Finite Group Databasebeta

← #11887177

Subgroups — #11887177

Order 1152.

1224 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596300C211yesmaximal, characteristic
2#10596110C211yesmaximal, characteristic, complemented
3C32×#452C211yesmaximal, characteristic, complemented
4#10592932C211yesmaximal, characteristic, complemented
5#10592887C211yesmaximal, characteristic, complemented
6#10592884C211yesmaximal, characteristic, complemented
7#10592868C211yesmaximal, characteristic, complemented
8SD16×C3⋊Dic3C211yesmaximal, characteristic, complemented
9#10592747C211yesmaximal, characteristic, complemented
10#10592504C211yesmaximal, characteristic, complemented
11#10592499C211yesmaximal, characteristic, complemented
12C3⋊D3×#102C211yesmaximal, characteristic, complemented
13#10592465C211yesmaximal, characteristic, complemented
14#10592436C211yesmaximal, characteristic, complemented
15#10592424C211yesmaximal, characteristic, complemented
16#80246—34nomaximal
17C2×C32×#41C2211yescharacteristic
18C2×D4×C3⋊D3C2211yescharacteristic
19C2×C32×Q16C2211yescharacteristic, complemented
20C2×C32×SD16C2211yescharacteristic
21C32×C22⋊Q8C2211yescharacteristic, complemented
22#64156C2211yescharacteristic
23#64155C2211yescharacteristic
24#64155C2211yescharacteristic
25Q8×C3⋊Dic3C2211yescharacteristic, complemented
26C2×#3755C2211yescharacteristic
27#64148—21no—
28#64146—21no—
29D4×C3⋊Dic3C2211yescharacteristic
30#64139C2211yescharacteristic
31#64138—21no—
32C4×#638C2211yescharacteristic
33C2×#3752—21no—
34#64135C2211yescharacteristic, complemented
35C22×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
36C2×C8×C3⋊D3C2211yescharacteristic
37#64101—21no—
38C4⋊C4×C3⋊D3C2211yescharacteristic
39#64099C2211yescharacteristic, complemented
40#64098—21no—
41#64092—21no—
42#64088—21no—
43C32×#103C2211yescharacteristic, complemented
44C32×#102C2211yescharacteristic
45C32×C22⋊C8C2211yescharacteristic, complemented
46#63662C2211yescharacteristic, complemented
47#63662C2211yescharacteristic, complemented
48#63662C2211yescharacteristic, complemented
49#63659C2211yescharacteristic
50#63659C2211yescharacteristic
51#63648C2211yescharacteristic
52#63648C2211yescharacteristic
53#63647C2211yescharacteristic
54#63645C2211yescharacteristic, complemented
55#63644C2211yescharacteristic, complemented
56#63641C2211yescharacteristic, complemented
57C8×C3⋊Dic3C2211yescharacteristic, complemented
58#63636C2211yescharacteristic
59#63635C2211yescharacteristic, complemented
60#5829—34no—
61#5639—34no—
62C3×#452D314yescomplemented
63#5233—34no—
64#5188—34no—
65#5185—34no—
66#5169—34no—
67Dic3×SD16—34no—
68#5048—34no—
69#4805—34no—
70#4800—34no—
71D3×#102—34no—
72#4766—34no—
73#4737—34no—
74#67⋊C8—34no—
75C23×C3⋊D3—21no—
76C32×#41—22no—
77C32×#41—41no—
78C2×C32×Q8C2311yescharacteristic
79C2×C32×Q8C2311yescharacteristic
80C2×C32×D4—21no—
81C2×C32×D4C2311yescharacteristic
82C22×C32×C4—21no—
83C22×C32×C4C2311yescharacteristic
84C2×#638—21no—
85C2×#638—21no—
86C2×#638D411yescharacteristic
87C22×C3⋊Dic3—21no—
88D4×C3⋊D3—22no—
89D4×C3⋊D3—41no—
90C2×C4⋊(C3⋊D3)C2311yescharacteristic
91C2×C4×C3⋊D3—21no—
92C2×C4×C3⋊D3C2311yescharacteristic
93C32×Q16—41no—
94C32×SD16—22no—
95C2×C32×C8C2311yescharacteristic
96C2×C32×C8C2311yescharacteristic
97C32×C4⋊C4—21no—
98C32×C4⋊C4C2311yescharacteristic
99C32×C22⋊C4—21no—
100#3757—21no—
101#3757—21no—
102#3757D411yescharacteristic
103#3755—41no—
104#3752—21no—
105#3752—21no—
106#3752—21no—
107#3752—21no—
108#3752—21no—
109#3752D411yescharacteristic
110C4⋊(C3⋊Dic3)—21no—
111C4⋊(C3⋊Dic3)—21no—
112C4⋊(C3⋊Dic3)C2311yescharacteristic
113C4⋊(C3⋊Dic3)C2311yescharacteristic
114C4⋊(C3⋊Dic3)C2311yescharacteristic
115#3750—21no—
116#3750—21no—
117#3750—21no—
118#3750—21no—
119#3750D411yescharacteristic
120C4×C3⋊Dic3—21no—
121C4×C3⋊Dic3—21no—
122C4×C3⋊Dic3C2311yescharacteristic
123C2×C3⋊(C3⋊C8)—41no—
124C2×C3⋊(C3⋊C8)C2311yescharacteristic
125C2×C3⋊(C3⋊C8)C2311yescharacteristic
126C8×C3⋊D3—41no—
127#3296—91noSylow
128C2×C3×#41C2×D314yes—
129C2×D3×D4—34no—
130C2×C3×Q16C2×D314yescomplemented
131C2×C3×SD16C2×D314yes—
132C3×C22⋊Q8C2×D314yescomplemented
133#947—34no—
134#946—34no—
135#946—34no—
136Q8×Dic3—34no—
137C2×#215—34no—
138#939—64no—
139#937—64no—
140D4×Dic3—34no—
141#930—34no—
142#929—64no—
143C4×#67—34no—
144C2×(C2×D3)⋊C4—64no—
145#926—34no—
146C22×C3⋊C8—34no—
147C2×D3×C8—34no—
148#892—64no—
149D3×C4⋊C4—34no—
150#890—34no—
151#889—64no—
152#883—64no—
153#879—64no—
154C3×#103C2×D314yescomplemented
155C3×#102C2×D314yes—
156C3×C22⋊C8C2×D314yescomplemented
157#835—34no—
158#835—34no—
159#835—34no—
160#832—34no—
161#832—34no—
162#821—34no—
163#821—34no—
164(C2×D3)⋊C8—34no—
165#818—34no—
166#817—34no—
167Dic3⋊C8—34no—
168C8×Dic3—34no—
169#809—34no—
170#808—34no—
171C23×C32—21no—
172C23×C32C2×D411yescharacteristic
173C22×C3⋊D3—21no—
174C22×C3⋊D3—42no—
175C22×C3⋊D3—42no—
176C22×C3⋊D3—81no—
177C22×C3⋊D3C2×D411yescharacteristic
178C32×Q8—21no—
179C32×Q8—21no—
180C32×Q8—21no—
181C32×Q8—21no—
182C32×D4—21no—
183C32×D4—42no—
184C32×D4#4112yes—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—42no—
191C2×C32×C4C2×D411yescharacteristic
192C2×C32×C4C2411yescharacteristic
193#638—22no—
194#638—42no—
195#638—81no—
196C2×C3⋊Dic3—21no—
197C2×C3⋊Dic3—21no—
198C2×C3⋊Dic3—21no—
199C2×C3⋊Dic3—81no—
200C2×C3⋊Dic3C2×D411yescharacteristic
201C2×C3⋊Dic3C2×D411yescharacteristic
202C2×C3⋊Dic3C2×D411yescharacteristic
203C4⋊(C3⋊D3)—21no—
204C4⋊(C3⋊D3)—21no—
205C4×C3⋊D3—21no—
206C4×C3⋊D3—21no—
207C4×C3⋊D3—42no—
208C32×C8—21no—
209C32×C8—21no—
210C3⋊(C3⋊C8)—21no—
211C3⋊(C3⋊C8)—21no—
212#552—91no—
213#546—91no—
214#488—91no—
215#483—91no—
216#482—91no—
217#478—91no—
218#467—91no—
219#465—91no—
220#464—91no—
221#452C3⋊D311yescharacteristic, complemented
222#450—91no—
223C4×SD16—91no—
224C8×D4—91no—
225#416—91no—
226C2×#102—91no—
227C23×D3—64no—
228C3×#41—28no—
229C3×#41—44no—
230C2×C3×Q8C22×D314yes—
231C2×C3×Q8C22×D314yes—
232C2×C3×D4—24no—
233C2×C3×D4C22×D314yes—
234C22×C3×C4—24no—
235C22×C3×C4C22×D314yes—
236C2×#67—34no—
237C2×#67—64no—
238C2×#67—64no—
239C22×Dic3—64no—
240D3×D4—68no—
241D3×D4—124no—
242C2×D12—34no—
243C2×C4×D3—34no—
244C2×C4×D3—64no—
245C3×Q16—44no—
246C3×SD16—28no—
247C2×C3×C8C22×D314yes—
248C2×C3×C8C22×D314yes—
249C3×C4⋊C4—24no—
250C3×C4⋊C4C22×D314yes—
251C3×C22⋊C4—24no—
252#217—34no—
253#217—64no—
254#217—64no—
255#215—124no—
256(C2×D3)⋊C4—34no—
257(C2×D3)⋊C4—64no—
258(C2×D3)⋊C4—64no—
259(C2×D3)⋊C4—64no—
260(C2×D3)⋊C4—64no—
261(C2×D3)⋊C4—64no—
262C4⋊Dic3—34no—
263C4⋊Dic3—34no—
264C4⋊Dic3—34no—
265C4⋊Dic3—64no—
266C4⋊Dic3—64no—
267Dic3⋊C4—34no—
268Dic3⋊C4—64no—
269Dic3⋊C4—64no—
270Dic3⋊C4—64no—
271Dic3⋊C4—64no—
272C4×Dic3—34no—
273C4×Dic3—64no—
274C4×Dic3—64no—
275C2×C3⋊C8—34no—
276C2×C3⋊C8—34no—
277C2×C3⋊C8—124no—
278D3×C8—124no—
279C22×C32—22no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—41no—
284C22×C32—41no—
285C22×C32C22×D411yescharacteristic
286C2×C3⋊D3—21no—
287C2×C3⋊D3—21no—
288C2×C3⋊D3—41no—
289C2×C3⋊D3—41no—
290C2×C3⋊D3—41no—
291C2×C3⋊D3—84no—
292C2×C3⋊D3#13512yes—
293C32×C4—21no—
294C32×C4—41no—
295C32×C4—41no—
296C32×C4—41no—
297C32×C4C2×#4111yescharacteristic
298C32×C4H(5,𝔽2)11yescharacteristic
299C3⋊Dic3—21no—
300C3⋊Dic3—21no—
301C3⋊Dic3—21no—
302C3⋊Dic3—41no—
303C3⋊Dic3—41no—
304C3⋊Dic3—41no—
305C2×#41C2×C3⋊D311yescharacteristic
306C22×D4—91no—
307C2×Q16C2×C3⋊D311yescharacteristic, complemented
308C2×SD16—91no—
309C2×SD16C2×C3⋊D311yescharacteristic
310C22×C8—91no—
311C22×C8—91no—
312#126—181no—
313#125—91no—
314#125—181no—
315#124—91no—
316#123—181no—
317#123—181no—
318C22⋊Q8—91no—
319C22⋊Q8—91no—
320C22⋊Q8C2×C3⋊D311yescharacteristic, complemented
321#121—91no—
322#121—181no—
323#120—181no—
324C4×Q8—91no—
325C4×D4—91no—
326C4×D4—91no—
327#117—91no—
328#117—181no—
329C2×C4⋊C4—91no—
330C2×C22⋊C4—181no—
331#107—91no—
332#106—91no—
333#106—91no—
334C4⋊C8—91no—
335#103—91no—
336#103—91no—
337#103—91no—
338#103C2×C3⋊D311yescharacteristic, complemented
339#102—91no—
340#102—91no—
341#102—91no—
342#102—91no—
343#102—91no—
344#102C2×C3⋊D311yescharacteristic
345C22⋊C8—91no—
346C22⋊C8C2×C3⋊D311yescharacteristic, complemented
347C4×C8—91no—
348C23×C3—24no—
349C23×C3D3×D414yes—
350C22×D3—34no—
351C22×D3—64no—
352C22×D3—128no—
353C22×D3—128no—
354C22×D3—244no—
355C3×Q8—24no—
356C3×Q8—24no—
357C3×Q8—24no—
358C3×Q8—24no—
359C3×D4—24no—
360C3×D4—48no—
361C3×D4#23718yes—
362C2×C3×C4—24no—
363C2×C3×C4—24no—
364C2×C3×C4—24no—
365C2×C3×C4—24no—
366C2×C3×C4—24no—
367C2×C3×C4—48no—
368C2×C3×C4D3×D414yes—
369C2×C3×C4C23×D314yes—
370#67—68no—
371#67—128no—
372#67—244no—
373C2×Dic3—34no—
374C2×Dic3—34no—
375C2×Dic3—34no—
376C2×Dic3—64no—
377C2×Dic3—64no—
378C2×Dic3—64no—
379C2×Dic3—244no—
380D12—64no—
381D12—64no—
382C4×D3—64no—
383C4×D3—64no—
384C4×D3—128no—
385C3×C8—24no—
386C3×C8—24no—
387C3⋊C8—64no—
388C3⋊C8—64no—
389C2×C32—42no—
390C2×C32—41no—
391C2×C32#54611yescharacteristic
392C2×C32C2×#13511yescharacteristic
393C2×C32#57711yescharacteristic
394C3⋊D3—22no—
395C3⋊D3—81no—
396C24—181no—
397#41—22no—
398#41—41no—
399C2×Q8C22×C3⋊D311yescharacteristic
400C2×Q8C22×C3⋊D311yescharacteristic
401C2×D4—21no—
402C2×D4—91no—
403C2×D4—91no—
404C2×D4—181no—
405C2×D4—182no—
406C2×D4—181no—
407C2×D4—361no—
408C2×D4C22×C3⋊D311yescharacteristic
409C22×C4—21no—
410C22×C4—91no—
411C22×C4—181no—
412C22×C4—181no—
413C22×C4C22×C3⋊D311yescharacteristic
414Q16—41no—
415SD16—22no—
416SD16—361no—
417C2×C8—91no—
418C2×C8—91no—
419C2×C8—361no—
420C2×C8—361no—
421C2×C8C22×C3⋊D311yescharacteristic
422C2×C8C22×C3⋊D311yescharacteristic
423C4⋊C4—21no—
424C4⋊C4—91no—
425C4⋊C4—91no—
426C4⋊C4—91no—
427C4⋊C4—91no—
428C4⋊C4—181no—
429C4⋊C4—181no—
430C4⋊C4—181no—
431C4⋊C4—181no—
432C4⋊C4—181no—
433C4⋊C4—181no—
434C4⋊C4C22×C3⋊D311yescharacteristic
435C22⋊C4—21no—
436C22⋊C4—91no—
437C22⋊C4—91no—
438C22⋊C4—181no—
439C22⋊C4—181no—
440C22⋊C4—181no—
441C22⋊C4—181no—
442C22⋊C4—181no—
443C22⋊C4—181no—
444C22⋊C4—181no—
445C42—91no—
446C42—181no—
447C42—181no—
448C22×C3—24no—
449C22×C3—24no—
450C22×C3—28no—
451C22×C3—24no—
452C22×C3—44no—
453C22×C3—44no—
454C22×C3C2×D3×D414yes—
455C2×D3—38no—
456C2×D3—64no—
457C2×D3—64no—
458C2×D3—124no—
459C2×D3—124no—
460C2×D3—124no—
461C2×D3—2416no—
462C3×C4—24no—
463C3×C4—44no—
464C3×C4—44no—
465C3×C4—44no—
466C3×C4C2×#23714yes—
467C3×C4#100414yes—
468Dic3—64no—
469Dic3—64no—
470Dic3—64no—
471Dic3—124no—
472Dic3—124no—
473Dic3—124no—
474C32#329611yescharacteristic, complemented, Sylow
475C23—21no—
476C23—91no—
477C23—181no—
478C23—362no—
479C23—362no—
480C23—721no—
481C23D4×C3⋊D311yescharacteristic
482Q8—21no—
483Q8—21no—
484Q8—21no—
485Q8—21no—
486D4—21no—
487D4—42no—
488D4—182no—
489D4—181no—
490D4—181no—
491D4—362no—
492D4—721no—
493D4#383012yes—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—42no—
500C2×C4—91no—
501C2×C4—91no—
502C2×C4—91no—
503C2×C4—181no—
504C2×C4—181no—
505C2×C4—181no—
506C2×C4—181no—
507C2×C4—181no—
508C2×C4—362no—
509C2×C4—721no—
510C2×C4D4×C3⋊D311yescharacteristic
511C2×C4C23×C3⋊D311yescharacteristic
512C8—21no—
513C8—21no—
514C8—181no—
515C8—181no—
516C2×C3—48no—
517C2×C3—44no—
518C2×C3#559114yes—
519C2×C3D3×#13514yes—
520C2×C3#577814yes—
521D3—68no—
522D3—244no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—22no—
527C22—41no—
528C22—41no—
529C22—92no—
530C22—181no—
531C22—181no—
532C22—361no—
533C22—361no—
534C22—361no—
535C22—724no—
536C22C2×D4×C3⋊D311yescharacteristic
537C4—21no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—181no—
542C4—181no—
543C4—181no—
544C4—361no—
545C4—361no—
546C4—361no—
547C4C2×#383011yescharacteristic
548C4#6436111yescharacteristic
549C3#8024614yescomplemented
550C2—41no—
551C2—42no—
552C2—182no—
553C2—721no—
554C2#1059606211yescharacteristic
555C2C3⋊D3×#13511yescharacteristic
556C2#1059624911yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#452C2
socleC22×C32C22×D4
prime radical#10596300C2
hypercenter#452C3⋊D3
hypocenterC32#3296
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. #10596110C2semidirect
3. C32×#452C2semidirect
4. #10592932C2semidirect
5. #10592887C2semidirect
6. #10592884C2semidirect
7. #10592868C2semidirect
8. SD16×C3⋊Dic3C2semidirect
9. #10592747C2semidirect
10. #10592504C2semidirect
11. #10592499C2semidirect
12. C3⋊D3×#102C2semidirect
13. #10592465C2semidirect
14. #10592436C2semidirect
15. #10592424C2semidirect
19. C2×C32×Q16C22semidirect
21. C32×C22⋊Q8C22semidirect
25. Q8×C3⋊Dic3C22semidirect
34. #64135C22semidirect
35. C22×C3⋊(C3⋊C8)C22semidirect
39. #64099C22semidirect
43. C32×#103C22semidirect
45. C32×C22⋊C8C22semidirect
46. #63662C22semidirect
47. #63662C22semidirect
48. #63662C22semidirect
54. #63645C22semidirect
55. #63644C22semidirect
56. #63641C22semidirect
57. C8×C3⋊Dic3C22semidirect
59. #63635C22semidirect
62. C3×#452D3semidirect
130. C2×C3×Q16C2×D3semidirect
132. C3×C22⋊Q8C2×D3semidirect
154. C3×#103C2×D3semidirect
156. C3×C22⋊C8C2×D3semidirect
221. #452C3⋊D3semidirect
307. C2×Q16C2×C3⋊D3semidirect
320. C22⋊Q8C2×C3⋊D3semidirect
338. #103C2×C3⋊D3semidirect
346. C22⋊C8C2×C3⋊D3semidirect
474. C32#3296semidirect
549. C3#80246semidirect

Maximal subgroups

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

subgroupindex
#105963002
#105961102
C32×#4522
#105929322
#105928872
#105928842
#105928682
SD16×C3⋊Dic32
#105927472
#105925042
#105924992
C3⋊D3×#1022
#105924652
#105924362
#105924242
#802463