Finite Group Databasebeta

← #10868014

Subgroups — #10868014

Order 768.

742 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74365C211yesmaximal, characteristic, complemented
2#73649C211yesmaximal, characteristic
3C3×#1947C211yesmaximal, characteristic, complemented
4#70634C211yesmaximal, characteristic
5#70634C211yesmaximal, characteristic
6#70612C211yesmaximal, characteristic, complemented
7#70610C211yesmaximal, characteristic, complemented
8#70484C211yesmaximal, characteristic, complemented
9#70449C211yesmaximal, characteristic, complemented
10#69585C211yesmaximal, characteristic, complemented
11#69459C211yesmaximal, characteristic, complemented
12#68895C211yesmaximal, characteristic, complemented
13#68877C211yesmaximal, characteristic, complemented
14#68813C211yesmaximal, characteristic, complemented
15#68780C211yesmaximal, characteristic, complemented
16#27041—31nomaximal, Sylow
17C2×#895C2211yescharacteristic
18C2×#893C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C3×#384C2211yescharacteristic
24C3×C4×C4⋊C4C2211yescharacteristic
25C2×#821C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×C8×Dic3C2211yescharacteristic
31#4994—21no—
32#4990—21no—
33#4987C2211yescharacteristic
34#4987C2211yescharacteristic
35#4986—21no—
36#4984C2211yescharacteristic
37#4984C2211yescharacteristic
38#4978—21no—
39Dic3×C4⋊C4C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#808C2211yescharacteristic
43#4939—21no—
44#4939C2211yescharacteristic
45C4×(C2×D3)⋊C4—21no—
46C4×(C2×D3)⋊C4C2211yescharacteristic
47#4928C2211yescharacteristic
48#4670—21no—
49#4665—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2497—31no—
61#2363—31no—
62#2071—31no—
63#2067—31no—
64#2033—31no—
65#2033—31no—
66#1947D311yescharacteristic, complemented
67#1942—31no—
68#1923—31no—
69#1921—31no—
70#1919—31no—
71#1864—31no—
72#1854—31no—
73#1825—31no—
74C4×#102—31no—
75C22×D12—21no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42—21no—
88C2×C3×C42C2311yescharacteristic
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4D411yescharacteristic
97C2×(C2×D3)⋊C4D411yescharacteristic
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×Dic3⋊C4—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#895—22no—
110#895—22no—
111#893—22no—
112#893—41no—
113#892—22no—
114#892—41no—
115C3×#107—22no—
116C3×#107—22no—
117C3×C4⋊C8—22no—
118C3×#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95D411yescharacteristic
126C3⋊#95D411yescharacteristic
127#821—22no—
128#821—41no—
129(C2×D3)⋊C8—22no—
130(C2×D3)⋊C8—41no—
131#818—41no—
132C8×Dic3—41no—
133#809—22no—
134#809—22no—
135#808—22no—
136C2×#121—31no—
137C2×C4×D4—31no—
138C2×#117—31no—
139C2×#107—31no—
140C2×#107C2×D311yescharacteristic
141C2×#107C2×D311yescharacteristic
142C2×#106—31no—
143C2×C4⋊C8C2×D311yescharacteristic
144C2×#102—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×C22⋊C8—31no—
149C2×C22⋊C8—31no—
150C2×C4×C8—31no—
151#400—61no—
152#396—61no—
153#390—31no—
154#390—61no—
155#389—31no—
156#389—31no—
157#389—31no—
158#388—61no—
159#388—61no—
160#387—61no—
161#386—61no—
162#384C2×D311yescharacteristic
163#380—31no—
164#380—31no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4C2×D311yescharacteristic
167C4×C22⋊C4—31no—
168C4×C22⋊C4—61no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340C2×D311yescharacteristic
176#340C2×D311yescharacteristic
177#336—31no—
178#336—31no—
179C23×D3—21no—
180C23×D3C2×D411yescharacteristic
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—24no—
200C2×D12—81no—
201C2×D12—81no—
202C2×D12#4112yes—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×C4×D3—81no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8#4112yes—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—81no—
225C3×C4⋊C4—81no—
226C3×C4⋊C4#4112yes—
227C3×C4⋊C4#4112yes—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—81no—
231(C2×D3)⋊C4—24no—
232(C2×D3)⋊C4—22no—
233(C2×D3)⋊C4—22no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—81no—
238(C2×D3)⋊C4—81no—
239(C2×D3)⋊C4—81no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—81no—
247C4⋊Dic3—81no—
248Dic3⋊C4—42no—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C4×Dic3—42no—
252C4×Dic3—81no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C22×D4—31no—
260C22×D4—61no—
261C23×C4—31no—
262C23×C4—61no—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267#121—62no—
268#121—62no—
269C4×D4—62no—
270C4×D4—121no—
271#117—62no—
272#117—121no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C22⋊C4—31no—
286C2×C22⋊C4—31no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C42—21no—
295C2×C42—31no—
296C2×C42—61no—
297C2×C42—61no—
298C2×C42C22×D311yescharacteristic
299#107—22no—
300#107—22no—
301#107—121no—
302#106—62no—
303C4⋊C8—22no—
304#102—62no—
305#102—62no—
306#102—62no—
307#102—121no—
308C22⋊C8—62no—
309C22⋊C8—121no—
310C4×C8—121no—
311#95—21no—
312#95—31no—
313#95—31no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320C23×C3C22×D411yescharacteristic
321C22×D3—22no—
322C22×D3—21no—
323C22×D3—21no—
324C22×D3—21no—
325C22×D3—21no—
326C22×D3—21no—
327C22×D3—21no—
328C22×D3—21no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4#13512yes—
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4H(5,𝔽2)11yescharacteristic
358C2×C3×C42−1+411yescharacteristic
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3#13712yes—
377D12—22no—
378D12—22no—
379D12—41no—
380D12—41no—
381D12—84no—
382C4×D3—41no—
383C4×D3—41no—
384C4×D3—41no—
385C4×D3—41no—
386C4×D3—84no—
387C3×C8—22no—
388C3×C8—41no—
389C3⋊C8—41no—
390C3⋊C8—41no—
391C24—31no—
392C24—61no—
393C2×D4—32no—
394C2×D4—64no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—61no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—241no—
401C2×D4—241no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—31no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—122no—
419C22×C4—122no—
420C22×C4—241no—
421C22×C4D3×D411yescharacteristic
422C22×C4D3×D411yescharacteristic
423C22×C4C23×D311yescharacteristic
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8#23512yes—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—122no—
451C4⋊C4—241no—
452C4⋊C4—241no—
453C4⋊C4#23712yes—
454C4⋊C4#23912yes—
455C22⋊C4—62no—
456C22⋊C4—64no—
457C22⋊C4—62no—
458C22⋊C4—122no—
459C22⋊C4—122no—
460C22⋊C4—122no—
461C22⋊C4—241no—
462C22⋊C4—241no—
463C22⋊C4—241no—
464C42—21no—
465C42—21no—
466C42—61no—
467C42—61no—
468C42—81no—
469C42—122no—
470C42—241no—
471C22×C3#52511yescharacteristic
472C22×C3#54611yescharacteristic
473C22×C3#54711yescharacteristic
474C22×C3C2×#13511yescharacteristic
475C22×C3C2×#13711yescharacteristic
476C22×C3#57611yescharacteristic
477C22×C3#57611yescharacteristic
478C2×D3—41no—
479C2×D3—41no—
480C2×D3—41no—
481C2×D3—42no—
482C2×D3—41no—
483C2×D3—42no—
484C2×D3—41no—
485C2×D3—42no—
486C2×D3—41no—
487C2×D3—41no—
488C2×D3—81no—
489C2×D3—81no—
490C2×D3—81no—
491C2×D3—81no—
492C3×C4—22no—
493C3×C4—41no—
494C3×C4—42no—
495C3×C4—42no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4#52911yescharacteristic
499C3×C4#53911yescharacteristic
500C3×C4#55311yescharacteristic
501C3×C4#55611yescharacteristic
502Dic3—22no—
503Dic3—41no—
504Dic3—41no—
505Dic3—81no—
506Dic3—81no—
507Dic3—81no—
508C23—61no—
509C23—61no—
510C23—61no—
511C23—61no—
512C23—61no—
513C23—62no—
514C23—61no—
515C23—61no—
516C23—121no—
517C23—121no—
518C23—121no—
519C23—121no—
520C23—121no—
521C23—121no—
522C23C2×D3×D411yescharacteristic
523D4—62no—
524D4—62no—
525D4—121no—
526D4—121no—
527D4—244no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—32no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—244no—
568C2×C4#91912yes—
569C2×C4C2×#23511yescharacteristic
570C2×C4C2×#23711yescharacteristic
571C2×C4C2×#23911yescharacteristic
572C2×C4#100711yescharacteristic
573C2×C4D3×#4111yescharacteristic
574C2×C4#100911yescharacteristic
575C8—22no—
576C8—41no—
577C8—121no—
578C8—121no—
579C2×C3#246611yescharacteristic
580C2×C3#310711yescharacteristic
581C2×C3#312211yescharacteristic
582C2×C3#330511yescharacteristic
583C2×C3#331411yescharacteristic
584C2×C3#333911yescharacteristic
585C2×C3#335711yescharacteristic
586D3—81no—
587D3—82no—
588C22—121no—
589C22—121no—
590C22—122no—
591C22—121no—
592C22—121no—
593C22—121no—
594C22—122no—
595C22—121no—
596C22—121no—
597C22—122no—
598C22—241no—
599C22—241no—
600C22—241no—
601C22—241no—
602C22#550711yescharacteristic
603C22#560611yescharacteristic
604C22#563011yescharacteristic
605C22#575711yescharacteristic
606C22C2×#91911yescharacteristic
607C22#577411yescharacteristic
608C22D3×#13711yescharacteristic
609C4—22no—
610C4—42no—
611C4—41no—
612C4—42no—
613C4—62no—
614C4—81no—
615C4—81no—
616C4—121no—
617C4—121no—
618C4—241no—
619C4—241no—
620C4—241no—
621C4#556511yescharacteristic
622C4#558011yescharacteristic
623C4#569211yescharacteristic
624C4#573211yescharacteristic
625C3#2704111yescharacteristic, complemented, Sylow
626C2—241no—
627C2—242no—
628C2#7602011yescharacteristic
629C2#7915711yescharacteristic
630C2#7923811yescharacteristic
631C2#8031711yescharacteristic
632C2#8045711yescharacteristic
633C2#8098211yescharacteristic
634C2#8112011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1947C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#1947D3
hypocenterC3#27041
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
1. #74365C2semidirect
3. C3×#1947C2semidirect
6. #70612C2semidirect
7. #70610C2semidirect
8. #70484C2semidirect
9. #70449C2semidirect
10. #69585C2semidirect
11. #69459C2semidirect
12. #68895C2semidirect
13. #68877C2semidirect
14. #68813C2semidirect
15. #68780C2semidirect
66. #1947D3semidirect
625. C3#27041semidirect

Maximal subgroups

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

subgroupindex
#743652
#736492
C3×#19472
#706342
#706122
#706102
#704842
#704492
#695852
#694592
#688952
#688772
#688132
#687802
#270413