Finite Group Databasebeta

← #10877181

Subgroups — #10877181

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×#4881C211yesmaximal, characteristic, complemented
2#78893C211yesmaximal, characteristic
3#78892C211yesmaximal, characteristic
4#78656C211yesmaximal, characteristic
5C3×#2078C211yesmaximal, characteristic, complemented
6#70626C211yesmaximal, characteristic, complemented
7#70300C211yesmaximal, characteristic, complemented
8#70299C211yesmaximal, characteristic, complemented
9#69995C211yesmaximal, characteristic, complemented
10#68991C211yesmaximal, characteristic, complemented
11#68990C211yesmaximal, characteristic, complemented
12#68985C211yesmaximal, characteristic, complemented
13#68984C211yesmaximal, characteristic, complemented
14#68940C211yesmaximal, characteristic, complemented
15#68924C211yesmaximal, characteristic, complemented
16#27676—31nomaximal, Sylow
17#5828C2211yescharacteristic
18C2×#948C2211yescharacteristic
19C22×ΓL(1,𝔽25)C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×D3×C4⋊C4C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C3×#417C2211yescharacteristic
24C3×#417C2211yescharacteristic
25C3×#408C2211yescharacteristic
26#5232C2211yescharacteristic
27#5232C2211yescharacteristic
28#5126C2211yescharacteristic
29#5126C2211yescharacteristic
30#5126C2211yescharacteristic
31C2×#822C2211yescharacteristic
32C2×#822C2211yescharacteristic
33#5122C2211yescharacteristic
34#5121C2211yescharacteristic
35#5121C2211yescharacteristic
36C2×#817C2211yescharacteristic
37#5102C2211yescharacteristic
38#4964C2211yescharacteristic
39#4964C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4881—21no—
43#4881—21no—
44#4881—21no—
45#4881—21no—
46#4815—21no—
47#4814—21no—
48#4811—21no—
49#4805—21no—
50#4784—21no—
51#4779—21no—
52#4777—21no—
53#4774—21no—
54#4730—21no—
55#4730—21no—
56#4729—21no—
57#4729—21no—
58C3×#343C2211yescharacteristic
59C3×#343C2211yescharacteristic
60C3×#341C2211yescharacteristic
61#4554C2211yescharacteristic
62#4554C2211yescharacteristic
63#4554C2211yescharacteristic
64#4554C2211yescharacteristic
65#4552C2211yescharacteristic
66#4530C2211yescharacteristic
67#4530C2211yescharacteristic
68#3093—31no—
69#3092—31no—
70#2930—31no—
71C2×#428—31no—
72#2090—31no—
73#2089—31no—
74#2079—31no—
75#2078D311yescharacteristic, complemented
76#1952—31no—
77#1951—31no—
78#1822—31no—
79#1821—31no—
80#1816—31no—
81#1815—31no—
82#1811—31no—
83C2×C3×#41C2311yescharacteristic
84C22×C4×D3—21no—
85C22×C4×D3C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C2×C3×M16C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92#948—22no—
93#948—22no—
94#946D411yescharacteristic
95#946D411yescharacteristic
96#938D411yescharacteristic
97#938D411yescharacteristic
98#937—21no—
99#935—21no—
100C4×#67—21no—
101C4×#67—21no—
102C2×(C2×D3)⋊C4—21no—
103#926C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×#208—21no—
109C2×#208C2311yescharacteristic
110C2×#208C2311yescharacteristic
111C2×#208C2311yescharacteristic
112C22×C3⋊C8—21no—
113C22×C3⋊C8C2311yescharacteristic
114C2×ΓL(1,𝔽25)—22no—
115C2×ΓL(1,𝔽25)—22no—
116C2×ΓL(1,𝔽25)D411yescharacteristic
117C2×ΓL(1,𝔽25)D411yescharacteristic
118C2×ΓL(1,𝔽25)D411yescharacteristic
119C2×ΓL(1,𝔽25)D411yescharacteristic
120#892—21no—
121#892—21no—
122#892—21no—
123#892—21no—
124D3×C4⋊C4—21no—
125D3×C4⋊C4—21no—
126D3×C4⋊C4—21no—
127D3×C4⋊C4—21no—
128C3×#107—21no—
129C3×#107—21no—
130C3×#103—21no—
131C3×#103—21no—
132C3×#102—21no—
133C3×#102—21no—
134C3×C22⋊C8—21no—
135C3×C22⋊C8—21no—
136#835—21no—
137#835—21no—
138#832—21no—
139#832—21no—
140#822—22no—
141#822—22no—
142(C2×D3)⋊C8—21no—
143(C2×D3)⋊C8—21no—
144(C2×D3)⋊C8—21no—
145(C2×D3)⋊C8—21no—
146(C2×D3)⋊C8—21no—
147(C2×D3)⋊C8—21no—
148#818—21no—
149#818—21no—
150#817—21no—
151#817—21no—
152#817—21no—
153#817—21no—
154#815—21no—
155Dic3⋊C8—21no—
156Dic3⋊C8—21no—
157C8×Dic3—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×M16—31no—
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×#108—31no—
184C2×#108—31no—
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×#107C2×D311yescharacteristic
195C2×#106—31no—
196C2×#106—31no—
197#424—31no—
198#417—31no—
199#417—31no—
200#417C2×D311yescharacteristic
201#417C2×D311yescharacteristic
202#408C2×D311yescharacteristic
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#343C2×D311yescharacteristic
214#343C2×D311yescharacteristic
215#341—31no—
216#341C2×D311yescharacteristic
217C23×D3#4111yescharacteristic
218C3×#41—21no—
219C3×#41—21no—
220C3×#41—21no—
221C3×#41—21no—
222C2×C3×Q8C2×D411yescharacteristic
223C2×C3×D4—21no—
224C2×C3×D4C2×D411yescharacteristic
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4C2411yescharacteristic
229C2×#67—21no—
230C2×#67—21no—
231C22×Dic3—21no—
232C22×Dic3—21no—
233C22×Dic3#4111yescharacteristic
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—21no—
237C2×C4×D3—21no—
238C2×C4×D3—41no—
239C2×C4×D3—41no—
240C2×C4×D3—82no—
241C2×C4×D3C2×D411yescharacteristic
242C2×C4×D3C2×D411yescharacteristic
243C2×C4×D3C2×D411yescharacteristic
244C2×C4×D3C2×D411yescharacteristic
245C3×M16—21no—
246C3×M16—21no—
247C3×M16—21no—
248C3×M16—21no—
249C3×M16—21no—
250C3×M16—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8C2×D411yescharacteristic
257C2×C3×C8C2×D411yescharacteristic
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262#217—21no—
263#217—21no—
264#217—21no—
265(C2×D3)⋊C4—21no—
266(C2×D3)⋊C4—21no—
267(C2×D3)⋊C4—41no—
268(C2×D3)⋊C4—41no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3C2×D411yescharacteristic
274C4⋊Dic3C2×D411yescharacteristic
275Dic3⋊C4—21no—
276Dic3⋊C4—21no—
277Dic3⋊C4—41no—
278Dic3⋊C4—41no—
279C4×Dic3—21no—
280C4×Dic3—41no—
281C4×Dic3—41no—
282#208—21no—
283#208—21no—
284#208—21no—
285#208—21no—
286#208—21no—
287#208—21no—
288#208—42no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—42no—
293C2×C3⋊C8C2×D411yescharacteristic
294C2×C3⋊C8C2×D411yescharacteristic
295C2×C3⋊C8#4112yes—
296C2×C3⋊C8#4112yes—
297ΓL(1,𝔽25)—22no—
298ΓL(1,𝔽25)—22no—
299ΓL(1,𝔽25)—22no—
300ΓL(1,𝔽25)—22no—
301C2×#41C22×D311yescharacteristic
302C23×C4—31no—
303C23×C4—61no—
304#131—62no—
305#131—62no—
306C2×M16—31no—
307C2×M16—31no—
308C2×M16—31no—
309C2×M16—31no—
310C2×M16—31no—
311C2×M16—31no—
312C2×M16—31no—
313C2×M16—62no—
314C2×M16—62no—
315C2×M16—61no—
316C2×M16C22×D311yescharacteristic
317C2×M16C22×D311yescharacteristic
318C2×M16C22×D311yescharacteristic
319C22×C8—31no—
320C22×C8—61no—
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—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4C22×D311yescharacteristic
343C2×C4⋊C4C22×D311yescharacteristic
344C2×C22⋊C4—61no—
345C2×C42—61no—
346#108—62no—
347#108—62no—
348#107—21no—
349#107—21no—
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—21no—
367#103—21no—
368#103—61no—
369#103—61no—
370#102—21no—
371#102—21no—
372#102—61no—
373#102—61no—
374C22⋊C8—21no—
375C22⋊C8—21no—
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×C3C2×#4111yescharacteristic
386C22×D3—21no—
387C22×D3—21no—
388C22×D3—41no—
389C22×D3—41no—
390C22×D3#12111yescharacteristic
391C22×D3#12111yescharacteristic
392C3×Q8—21no—
393C3×Q8—21no—
394C3×D4—21no—
395C3×D4—21no—
396C3×D4—41no—
397C3×D4—41no—
398C2×C3×C4—21no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—81no—
404C2×C3×C4—81no—
405C2×C3×C4C22×D411yescharacteristic
406C2×C3×C4C22×D411yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C42−1+411yescharacteristic
410C2×C3×C42−1+411yescharacteristic
411#67—81no—
412#67—81no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—81no—
420C2×Dic3—81no—
421C2×Dic3#12111yescharacteristic
422C2×Dic3#12111yescharacteristic
423C4×D3—21no—
424C4×D3—21no—
425C4×D3—21no—
426C4×D3—21no—
427C4×D3—21no—
428C4×D3—21no—
429C4×D3—21no—
430C4×D3—21no—
431C4×D3—82no—
432C4×D3—82no—
433C3×C8—21no—
434C3×C8—21no—
435C3×C8—41no—
436C3×C8—41no—
437C3×C8—41no—
438C3⋊C8—41no—
439C3⋊C8—41no—
440C3⋊C8—41no—
441C3⋊C8#12312yes—
442C3⋊C8#12312yes—
443C24—31no—
444#41—21no—
445#41—21no—
446#41—21no—
447#41—21no—
448C2×Q8D3×D411yescharacteristic
449C2×D4—21no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4D3×D411yescharacteristic
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—31no—
460C22×C4—31no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—121no—
468C22×C4—121no—
469C22×C4—242no—
470C22×C4C23×D311yescharacteristic
471M16—21no—
472M16—21no—
473M16—21no—
474M16—21no—
475M16—21no—
476M16—21no—
477M16—62no—
478M16—62no—
479M16—62no—
480M16—62no—
481M16—61no—
482M16—61no—
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—31no—
494C2×C8—31no—
495C2×C8—32no—
496C2×C8—32no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C2×C8—122no—
501C2×C8D3×D411yescharacteristic
502C2×C8D3×D411yescharacteristic
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—31no—
508C4⋊C4—31no—
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—41no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3C2×#12111yescharacteristic
533C22×C3#55211yescharacteristic
534C22×C3#55211yescharacteristic
535C2×D3—21no—
536C2×D3—21no—
537C2×D3—21no—
538C2×D3—21no—
539C2×D3—21no—
540C2×D3—21no—
541C2×D3—42no—
542C2×D3—42no—
543C3×C4—41no—
544C3×C4—81no—
545C3×C4—81no—
546C3×C4C2×#12311yescharacteristic
547C3×C4#53611yescharacteristic
548C3×C4#54711yescharacteristic
549C3×C4#54711yescharacteristic
550Dic3—21no—
551Dic3—21no—
552Dic3—41no—
553Dic3—41no—
554Dic3—81no—
555Dic3—81no—
556C23—21no—
557C23—31no—
558C23—31no—
559C23—61no—
560C23—61no—
561C23—121no—
562C23—121no—
563C23D3×#4111yescharacteristic
564Q8—21no—
565Q8—21no—
566D4—21no—
567D4—21no—
568D4—41no—
569D4—41no—
570D4—241no—
571D4—241no—
572C2×C4—21no—
573C2×C4—31no—
574C2×C4—31no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—242no—
596C2×C4—242no—
597C2×C4—241no—
598C2×C4—241no—
599C2×C4C2×D3×D411yescharacteristic
600C2×C4C2×D3×D411yescharacteristic
601C2×C4D3×#4111yescharacteristic
602C2×C4D3×#4111yescharacteristic
603C2×C4#101011yescharacteristic
604C2×C4#101011yescharacteristic
605C8—21no—
606C8—21no—
607C8—32no—
608C8—32no—
609C8—41no—
610C8—41no—
611C8—41no—
612C8—121no—
613C8—121no—
614C8—121no—
615C2×C3—21no—
616C2×C3—21no—
617C2×C3—81no—
618C2×C3#248611yescharacteristic
619C2×C3#306811yescharacteristic
620C2×C3#306911yescharacteristic
621D3—41no—
622D3—41no—
623C22—21no—
624C22—21no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—61no—
629C22—61no—
630C22—61no—
631C22—61no—
632C22—61no—
633C22—61no—
634C22—122no—
635C22—122no—
636C22D3×#12111yescharacteristic
637C22#570811yescharacteristic
638C22#570811yescharacteristic
639C4—41no—
640C4—61no—
641C4—61no—
642C4—81no—
643C4—81no—
644C4—121no—
645C4—121no—
646C4—241no—
647C4—241no—
648C4D3×#12311yescharacteristic
649C4#565511yescharacteristic
650C4#565511yescharacteristic
651C4#567511yescharacteristic
652C3#2767611yescharacteristic, complemented, Sylow
653C2—21no—
654C2—21no—
655C2—81no—
656C2—121no—
657C2—121no—
658C2#7748511yescharacteristic
659C2#8093811yescharacteristic
660C2#8094411yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2078C2
socleC22×C3C2×#121
prime radical#5828C22
hypercenter#2078D3
hypocenterC3#27676
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2078

Complements

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

NH ≅ G/Ntype
1. C2×#4881C2semidirect
5. C3×#2078C2semidirect
6. #70626C2semidirect
7. #70300C2semidirect
8. #70299C2semidirect
9. #69995C2semidirect
10. #68991C2semidirect
11. #68990C2semidirect
12. #68985C2semidirect
13. #68984C2semidirect
14. #68940C2semidirect
15. #68924C2semidirect
75. #2078D3semidirect
652. C3#27676semidirect

Maximal subgroups

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

subgroupindex
C2×#48812
#788932
#788922
#786562
C3×#20782
#706262
#703002
#702992
#699952
#689912
#689902
#689852
#689842
#689402
#689242
#276763