Finite Group Databasebeta

← #72228295

Subgroups — #72228295

Order 2304.

4560 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 4560 conjugacy classes of subgroups — too many to draw. Only its 177 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
1#11923673C211yesmaximal, characteristic, complemented
2(2−1+4⋊C3)⋊A4C213yesmaximal, complemented
3C2×(#41⋊C3)⋊A4C213yesmaximal, complemented
4#11710937C312yesmaximal, complemented
5#11710936C312yesmaximal, complemented
6C3×#5945—42nomaximal
7(#41⋊C3)⋊A4C2211yescharacteristic, complemented
8(#41⋊C3)⋊A4C2213yescomplemented
9C2×Ω+(4,𝔽3)C2213yescomplemented
10#87548C2×C312yescomplemented
11#87547C2×C316yescomplemented
12C2×#5950C2×C316yescomplemented
13#87537—36no—
14#87537C2×C312yescomplemented
15#87536—32no—
16#87536C2×C316yescomplemented
17C2×#5945—32no—
18C2×#5945—36no—
19C2×#5945C2×C316yescomplemented
20C3×2−1+4⋊C3—46no—
21C3×#994—42no—
22C2×C3×#41⋊C3—46no—
23Ω+(4,𝔽3)C2311yescharacteristic, complemented
24#63103C3211yescharacteristic, complemented, Sylow
25C3×#585—42no—
26#5950C22×C312yescomplemented
27#5950C22×C316yescomplemented
28C2×#997C22×C316yescomplemented
29#5945—312no—
30#5945—36no—
31#5945—312no—
32#5945—32no—
33#5945—36no—
34#5945—36no—
35#5945—36no—
36#5945—44no—
37#5945A412yescomplemented
38#5945C22×C312yescomplemented
39#5945C22×C316yescomplemented
40C2×2−1+4⋊C3—312no—
41C2×2−1+4⋊C3—36no—
42C2×2−1+4⋊C3—36no—
43C2×2−1+4⋊C3—36no—
44C2×#994—32no—
45C2×#994—36no—
46C2×#994C22×C316yescomplemented
47A4×#41—122no—
48C22×#41⋊C3—34no—
49C22×#41⋊C3—36no—
50C32×#41—161no—
51C3×#41⋊C3—42no—
52C3×#41⋊C3—46no—
53C2×C3×SL(2,𝔽3)—46no—
542−1+6—36no—
552−1+6—91no—
562−1+6C2×C3211yescharacteristic, complemented
57H(7,𝔽2)—32no—
58H(7,𝔽2)—93no—
59H(7,𝔽2)C2×C3213yescomplemented
60C2×#585—36no—
61C2×#585—32no—
62C2×#585—91no—
63C2×#585—93no—
64C2×#585C2×C3213yescomplemented
65C3×2−1+4—46no—
66C3×2−1+4—122no—
67C3×H(5,𝔽2)—42no—
68C3×H(5,𝔽2)—126no—
69C2×C3×#41—46no—
70C2×C3×#41—122no—
71C2×C3×#41—126no—
72#997C23×C312yes—
732−1+4⋊C3—312no—
742−1+4⋊C3—36no—
752−1+4⋊C3—312no—
762−1+4⋊C3—36no—
772−1+4⋊C3—32no—
782−1+4⋊C3—412no—
792−1+4⋊C3C2×A416yescomplemented
80#994—36no—
81#994—36no—
82#994—44no—
83#994C2×A412yescomplemented
84#994C23×C312yes—
85C2×#41⋊C3—312no—
86C2×#41⋊C3—36no—
87C2×#41⋊C3—32no—
88C2×#41⋊C3—36no—
89C2×#41⋊C3—36no—
90C2×#41⋊C3—36no—
91C2×#41⋊C3—412no—
92C2×#41⋊C3C2×A416yescomplemented
93Q8×A4—122no—
94C22×SL(2,𝔽3)—34no—
95C22×SL(2,𝔽3)—36no—
96D4×A4—126no—
97C2×C4×A4—126no—
98C32×Q8—161no—
99C32×D4—163no—
100C2×C32×C4—163no—
101C3×SL(2,𝔽3)—42no—
102#585—312no—
103#585—312no—
104#585—36no—
105#585—32no—
106#585—36no—
107#585—36no—
108#585—36no—
109#585—96no—
110#585—92no—
111#585—94no—
112#585—94no—
113#585—912no—
114#585—96no—
115#585—912no—
116#585—924no—
117#585—96no—
118#585—92no—
119#585—912no—
120#585—912no—
121#585—912no—
122#585—93no—
123#585—96no—
124#585—93no—
125#585—93no—
126#585—93no—
127#585C3×A412yes—
128#585C22×C3211yescharacteristic
129#585C22×C3213yes—
130C2×2−1+4—312no—
131C2×2−1+4—36no—
132C2×2−1+4—36no—
133C2×2−1+4—36no—
134C2×2−1+4—91no—
135C2×2−1+4—92no—
136C2×2−1+4—93no—
137C2×2−1+4—96no—
138C2×2−1+4—912no—
139C2×2−1+4—92no—
140C2×2−1+4—96no—
141C2×H(5,𝔽2)—36no—
142C2×H(5,𝔽2)—32no—
143C2×H(5,𝔽2)—93no—
144C2×H(5,𝔽2)—96no—
145C2×H(5,𝔽2)—91no—
146C2×H(5,𝔽2)—912no—
147C2×H(5,𝔽2)—912no—
148C2×H(5,𝔽2)—96no—
149C2×H(5,𝔽2)—96no—
150C2×H(5,𝔽2)—912no—
151C2×H(5,𝔽2)—96no—
152C2×H(5,𝔽2)—93no—
153C2×H(5,𝔽2)C22×C3213yes—
154C22×#41—32no—
155C22×#41—34no—
156C22×#41—36no—
157C22×#41—93no—
158C22×#41—96no—
159C22×#41—91no—
160C22×#41—96no—
161C22×#41—93no—
162C22×#41—912no—
163C22×A4—126no—
164C3×#41—42no—
165C3×#41—42no—
166C3×#41—46no—
167C3×#41—126no—
168C3×#41—126no—
169C3×#41—122no—
170C3×#41—1212no—
171C3×#41—1212no—
172C3×#41—126no—
173C3×#41—126no—
174C3×#41—162no—
175C2×C3×Q8—46no—
176C2×C3×Q8—122no—
177C2×C3×Q8—126no—
178C2×C3×D4—126no—
179C2×C3×D4—126no—
180C2×C3×D4—1212no—
181C2×C3×D4—126no—
182C22×C3×C4—126no—
183C22×C3×C4—124no—
184#41⋊C3—32no—
185#41⋊C3—36no—
186#41⋊C3—44no—
187#41⋊C3—412no—
188#41⋊C3C22×A412yes—
189#41⋊C3C22×A416yes—
190C2×SL(2,𝔽3)—36no—
191C2×SL(2,𝔽3)—32no—
192C2×SL(2,𝔽3)—412no—
193C2×SL(2,𝔽3)C22×A416yes—
194C4×A4—122no—
195C4×A4—126no—
196C22×C32—163no—
197C32×C4—161no—
198C32×C4—163no—
1992−1+4—32no—
2002−1+4—312no—
2012−1+4—36no—
2022−1+4—312no—
2032−1+4—36no—
2042−1+4—32no—
2052−1+4—96no—
2062−1+4—94no—
2072−1+4—96no—
2082−1+4—96no—
2092−1+4—912no—
2102−1+4—912no—
2112−1+4—96no—
2122−1+4—912no—
2132−1+4—924no—
2142−1+4—924no—
2152−1+4—912no—
2162−1+4—924no—
2172−1+4—924no—
2182−1+4—96no—
2192−1+4—92no—
2202−1+4—96no—
2212−1+4—96no—
2222−1+4—912no—
2232−1+4—96no—
2242−1+4C2×C3×A416yes—
225H(5,𝔽2)—36no—
226H(5,𝔽2)—36no—
227H(5,𝔽2)—36no—
228H(5,𝔽2)—96no—
229H(5,𝔽2)—924no—
230H(5,𝔽2)—912no—
231H(5,𝔽2)—912no—
232H(5,𝔽2)—96no—
233H(5,𝔽2)—92no—
234H(5,𝔽2)—912no—
235H(5,𝔽2)—912no—
236H(5,𝔽2)—924no—
237H(5,𝔽2)—912no—
238H(5,𝔽2)—924no—
239H(5,𝔽2)—948no—
240H(5,𝔽2)—912no—
241H(5,𝔽2)—924no—
242H(5,𝔽2)—94no—
243H(5,𝔽2)—96no—
244H(5,𝔽2)—96no—
245H(5,𝔽2)—912no—
246H(5,𝔽2)—912no—
247H(5,𝔽2)—924no—
248H(5,𝔽2)—912no—
249H(5,𝔽2)—912no—
250H(5,𝔽2)—912no—
251H(5,𝔽2)—924no—
252H(5,𝔽2)—93no—
253H(5,𝔽2)—96no—
254H(5,𝔽2)—93no—
255H(5,𝔽2)—91no—
256H(5,𝔽2)C2×C3×A412yes—
257H(5,𝔽2)C23×C3211yescharacteristic
258C2×#41—36no—
259C2×#41—32no—
260C2×#41—312no—
261C2×#41—32no—
262C2×#41—36no—
263C2×#41—36no—
264C2×#41—36no—
265C2×#41—36no—
266C2×#41—91no—
267C2×#41—93no—
268C2×#41—96no—
269C2×#41—96no—
270C2×#41—96no—
271C2×#41—912no—
272C2×#41—96no—
273C2×#41—96no—
274C2×#41—96no—
275C2×#41—96no—
276C2×#41—96no—
277C2×#41—96no—
278C2×#41—912no—
279C2×#41—92no—
280C2×#41—96no—
281C2×#41—93no—
282C2×#41—91no—
283C2×#41—93no—
284C2×#41—912no—
285C2×#41—912no—
286C2×#41—912no—
287C2×#41—912no—
288C2×#41—912no—
289C2×#41—93no—
290C2×#41—96no—
291C2×#41—93no—
292C2×#41—96no—
293C2×#41—96no—
294C2×#41—92no—
295C2×#41—912no—
296C2×#41—912no—
297C2×#41—912no—
298C2×#41—924no—
299C2×#41—924no—
300C2×#41—924no—
301C2×#41—96no—
302C2×#41—924no—
303C2×#41—948no—
304C2×#41—912no—
305C2×#41—924no—
306C2×#41—924no—
307C2×#41—91no—
308C2×#41—912no—
309C2×#41—912no—
310C2×#41—912no—
311C2×#41—912no—
312C2×#41—912no—
313C2×#41—96no—
314C2×#41—96no—
315C2×#41—96no—
316C2×#41—96no—
317C2×#41—96no—
318C2×#41—912no—
319C2×#41—912no—
320C2×#41C2×C3×A416yes—
321C22×Q8—32no—
322C22×Q8—34no—
323C22×Q8—36no—
324C22×Q8—91no—
325C22×Q8—96no—
326C22×Q8—93no—
327C22×Q8—93no—
328C22×Q8—912no—
329C22×Q8—96no—
330C22×D4—36no—
331C22×D4—93no—
332C22×D4—96no—
333C22×D4—912no—
334C22×D4—96no—
335C22×D4—912no—
336C22×D4—96no—
337C22×D4—94no—
338C22×D4—924no—
339C22×D4—96no—
340C22×D4—96no—
341C22×D4—912no—
342C22×D4—93no—
343C22×D4—93no—
344C23×C4—36no—
345C23×C4—93no—
346C23×C4—94no—
347C23×C4—96no—
348C23×C3—126no—
349C23×C3—124no—
350C2×A4—122no—
351C2×A4—246no—
352C3×Q8—42no—
353C3×Q8—42no—
354C3×Q8—126no—
355C3×Q8—126no—
356C3×Q8—162no—
357C3×D4—46no—
358C3×D4—126no—
359C3×D4—1212no—
360C3×D4—122no—
361C3×D4—126no—
362C3×D4—1212no—
363C3×D4—166no—
364C2×C3×C4—46no—
365C2×C3×C4—122no—
366C2×C3×C4—126no—
367C2×C3×C4—126no—
368C2×C3×C4—126no—
369C2×C3×C4—126no—
370C2×C3×C4—1212no—
371C2×C3×C4—166no—
372SL(2,𝔽3)—44no—
373SL(2,𝔽3)C23×A412yes—
374C2×C32—161no—
375C2×C32—323no—
376C24—36no—
377C24—93no—
378C24—94no—
379C24—96no—
380#41—312no—
381#41—32no—
382#41—36no—
383#41—312no—
384#41—36no—
385#41—36no—
386#41—36no—
387#41—32no—
388#41—36no—
389#41—96no—
390#41—912no—
391#41—96no—
392#41—96no—
393#41—912no—
394#41—924no—
395#41—924no—
396#41—912no—
397#41—912no—
398#41—912no—
399#41—92no—
400#41—96no—
401#41—96no—
402#41—96no—
403#41—96no—
404#41—96no—
405#41—912no—
406#41—912no—
407#41—924no—
408#41—924no—
409#41—912no—
410#41—912no—
411#41—912no—
412#41—94no—
413#41—924no—
414#41—924no—
415#41—924no—
416#41—924no—
417#41—948no—
418#41—924no—
419#41—912no—
420#41—912no—
421#41—91no—
422#41—93no—
423#41—912no—
424#41—912no—
425#41—912no—
426#41—94no—
427#41—912no—
428#41—93no—
429#41—92no—
430#41—96no—
431#41—96no—
432#41—96no—
433#41—96no—
434#41—96no—
435#41—924no—
436#41A4211yescharacteristic
437#41C22×C3×A412yes—
438#41C22×C3×A416yes—
439C2×Q8—36no—
440C2×Q8—32no—
441C2×Q8—36no—
442C2×Q8—32no—
443C2×Q8—96no—
444C2×Q8—96no—
445C2×Q8—912no—
446C2×Q8—96no—
447C2×Q8—93no—
448C2×Q8—912no—
449C2×Q8—912no—
450C2×Q8—91no—
451C2×Q8—93no—
452C2×Q8—96no—
453C2×Q8—912no—
454C2×Q8—96no—
455C2×Q8—924no—
456C2×Q8—91no—
457C2×Q8—96no—
458C2×Q8—96no—
459C2×Q8—912no—
460C2×Q8C22×C3×A416yes—
461C2×D4—36no—
462C2×D4—312no—
463C2×D4—36no—
464C2×D4—36no—
465C2×D4—96no—
466C2×D4—96no—
467C2×D4—912no—
468C2×D4—96no—
469C2×D4—96no—
470C2×D4—93no—
471C2×D4—912no—
472C2×D4—92no—
473C2×D4—96no—
474C2×D4—96no—
475C2×D4—96no—
476C2×D4—912no—
477C2×D4—912no—
478C2×D4—924no—
479C2×D4—924no—
480C2×D4—912no—
481C2×D4—912no—
482C2×D4—96no—
483C2×D4—92no—
484C2×D4—96no—
485C2×D4—912no—
486C2×D4—912no—
487C2×D4—912no—
488C2×D4—912no—
489C2×D4—912no—
490C2×D4—948no—
491C2×D4—924no—
492C2×D4—924no—
493C2×D4—924no—
494C2×D4—96no—
495C2×D4—96no—
496C2×D4—93no—
497C2×D4—96no—
498C2×D4—912no—
499C2×D4—912no—
500C2×D4—91no—
501C2×D4—93no—
502C22×C4—34no—
503C22×C4—36no—
504C22×C4—32no—
505C22×C4—36no—
506C22×C4—91no—
507C22×C4—96no—
508C22×C4—93no—
509C22×C4—93no—
510C22×C4—93no—
511C22×C4—96no—
512C22×C4—93no—
513C22×C4—96no—
514C22×C4—912no—
515C22×C4—912no—
516C22×C4—912no—
517C22×C4—94no—
518C22×C4—912no—
519C22×C4—924no—
520C22×C4—96no—
521C22×C4—93no—
522C22×C4—96no—
523C22×C4—96no—
524C22×C4—96no—
525C22×C3—46no—
526C22×C3—122no—
527C22×C3—126no—
528C22×C3—166no—
529C22×C3—486no—
530C22×C3—484no—
531A4—122no—
532C3×C4—46no—
533C3×C4—42no—
534C3×C4—122no—
535C3×C4—126no—
536C3×C4—166no—
537C3×C4—162no—
538C32—161noSylow
539C23—34no—
540C23—36no—
541C23—32no—
542C23—91no—
543C23—93no—
544C23—96no—
545C23—912no—
546C23—96no—
547C23—93no—
548C23—246no—
549C23—723no—
550C23—724no—
551C23—726no—
552Q8—36no—
553Q8—36no—
554Q8—96no—
555Q8—96no—
556Q8—92no—
557Q8—912no—
558Q8—94no—
559Q8—93no—
560Q8C2×A4211yescharacteristic
561Q8C23×C3×A412yes—
562D4—32no—
563D4—36no—
564D4—312no—
565D4—312no—
566D4—36no—
567D4—912no—
568D4—92no—
569D4—96no—
570D4—912no—
571D4—96no—
572D4—912no—
573D4—94no—
574D4—912no—
575D4—912no—
576D4—912no—
577D4—96no—
578D4—93no—
579D4C2×A4213yes—
580C2×C4—312no—
581C2×C4—36no—
582C2×C4—36no—
583C2×C4—36no—
584C2×C4—32no—
585C2×C4—36no—
586C2×C4—91no—
587C2×C4—96no—
588C2×C4—93no—
589C2×C4—92no—
590C2×C4—96no—
591C2×C4—96no—
592C2×C4—96no—
593C2×C4—924no—
594C2×C4—912no—
595C2×C4—912no—
596C2×C4—93no—
597C2×C4—93no—
598C2×C4—93no—
599C2×C4—91no—
600C2×C4—93no—
601C2×C4—92no—
602C2×C4—96no—
603C2×C4C2×A4213yes—
604C2×C3—42no—
605C2×C3—86no—
606C2×C3—162no—
607C2×C3—242no—
608C2×C3—246no—
609C2×C3—326no—
610C22—36no—
611C22—32no—
612C22—91no—
613C22—93no—
614C22—126no—
615C22—124no—
616C22—122no—
617C22—363no—
618C22—361no—
619C22—363no—
620C22—366no—
621C22—3612no—
622C22—366no—
623C22C22×A4213yes—
624C4—32no—
625C4—36no—
626C4—91no—
627C4—93no—
628C4C22×A4213yes—
629C4C22×A4211yescharacteristic
630C3—42no—
631C3—162no—
632C2—23no—
633C2—62no—
634C2—66no—
635C2—181no—
636C2—183no—
637C2C23×A4211yescharacteristic
subgroupG/H
centerC4C22×A42
commutator, cosocle, hypocenterH(5,𝔽2)C23×C32
Frattini, socleC2C23×A42
Fitting#63103C32
hypercenter#41A42
LCSG ▷ H(5,𝔽2)
UCSC1 ◁ C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #11923673C2semidirect
2. (2−1+4⋊C3)⋊A4C2semidirect
3. C2×(#41⋊C3)⋊A4C2semidirect
4. #11710937C3semidirect
5. #11710936C3semidirect
7. (#41⋊C3)⋊A4C22semidirect
8. (#41⋊C3)⋊A4C22semidirect
9. C2×Ω+(4,𝔽3)C22semidirect
10. #87548C2×C3semidirect
11. #87547C2×C3semidirect
12. C2×#5950C2×C3semidirect
14. #87537C2×C3semidirect
16. #87536C2×C3semidirect
19. C2×#5945C2×C3semidirect
23. Ω+(4,𝔽3)C23semidirect
24. #63103C32semidirect
26. #5950C22×C3semidirect
27. #5950C22×C3semidirect
28. C2×#997C22×C3semidirect
37. #5945A4semidirect
38. #5945C22×C3semidirect
39. #5945C22×C3semidirect
46. C2×#994C22×C3semidirect
56. 2−1+6C2×C32semidirect
59. H(7,𝔽2)C2×C32semidirect
64. C2×#585C2×C32semidirect
79. 2−1+4⋊C3C2×A4semidirect
83. #994C2×A4semidirect
92. C2×#41⋊C3C2×A4semidirect

Maximal subgroups

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

subgroupindex
#119236732
(2−1+4⋊C3)⋊A42
C2×(#41⋊C3)⋊A42
#117109373
#117109363
C3×#59454