Finite Group Databasebeta

← #10877182

Subgroups — #10877182

Order 768.

704 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83756C211yesmaximal, characteristic
2C2×#5121C211yesmaximal, characteristic, complemented
3#78623C211yesmaximal, characteristic
4#78622C211yesmaximal, characteristic
5C3×#1811C211yesmaximal, characteristic, complemented
6#70721C211yesmaximal, characteristic, complemented
7#70720C211yesmaximal, characteristic, complemented
8#70701C211yesmaximal, characteristic, complemented
9#70700C211yesmaximal, characteristic, complemented
10#69974C211yesmaximal, characteristic, complemented
11#69973C211yesmaximal, characteristic, complemented
12#68924C211yesmaximal, characteristic, complemented
13#68923C211yesmaximal, characteristic, complemented
14#68922C211yesmaximal, characteristic, complemented
15#68921C211yesmaximal, characteristic, complemented
16#27676—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18#5791C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C2×#901C2211yescharacteristic
22C2×C3×#108C2211yescharacteristic
23C2×C3×#108C2211yescharacteristic
24C3×#407C2211yescharacteristic
25C3×#407C2211yescharacteristic
26#5207C2211yescharacteristic
27#5133C2211yescharacteristic
28#5133C2211yescharacteristic
29#5133C2211yescharacteristic
30#5133C2211yescharacteristic
31#5128—21no—
32#5128—21no—
33#5127—21no—
34#5127—21no—
35#5121—21no—
36#5121—21no—
37#5121—21no—
38#5121—21no—
39#5121C2211yescharacteristic
40#5121C2211yescharacteristic
41#5121C2211yescharacteristic
42#5121C2211yescharacteristic
43#5110C2211yescharacteristic
44C2×#818C2211yescharacteristic
45C2×#818C2211yescharacteristic
46C2×#817C2211yescharacteristic
47C2×#817C2211yescharacteristic
48#5102C2211yescharacteristic
49#5101C2211yescharacteristic
50#4737—21no—
51#4737—21no—
52#4736—21no—
53#4735—21no—
54#4723—21no—
55#4723—21no—
56#4722—21no—
57#4721—21no—
58C3×#341C2211yescharacteristic
59C3×#341C2211yescharacteristic
60#4554C2211yescharacteristic
61#4554C2211yescharacteristic
62#4554C2211yescharacteristic
63#4554C2211yescharacteristic
64#4554C2211yescharacteristic
65#4554C2211yescharacteristic
66#4554C2211yescharacteristic
67#4554C2211yescharacteristic
68#3093—31no—
69#3092—31no—
70#2930—31no—
71C2×#428—31no—
72#2090—31no—
73#2089—31no—
74#2079—31no—
75#2078—31no—
76#1952—31no—
77#1951—31no—
78#1822—31no—
79#1821—31no—
80#1816—31no—
81#1815—31no—
82#1811D311yescharacteristic, complemented
83C23×C3×C4C2311yescharacteristic
84C23×Dic3—21no—
85C2×#235C2311yescharacteristic
86C2×C3×M16—22no—
87C2×C3×M16—22no—
88C2×C3×M16D411yescharacteristic
89C2×C3×M16D411yescharacteristic
90C2×C3×M16D411yescharacteristic
91C2×C3×M16D411yescharacteristic
92C2×C3×M16C2311yescharacteristic
93C2×C3×M16C2311yescharacteristic
94C2×C3×M16C2311yescharacteristic
95C2×C3×M16C2311yescharacteristic
96C22×C3×C8C2311yescharacteristic
97C22×C3×C8C2311yescharacteristic
98#953—21no—
99C2×#217—21no—
100#930D411yescharacteristic
101#930D411yescharacteristic
102#929—21no—
103C4×#67—21no—
104C4×#67—21no—
105#926—21no—
106#926—21no—
107#926—21no—
108#926—21no—
109#926C2311yescharacteristic
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119#924D411yescharacteristic
120#924D411yescharacteristic
121C2×C4×Dic3—21no—
122C2×#208C2311yescharacteristic
123C2×#208C2311yescharacteristic
124#901—22no—
125#901—22no—
126C2×ΓL(1,𝔽25)—21no—
127C2×D3×C8—21no—
128C3×#108—22no—
129C3×#108—22no—
130C3×C22⋊C8—21no—
131C3×C22⋊C8—21no—
132C3×C22⋊C8—21no—
133C3×C22⋊C8—21no—
134#830—21no—
135#830—21no—
136#821—21no—
137#821—21no—
138#821—21no—
139#821—21no—
140(C2×D3)⋊C8—21no—
141(C2×D3)⋊C8—21no—
142#818—21no—
143#818—21no—
144#818—21no—
145#818—21no—
146#818—21no—
147#818—21no—
148#818—21no—
149#818—21no—
150#817—21no—
151#817—21no—
152#817—21no—
153#817—21no—
154#817—21no—
155#817—21no—
156#817—21no—
157#817—21no—
158#816—21no—
159#816—21no—
160#816—21no—
161#816—21no—
162#815—21no—
163Dic3⋊C8—21no—
164Dic3⋊C8—21no—
165C8×Dic3—21no—
166C2×#131—31no—
167C22×M16C2×D311yescharacteristic
168#525—31no—
169C2×#117—31no—
170C22×C4⋊C4—31no—
171#485—61no—
172#485—61no—
173#484—61no—
174#483—61no—
175#482—61no—
176#482—61no—
177#481—61no—
178#480—61no—
179#436—61no—
180#436—61no—
181#435—61no—
182#435—61no—
183C2×#108C2×D311yescharacteristic
184C2×#108C2×D311yescharacteristic
185#428—31no—
186#428—31no—
187#428—31no—
188#428—31no—
189#428—61no—
190#428—61no—
191#428—61no—
192#428—61no—
193C2×#107—31no—
194C2×#107—31no—
195C2×#106—31no—
196C2×#106—31no—
197#424—31no—
198#417—31no—
199#417—31no—
200#417—31no—
201#417—31no—
202#408—31no—
203#407—31no—
204#407C2×D311yescharacteristic
205#407C2×D311yescharacteristic
206#405—31no—
207#343—31no—
208#343—31no—
209#343—31no—
210#343—31no—
211#343—31no—
212#343—31no—
213#343—31no—
214#343—31no—
215#341C2×D311yescharacteristic
216#341C2×D311yescharacteristic
217C24×C3#4111yescharacteristic
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4#4111yescharacteristic
227C22×C3×C4C2411yescharacteristic
228C2×#67—21no—
229C2×#67—21no—
230C2×#67—21no—
231C22×Dic3—21no—
232C22×Dic3—21no—
233C22×Dic3—21no—
234C22×Dic3—21no—
235C22×Dic3—41no—
236C22×Dic3—41no—
237C22×Dic3—82no—
238#235—21no—
239#235—21no—
240#235—21no—
241#235—21no—
242C2×D12C2×D411yescharacteristic
243C2×C4×D3—21no—
244C2×Dic6C2×D411yescharacteristic
245C3×M16—22no—
246C3×M16—22no—
247C3×M16—22no—
248C3×M16—22no—
249C3×M16—21no—
250C3×M16—21no—
251C3×M16—21no—
252C3×M16—21no—
253C3×M16—21no—
254C3×M16—21no—
255C3×M16—21no—
256C3×M16—21no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C2×C3×C8—21no—
263C2×C3×C8C2×D411yescharacteristic
264C2×C3×C8C2×D411yescharacteristic
265C2×C3×C8C2×D411yescharacteristic
266C2×C3×C8C2×D411yescharacteristic
267C2×C3×C8#4112yes—
268C2×C3×C8#4112yes—
269#217—21no—
270#217—21no—
271#217—21no—
272#217—41no—
273#217—41no—
274(C2×D3)⋊C4—21no—
275(C2×D3)⋊C4—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282C4⋊Dic3—21no—
283C4⋊Dic3—21no—
284C4⋊Dic3—41no—
285C4⋊Dic3—41no—
286C4⋊Dic3C2×D411yescharacteristic
287C4⋊Dic3C2×D411yescharacteristic
288Dic3⋊C4—21no—
289Dic3⋊C4—21no—
290C4×Dic3—21no—
291C4×Dic3—41no—
292C4×Dic3—41no—
293#208—21no—
294#208—21no—
295#208—21no—
296#208—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299ΓL(1,𝔽25)—42no—
300D3×C8—42no—
301C2×#41—31no—
302C23×C4—61no—
303C23×C4C22×D311yescharacteristic
304#131—62no—
305#131—62no—
306C2×M16—22no—
307C2×M16—22no—
308C2×M16—31no—
309C2×M16—31no—
310C2×M16—61no—
311C2×M16D1211yescharacteristic
312C2×M16D1211yescharacteristic
313C2×M16D1211yescharacteristic
314C2×M16D1211yescharacteristic
315C2×M16C22×D311yescharacteristic
316C2×M16C22×D311yescharacteristic
317C2×M16C22×D311yescharacteristic
318C2×M16C22×D311yescharacteristic
319C22×C8—61no—
320C22×C8C22×D311yescharacteristic
321C22×C8C22×D311yescharacteristic
322#123—61no—
323C22⋊Q8—31no—
324C22⋊Q8—31no—
325#121—31no—
326#121—31no—
327#120—61no—
328C4×D4—61no—
329C4×D4—61no—
330#117—31no—
331#117—61no—
332#117—61no—
333#117—61no—
334#117—61no—
335C2×C4⋊C4—31no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C42—61no—
346#108—22no—
347#108—22no—
348#107—61no—
349#107—61no—
350#107—61no—
351#107—61no—
352#107—61no—
353#107—61no—
354#107—61no—
355#107—61no—
356#106—61no—
357#106—61no—
358#106—61no—
359#106—61no—
360#106—61no—
361#106—61no—
362#106—61no—
363#106—61no—
364C4⋊C8—61no—
365C4⋊C8—61no—
366#103—61no—
367#103—61no—
368#103—61no—
369#103—61no—
370#102—61no—
371#102—61no—
372#102—61no—
373#102—61no—
374C22⋊C8—21no—
375C22⋊C8—21no—
376C22⋊C8—21no—
377C22⋊C8—21no—
378C22⋊C8—61no—
379C22⋊C8—61no—
380C22⋊C8—61no—
381C22⋊C8—61no—
382#97—61no—
383C4×C8—61no—
384C23×C3—21no—
385C23×C3—21no—
386C23×C3—41no—
387C23×C3—41no—
388C23×C3#12111yescharacteristic
389C23×C3#12111yescharacteristic
390C23×C3C2×#4111yescharacteristic
391C22×D3—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4#12111yescharacteristic
403C2×C3×C4#12111yescharacteristic
404C2×C3×C4C22×D411yescharacteristic
405C2×C3×C4C22×D411yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C42−1+411yescharacteristic
409C2×C3×C42−1+411yescharacteristic
410#67—41no—
411#67—41no—
412#67—81no—
413#67—81no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—82no—
422C2×Dic3—82no—
423C2×Dic3—81no—
424C2×Dic3—81no—
425C2×Dic3—81no—
426C2×Dic3—81no—
427D12—21no—
428D12—21no—
429C4×D3—41no—
430C4×D3—41no—
431Dic6—21no—
432Dic6—21no—
433C3×C8—21no—
434C3×C8—21no—
435C3×C8—41no—
436C3×C8—41no—
437C3×C8—41no—
438C3×C8—41no—
439C3×C8#12312yes—
440C3×C8#12312yes—
441C3⋊C8—41no—
442C3⋊C8—41no—
443C24#23911yescharacteristic
444#41—61no—
445#41—61no—
446#41—61no—
447#41—61no—
448C2×Q8—31no—
449C2×D4—31no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—121no—
463C22×C4—121no—
464C22×C4—242no—
465C22×C4C2×D1211yescharacteristic
466C22×C4C2×D1211yescharacteristic
467C22×C4D3×D411yescharacteristic
468C22×C4D3×D411yescharacteristic
469C22×C4#23911yescharacteristic
470C22×C4C23×D311yescharacteristic
471M16—22no—
472M16—22no—
473M16—22no—
474M16—22no—
475M16—21no—
476M16—21no—
477M16—21no—
478M16—21no—
479M16—21no—
480M16—21no—
481M16—21no—
482M16—21no—
483M16—61no—
484M16—61no—
485M16—61no—
486M16—61no—
487M16—122no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—122no—
497C2×C8C2×D1211yescharacteristic
498C2×C8C2×D1211yescharacteristic
499C2×C8C2×D1211yescharacteristic
500C2×C8C2×D1211yescharacteristic
501C2×C8#23712yes—
502C2×C8#23712yes—
503C4⋊C4—31no—
504C4⋊C4—31no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—121no—
516C4⋊C4—121no—
517C22⋊C4—61no—
518C22⋊C4—61no—
519C22⋊C4—61no—
520C22⋊C4—61no—
521C22⋊C4—61no—
522C22⋊C4—121no—
523C22⋊C4—121no—
524C42—61no—
525C42—121no—
526C42—121no—
527C22×C3—21no—
528C22×C3—21no—
529C22×C3—21no—
530C22×C3—21no—
531C22×C3—21no—
532C22×C3—21no—
533C22×C3—21no—
534C22×C3—21no—
535C22×C3—42no—
536C22×C3—42no—
537C22×C3C2×#12111yescharacteristic
538C22×C3#55211yescharacteristic
539C22×C3#55211yescharacteristic
540C2×D3—41no—
541C2×D3—41no—
542C2×D3—41no—
543C3×C4—21no—
544C3×C4—21no—
545C3×C4C2×#12311yescharacteristic
546C3×C4#53611yescharacteristic
547C3×C4#54711yescharacteristic
548C3×C4#54711yescharacteristic
549Dic3—41no—
550Dic3—41no—
551Dic3—41no—
552Dic3—81no—
553Dic3—81no—
554Dic3—81no—
555Dic3—81no—
556C23—21no—
557C23—21no—
558C23—41no—
559C23—41no—
560C23—61no—
561C23#89511yescharacteristic
562C23#89511yescharacteristic
563C23C2×#23911yescharacteristic
564Q8—61no—
565Q8—61no—
566D4—61no—
567D4—61no—
568D4—121no—
569D4—121no—
570D4—241no—
571D4—241no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—242no—
592C2×C4—242no—
593C2×C4—241no—
594C2×C4—241no—
595C2×C4—241no—
596C2×C4—241no—
597C2×C4#89511yescharacteristic
598C2×C4#89511yescharacteristic
599C2×C4C22×D1211yescharacteristic
600C2×C4C2×D3×D411yescharacteristic
601C2×C4C2×#23711yescharacteristic
602C2×C4C2×#23711yescharacteristic
603C2×C4#101011yescharacteristic
604C2×C4#101011yescharacteristic
605C8—21no—
606C8—21no—
607C8—41no—
608C8—41no—
609C8—41no—
610C8—41no—
611C8—121no—
612C8—121no—
613C8#88612yes—
614C8#88612yes—
615C2×C3—21no—
616C2×C3—21no—
617C2×C3—41no—
618C2×C3—41no—
619C2×C3#248611yescharacteristic
620C2×C3#306811yescharacteristic
621C2×C3#306911yescharacteristic
622D3—81no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—42no—
632C22—42no—
633C22—121no—
634C22—121no—
635C22—121no—
636C22C2×#89511yescharacteristic
637C22#564211yescharacteristic
638C22#564211yescharacteristic
639C4—21no—
640C4—21no—
641C4—121no—
642C4—121no—
643C4—121no—
644C4—241no—
645C4—241no—
646C4—241no—
647C4—241no—
648C4C2×#88611yescharacteristic
649C4#552611yescharacteristic
650C4#561111yescharacteristic
651C4#561111yescharacteristic
652C3#2767611yescharacteristic, complemented, Sylow
653C2—21no—
654C2—21no—
655C2—41no—
656C2—41no—
657C2—241no—
658C2#7402711yescharacteristic
659C2#7926111yescharacteristic
660C2#7926211yescharacteristic
subgroupG/H
centerC22C2×#895
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1811C2
socleC22×C3C2×#121
prime radical#5791C22
hypercenter#1811D3
hypocenterC3#27676
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1811

Complements

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

NH ≅ G/Ntype
2. C2×#5121C2semidirect
5. C3×#1811C2semidirect
6. #70721C2semidirect
7. #70720C2semidirect
8. #70701C2semidirect
9. #70700C2semidirect
10. #69974C2semidirect
11. #69973C2semidirect
12. #68924C2semidirect
13. #68923C2semidirect
14. #68922C2semidirect
15. #68921C2semidirect
82. #1811D3semidirect
652. C3#27676semidirect

Maximal subgroups

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

subgroupindex
#837562
C2×#51212
#786232
#786222
C3×#18112
#707212
#707202
#707012
#707002
#699742
#699732
#689242
#689232
#689222
#689212
#276763