3—124no—
431Dic3—124no—
432C32#300711yescharacteristic, complemented, Sylow
433C23—24no—
434C23—22no—
435C23—42no—
436C23—42no—
437C23—91no—
438C23—91no—
439C23—182no—
440C23—182no—
441C23—361no—
442C23D4×C3⋊D311yescharacteristic
443Q8—22no—
444Q8—94no—
445Q8—184no—
446Q8—182no—
447Q8—182no—
448Q8—362no—
449D4—24no—
450D4—22no—
451D4—188no—
452D4—188no—
453D4D4×C3⋊D314yes—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—91no—
458C2×C4—94no—
459C2×C4—91no—
460C2×C4—182no—
461C2×C4—182no—
462C2×C4—182no—
463C2×C4—188no—
464C2×C4—184no—
465C2×C4—188no—
466C2×C4—182no—
467C2×C4—182no—
468C2×C4—181no—
469C2×C4—181no—
470C2×C4—362no—
471C2×C4—362no—
472C2×C4D4×C3⋊D311yescharacteristic
473C2×C4C23×C3⋊D311yescharacteristic
474C8—22no—
475C8—182no—
476C2×C3—28no—
477C2×C3—416no—
478C2×C3D3×#12014yes—
479C2×C3C2×#91514yes—
480C2×C3#577314yes—
481D3—68no—
482D3—124no—
483C22—21no—
484C22—21no—
485C22—24no—
486C22—48no—
487C22—42no—
488C22—42no—
489C22—92no—
490C22—181no—
491C22—181no—
492C22—181no—
493C22—181no—
494C22—181no—
495C22—364no—
496C22#6412412yescomplemented
497C22C2×D4×C3⋊D311yescharacteristic
498C4—21no—
499C4—41no—
500C4—41no—
501C4—92no—
502C4—181no—
503C4—184no—
504C4—361no—
505C4—361no—
506C4C2×D4×C3⋊D312yes—
507C3#8014714yescomplemented
508C2—22no—
509C2—44no—
510C2—182no—
511C2—361no—
512C2C3⋊D3×#12011yescharacteristic
513C2C2×#6412411yescharacteristic
514C2#1059624411yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#450C2
socleC22×C32C22×D4
prime radicalC22×#3830C2
hypercenter#450C3⋊D3
hypocenterC32#3007
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×#64124C2semidirect
3. C3⋊D3×C22⋊Q8C2semidirect
4. C32×#450C2semidirect
5. #10592917C2semidirect
6. #10592872C2semidirect
7. #10592749C2semidirect
8. #10592482C2semidirect
9. #10592470C2semidirect
10. #10592438C2semidirect
11. #10592425C2semidirect
16. C2×Q8×C3⋊D3C22semidirect
24. C2×C32×SD16C22semidirect
26. #64155C22semidirect
27. C2×#3756C22semidirect
28. C2×#3754C22semidirect
31. C2×#3746C22semidirect
32. C2×#3743C22semidirect
33. #64106C22semidirect
36. C4⋊C4×C3⋊D3C22semidirect
39. C32×#102C22semidirect
41. #63659C22semidirect
43. #63647C22semidirect
44. #63643C22semidirect
45. #63637C22semidirect
49. C3×#450D3semidirect
121. C2×C3×SD16C2×D3semidirect
136. C3×#102C2×D3semidirect
194. #450C3⋊D3semidirect
255. C22×C32#137semidirect
288. C2×SD16C2×C3⋊D3semidirect
302. #102C2×C3⋊D3semidirect
414. C22×C3#915semidirect
432. C32#3007semidirect
496. C22#64124semidirect
507. C3#80147semidirect

Maximal subgroups

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

subgroupindex
C22×#38302
C2×#641242
C3⋊D3×C22⋊Q82
C32×#4502
#105929172
#105928722
#105927492
#105924822
#105924702
#105924382
#105924252
#801473