Finite Group Databasebeta

← #10877174

Subgroups — #10877174

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
1C2×C3×#428C211yesmaximal, characteristic, complemented
2#83756C211yesmaximal, characteristic
3#82216C211yesmaximal, characteristic
4#82139C211yesmaximal, characteristic
5#72000C211yesmaximal, characteristic, complemented
6#71589C211yesmaximal, characteristic, complemented
7#71562C211yesmaximal, characteristic, complemented
8#70647C211yesmaximal, characteristic, complemented
9#70646C211yesmaximal, characteristic, complemented
10#70626C211yesmaximal, characteristic, complemented
11#70625C211yesmaximal, characteristic, complemented
12#70593C211yesmaximal, characteristic, complemented
13#70592C211yesmaximal, characteristic, complemented
14#70573C211yesmaximal, characteristic, complemented
15#70572C211yesmaximal, characteristic, complemented
16#27676—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20#5791C2211yescharacteristic
21C2×#901C2211yescharacteristic
22C3×#428—21no—
23C3×#428—21no—
24C3×#428—21no—
25C3×#428—21no—
26C2×C3×#107C2211yescharacteristic
27C2×C3×#107C2211yescharacteristic
28C2×C3×#106C2211yescharacteristic
29C2×C3×#106C2211yescharacteristic
30#5207C2211yescharacteristic
31#5207C2211yescharacteristic
32#5207C2211yescharacteristic
33#5128—21no—
34#5128—21no—
35#5127—21no—
36#5127—21no—
37#5110C2211yescharacteristic
38C2×#819C2211yescharacteristic
39C2×#819C2211yescharacteristic
40#5102C2211yescharacteristic
41#5101C2211yescharacteristic
42#5003—21no—
43#5003—21no—
44#4996—21no—
45#4996—21no—
46#4971—21no—
47#4970—21no—
48#4968—21no—
49#4967—21no—
50#4966C2211yescharacteristic
51#4966C2211yescharacteristic
52#4966C2211yescharacteristic
53#4966C2211yescharacteristic
54#4964C2211yescharacteristic
55#4964C2211yescharacteristic
56#4964C2211yescharacteristic
57#4964C2211yescharacteristic
58#4552C2211yescharacteristic
59#4552C2211yescharacteristic
60#4530C2211yescharacteristic
61#4530C2211yescharacteristic
62#4530C2211yescharacteristic
63#4530C2211yescharacteristic
64#4530C2211yescharacteristic
65#4530C2211yescharacteristic
66#4530C2211yescharacteristic
67#4530C2211yescharacteristic
68#3093—31no—
69#3092—31no—
70#2930—31no—
71C2×#428D311yescharacteristic, complemented
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#1811—31no—
83C23×C3×C4—21no—
84C23×C3×C4C2311yescharacteristic
85C2×#235C2311yescharacteristic
86C2×C3×M16—22no—
87C2×C3×M16—22no—
88C2×C3×M16D411yescharacteristic
89C2×C3×M16D411yescharacteristic
90C2×C3×M16D411yescharacteristic
91C2×C3×M16D411yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C22×C3×C8C2311yescharacteristic
94C3×#117—21no—
95C3×#117—21no—
96C3×#117—21no—
97C3×#117—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—21no—
108C2×C3×C42—21no—
109#953—21no—
110#930D411yescharacteristic
111#930D411yescharacteristic
112#929—21no—
113C4×#67—21no—
114C4×#67—21no—
115#926C2311yescharacteristic
116#924D411yescharacteristic
117#924D411yescharacteristic
118C2×#208C2311yescharacteristic
119C2×#208C2311yescharacteristic
120C2×#208C2311yescharacteristic
121C2×#208C2311yescharacteristic
122C2×#208C2311yescharacteristic
123C2×#208C2311yescharacteristic
124#901—22no—
125#901—22no—
126C2×ΓL(1,𝔽25)—21no—
127C2×D3×C8—21no—
128C3×#107—21no—
129C3×#107—21no—
130C3×#107—21no—
131C3×#107—21no—
132C3×#106—21no—
133C3×#106—21no—
134C3×#106—21no—
135C3×#106—21no—
136#830—21no—
137#830—21no—
138#830—21no—
139#830—21no—
140#830—21no—
141#830—21no—
142(C2×D3)⋊C8—21no—
143(C2×D3)⋊C8—21no—
144#819—22no—
145#819—22no—
146#815—21no—
147Dic3⋊C8—21no—
148Dic3⋊C8—21no—
149C8×Dic3—21no—
150#810—21no—
151#810—21no—
152#810—21no—
153#810—21no—
154#809—21no—
155#809—21no—
156#809—21no—
157#809—21no—
158#808—21no—
159#808—21no—
160#808—21no—
161#808—21no—
162#807—21no—
163#807—21no—
164#807—21no—
165#807—21no—
166C2×#131—31no—
167C22×M16C2×D311yescharacteristic
168#525—31no—
169C2×#117C2×D311yescharacteristic
170C22×C4⋊C4C2×D311yescharacteristic
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×#108—31no—
184C2×#108—31no—
185#428—21no—
186#428—21no—
187#428—21no—
188#428—21no—
189#428—31no—
190#428—31no—
191#428—31no—
192#428—31no—
193C2×#107C2×D311yescharacteristic
194C2×#107C2×D311yescharacteristic
195C2×#106C2×D311yescharacteristic
196C2×#106C2×D311yescharacteristic
197#424—31no—
198#417—31no—
199#417—31no—
200#417—31no—
201#417—31no—
202#408—31no—
203#407—31no—
204#407—31no—
205#407—31no—
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#341—31no—
216#341—31no—
217C24×C3#4111yescharacteristic
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—41no—
227C22×C3×C4—41no—
228C22×C3×C4—82no—
229C22×C3×C4C2×D411yescharacteristic
230C22×C3×C4C2×D411yescharacteristic
231C22×C3×C4C2×D411yescharacteristic
232C22×C3×C4C2×D411yescharacteristic
233C22×C3×C4#4111yescharacteristic
234C22×C3×C4C2411yescharacteristic
235C2×#67—21no—
236C2×#67—21no—
237C2×#67—21no—
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—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8C2×D411yescharacteristic
252C2×C3×C8C2×D411yescharacteristic
253C2×C3×C8C2×D411yescharacteristic
254C2×C3×C8C2×D411yescharacteristic
255C2×C3×C8#4112yes—
256C2×C3×C8#4112yes—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—41no—
266C3×C4⋊C4—41no—
267C3×C22⋊C4—41no—
268C3×C22⋊C4—41no—
269C3×C42—41no—
270C3×C42—41no—
271#217—21no—
272#217—21no—
273#217—21no—
274(C2×D3)⋊C4—21no—
275(C2×D3)⋊C4—21no—
276C4⋊Dic3C2×D411yescharacteristic
277C4⋊Dic3C2×D411yescharacteristic
278Dic3⋊C4—21no—
279Dic3⋊C4—21no—
280C4×Dic3—21no—
281#208—21no—
282#208—21no—
283#208—21no—
284#208—21no—
285#208—21no—
286#208—21no—
287#208—21no—
288#208—21no—
289#208—21no—
290#208—21no—
291#208—21no—
292#208—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299ΓL(1,𝔽25)—42no—
300D3×C8—42no—
301C2×#41—31no—
302C23×C4—21no—
303C23×C4C22×D311yescharacteristic
304#131—62no—
305#131—62no—
306C2×M16—22no—
307C2×M16—22no—
308C2×M16—31no—
309C2×M16—31no—
310C2×M16—31no—
311C2×M16—31no—
312C2×M16—31no—
313C2×M16—31no—
314C2×M16—61no—
315C2×M16#6711yescharacteristic
316C2×M16#6711yescharacteristic
317C2×M16#6711yescharacteristic
318C2×M16#6711yescharacteristic
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—21no—
331#117—21no—
332#117—21no—
333#117—21no—
334#117—31no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C4⋊C4C22×D311yescharacteristic
343C2×C4⋊C4C22×D311yescharacteristic
344C2×C22⋊C4—21no—
345C2×C42—21no—
346#108—62no—
347#108—62no—
348#107—21no—
349#107—21no—
350#107—21no—
351#107—21no—
352#107—61no—
353#107—61no—
354#107—61no—
355#107—61no—
356#106—21no—
357#106—21no—
358#106—21no—
359#106—21no—
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—61no—
375C22⋊C8—61no—
376C22⋊C8—61no—
377C22⋊C8—61no—
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—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—82no—
407C2×C3×C4—82no—
408C2×C3×C4—81no—
409C2×C3×C4—81no—
410C2×C3×C4—81no—
411C2×C3×C4—81no—
412C2×C3×C4#12111yescharacteristic
413C2×C3×C4#12111yescharacteristic
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4C22×D411yescharacteristic
416C2×C3×C4C2×#4111yescharacteristic
417C2×C3×C4C2×#4111yescharacteristic
418C2×C3×C42−1+411yescharacteristic
419C2×C3×C42−1+411yescharacteristic
420#67—41no—
421#67—41no—
422#67—81no—
423#67—81no—
424C2×Dic3—21no—
425C2×Dic3—21no—
426C2×Dic3—21no—
427D12—21no—
428D12—21no—
429C4×D3—41no—
430C4×D3—41no—
431Dic6—21no—
432Dic6—21no—
433C3×C8—21no—
434C3×C8—21no—
435C3×C8#12312yes—
436C3×C8#12312yes—
437C3⋊C8—41no—
438C3⋊C8—41no—
439C3⋊C8—41no—
440C3⋊C8—41no—
441C3⋊C8—41no—
442C3⋊C8—41no—
443C24#23711yescharacteristic
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—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—61no—
464C22×C4—82no—
465C22×C4D3×D411yescharacteristic
466C22×C4D3×D411yescharacteristic
467C22×C4#23711yescharacteristic
468C22×C4C2×#6711yescharacteristic
469C22×C4C2×#6711yescharacteristic
470C22×C4C23×D311yescharacteristic
471M16—22no—
472M16—22no—
473M16—22no—
474M16—22no—
475M16—61no—
476M16—61no—
477M16—61no—
478M16—61no—
479M16—61no—
480M16—61no—
481M16—61no—
482M16—61no—
483M16—61no—
484M16—61no—
485M16—61no—
486M16—61no—
487M16—122no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—122no—
497C2×C8#23512yes—
498C2×C8#23512yes—
499C2×C8C2×#6711yescharacteristic
500C2×C8C2×#6711yescharacteristic
501C2×C8C2×#6711yescharacteristic
502C2×C8C2×#6711yescharacteristic
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—31no—
512C4⋊C4—31no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—61no—
520C22⋊C4—61no—
521C22⋊C4—61no—
522C22⋊C4—61no—
523C22⋊C4—61no—
524C42—41no—
525C42—41no—
526C42—61no—
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×C4—81no—
546C3×C4—81no—
547C3×C4—81no—
548C3×C4—81no—
549C3×C4C2×#12311yescharacteristic
550C3×C4#53611yescharacteristic
551C3×C4#54711yescharacteristic
552C3×C4#54711yescharacteristic
553Dic3—41no—
554Dic3—41no—
555Dic3—41no—
556C23—21no—
557C23—21no—
558C23—41no—
559C23—41no—
560C23—61no—
561C23#93811yescharacteristic
562C23#93811yescharacteristic
563C23C2×#23711yescharacteristic
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—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4#93811yescharacteristic
598C2×C4#93811yescharacteristic
599C2×C4C2×#23511yescharacteristic
600C2×C4C2×#23511yescharacteristic
601C2×C4C2×D3×D411yescharacteristic
602C2×C4#100711yescharacteristic
603C2×C4#100711yescharacteristic
604C2×C4C22×#6711yescharacteristic
605C8—21no—
606C8—21no—
607C8—121no—
608C8—121no—
609C8—121no—
610C8—121no—
611C8—121no—
612C8—121no—
613C8#92912yes—
614C8#92912yes—
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—
636C22#551311yescharacteristic
637C22#551311yescharacteristic
638C22C2×#93811yescharacteristic
639C4—21no—
640C4—21no—
641C4—81no—
642C4—81no—
643C4—81no—
644C4—81no—
645C4—121no—
646C4—121no—
647C4—121no—
648C4#550711yescharacteristic
649C4#550711yescharacteristic
650C4C2×#92911yescharacteristic
651C4#581611yescharacteristic
652C3#2767611yescharacteristic, complemented, Sylow
653C2—21no—
654C2—21no—
655C2—41no—
656C2—41no—
657C2—241no—
658C2#8219611yescharacteristic
659C2#8396311yescharacteristic
660C2#8399811yescharacteristic
subgroupG/H
centerC22C2×#938
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#428C2
socleC22×C3C2×#121
prime radical#5791C22
hypercenterC2×#428D3
hypocenterC3#27676
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×C3×#428C2semidirect
5. #72000C2semidirect
6. #71589C2semidirect
7. #71562C2semidirect
8. #70647C2semidirect
9. #70646C2semidirect
10. #70626C2semidirect
11. #70625C2semidirect
12. #70593C2semidirect
13. #70592C2semidirect
14. #70573C2semidirect
15. #70572C2semidirect
71. C2×#428D3semidirect
652. C3#27676semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4282
#837562
#822162
#821392
#720002
#715892
#715622
#706472
#706462
#706262
#706252
#705932
#705922
#705732
#705722
#276763