Finite Group Databasebeta

← #10868103

Subgroups — #10868103

Order 768.

806 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76206C211yesmaximal, characteristic
2#73792C211yesmaximal, characteristic, complemented
3C3×#1921C211yesmaximal, characteristic, complemented
4#70634C211yesmaximal, characteristic
5#70615C211yesmaximal, characteristic, complemented
6#69620C211yesmaximal, characteristic, complemented
7#69474C211yesmaximal, characteristic, complemented
8#68902C211yesmaximal, characteristic, complemented
9#68900C211yesmaximal, characteristic, complemented
10#68875C211yesmaximal, characteristic, complemented
11#68872C211yesmaximal, characteristic, complemented
12#68822C211yesmaximal, characteristic
13#68818C211yesmaximal, characteristic, complemented
14#68671C211yesmaximal, characteristic, complemented
15#68658C211yesmaximal, characteristic, complemented
16#27047—31nomaximal, Sylow
17C2×#895C2211yescharacteristic
18C2×#893—21no—
19C2×D3×C4⋊C4C2211yescharacteristic
20C2×#876—21no—
21C2×C4×D12C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C3×#389C2211yescharacteristic
24C3×C4×C4⋊C4C2211yescharacteristic
25C2×#821C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×#817C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#4989—21no—
32#4988C2211yescharacteristic
33#4987C2211yescharacteristic
34#4987C2211yescharacteristic
35#4975C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37#4972C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42C4×(C2×D3)⋊C4C2211yescharacteristic
43#4933—21no—
44#4928C2211yescharacteristic
45C2×#803C2211yescharacteristic
46#4675—21no—
47#4669—21no—
48#4665—21no—
49#4658—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#336C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2496—31no—
61#2361—31no—
62#2071—31no—
63#2065—31no—
64#2033—31no—
65#2033—31no—
66#1947—31no—
67#1943—31no—
68#1924—31no—
69#1921D311yescharacteristic, complemented
70#1920—31no—
71#1864—31no—
72#1853—31no—
73#1824—31no—
74#1762—31no—
75C22×D12—21no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×C4×D3C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
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)⋊C4D411yescharacteristic
96C2×(C2×D3)⋊C4D411yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#895—22no—
110#895—22no—
111#893—44no—
112D3×C4⋊C4—22no—
113D3×C4⋊C4—41no—
114#876—44no—
115C4×D12—22no—
116C4×D12—41no—
117C3×#107—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#817—22no—
132Dic3⋊C8—22no—
133#809—22no—
134#809—22no—
135#808—41no—
136#807—22no—
137#803—41no—
138C2×#126—61no—
139C2×#121—31no—
140C2×C4×D4—31no—
141C2×C4×D4—61no—
142C22×C4⋊C4—31no—
143C2×#107—31no—
144C2×#107C2×D311yescharacteristic
145C2×#106—31no—
146C2×#106—31no—
147C2×C4⋊C8—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×#102—31no—
151C2×#102—31no—
152C2×C22⋊C8—31no—
153C2×C22⋊C8—31no—
154C2×#97—31no—
155#400—61no—
156#397—61no—
157#390—31no—
158#389—31no—
159#389—31no—
160#389C2×D311yescharacteristic
161#388—61no—
162#387—61no—
163#386—61no—
164#385—61no—
165#384—31no—
166#380—31no—
167#380—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4C2×D311yescharacteristic
170C4×C22⋊C4—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340C2×D311yescharacteristic
177#340C2×D311yescharacteristic
178#340C2×D311yescharacteristic
179#336—31no—
180#336C2×D311yescharacteristic
181C23×D3—21no—
182C23×D3C2×D411yescharacteristic
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2411yescharacteristic
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3C2×D411yescharacteristic
194C22×Dic3C2×D411yescharacteristic
195C22×Dic3C2×D411yescharacteristic
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—24no—
201C2×D12—21no—
202C2×D12—42no—
203C2×D12—42no—
204C2×D12#4112yes—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—42no—
212C2×C4×D3—42no—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×C4×D3—81no—
216C2×C4×D3—81no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4#4112yes—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—42no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—22no—
237(C2×D3)⋊C4—22no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—42no—
241(C2×D3)⋊C4—42no—
242(C2×D3)⋊C4—81no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3#4112yes—
249Dic3⋊C4—42no—
250Dic3⋊C4—42no—
251Dic3⋊C4—42no—
252Dic3⋊C4—42no—
253Dic3⋊C4—42no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C4×Dic3—42no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8#4112yes—
263C22×D4—31no—
264C22×D4—61no—
265C23×C4—31no—
266C23×C4—61no—
267C23×C4—61no—
268C22×C8—31no—
269C22×C8—31no—
270C22×C8C22×D311yescharacteristic
271C22×C8C22×D311yescharacteristic
272#126—124no—
273#121—62no—
274#121—62no—
275C4×D4—62no—
276C4×D4—121no—
277C4×D4—124no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—62no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—121no—
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C22⋊C4—31no—
294C2×C22⋊C4—31no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C42—21no—
302C2×C42—31no—
303C2×C42—61no—
304C2×C42C22×D311yescharacteristic
305#107—22no—
306#107—62no—
307#106—62no—
308#106—121no—
309C4⋊C8—62no—
310#102—62no—
311#102—62no—
312#102—62no—
313#102—121no—
314C22⋊C8—62no—
315C22⋊C8—121no—
316#97—121no—
317#95—21no—
318#95—21no—
319#95—31no—
320#95—31no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326C23×C3C22×D411yescharacteristic
327C22×D3—21no—
328C22×D3—21no—
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—21no—
332C22×D3—21no—
333C22×D3—22no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—41no—
340C22×D3AGL(1,ℤ/8)12yes—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4H(5,𝔽2)11yescharacteristic
364C2×C3×C42−1+411yescharacteristic
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—22no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3AGL(1,ℤ/8)12yes—
383D12—22no—
384D12—22no—
385D12—48no—
386D12—41no—
387D12—41no—
388C4×D3—41no—
389C4×D3—41no—
390C4×D3—41no—
391C4×D3—41no—
392C4×D3—84no—
393C4×D3—84no—
394C3×C8—41no—
395C3×C8—41no—
396C3⋊C8—22no—
397C3⋊C8—41no—
398C24—31no—
399C24—61no—
400C2×D4—32no—
401C2×D4—64no—
402C2×D4—61no—
403C2×D4—61no—
404C2×D4—61no—
405C2×D4—61no—
406C2×D4—61no—
407C2×D4—122no—
408C2×D4—122no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—122no—
426C22×C4—122no—
427C22×C4—122no—
428C22×C4—122no—
429C22×C4—241no—
430C22×C4—241no—
431C22×C4D3×D411yescharacteristic
432C22×C4D3×D411yescharacteristic
433C22×C4C23×D311yescharacteristic
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—32no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—32no—
454C4⋊C4—42no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—122no—
461C4⋊C4—122no—
462C4⋊C4—122no—
463C4⋊C4—122no—
464C4⋊C4—122no—
465C4⋊C4#23512yes—
466C22⋊C4—62no—
467C22⋊C4—62no—
468C22⋊C4—64no—
469C22⋊C4—122no—
470C22⋊C4—122no—
471C22⋊C4—122no—
472C22⋊C4—122no—
473C22⋊C4—241no—
474C42—21no—
475C42—21no—
476C42—42no—
477C42—61no—
478C42—61no—
479C42—122no—
480C22×C3#52511yescharacteristic
481C22×C3#54611yescharacteristic
482C22×C3#54711yescharacteristic
483C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
484C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
485C22×C3#57711yescharacteristic
486C22×C3#57811yescharacteristic
487C2×D3—22no—
488C2×D3—22no—
489C2×D3—22no—
490C2×D3—41no—
491C2×D3—41no—
492C2×D3—41no—
493C2×D3—41no—
494C2×D3—42no—
495C2×D3—42no—
496C2×D3—42no—
497C2×D3—81no—
498C2×D3—81no—
499C2×D3—81no—
500C2×D3—81no—
501C3×C4—42no—
502C3×C4—41no—
503C3×C4—41no—
504C3×C4—81no—
505C3×C4—81no—
506C3×C4—81no—
507C3×C4#52911yescharacteristic
508C3×C4#53911yescharacteristic
509C3×C4#55311yescharacteristic
510C3×C4#55611yescharacteristic
511Dic3—22no—
512Dic3—41no—
513Dic3—41no—
514Dic3—42no—
515Dic3—81no—
516Dic3—81no—
517C23—32no—
518C23—61no—
519C23—61no—
520C23—61no—
521C23—61no—
522C23—61no—
523C23—61no—
524C23—62no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23C2×D3×D411yescharacteristic
532D4—62no—
533D4—62no—
534D4—128no—
535D4—121no—
536D4—121no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—32no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—62no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—244no—
578C2×C4—244no—
579C2×C4C2×#23511yescharacteristic
580C2×C4#100711yescharacteristic
581C2×C4D3×#4111yescharacteristic
582C2×C4D3×#4111yescharacteristic
583C2×C4D3×#4111yescharacteristic
584C2×C4#100911yescharacteristic
585C8—41no—
586C8—41no—
587C8—62no—
588C8—121no—
589C2×C3#246611yescharacteristic
590C2×C3#310911yescharacteristic
591C2×C3#312011yescharacteristic
592C2×C3#330811yescharacteristic
593C2×C3#332411yescharacteristic
594C2×C3#334311yescharacteristic
595C2×C3#334511yescharacteristic
596D3—42no—
597D3—82no—
598C22—62no—
599C22—62no—
600C22—62no—
601C22—121no—
602C22—121no—
603C22—121no—
604C22—121no—
605C22—122no—
606C22—122no—
607C22—122no—
608C22—241no—
609C22—241no—
610C22—241no—
611C22—241no—
612C22#554711yescharacteristic
613C22#563111yescharacteristic
614C22#565411yescharacteristic
615C22#576611yescharacteristic
616C22#577011yescharacteristic
617C22D3×AGL(1,ℤ/8)11yescharacteristic
618C22D3×AGL(1,ℤ/8)11yescharacteristic
619C4—41no—
620C4—41no—
621C4—42no—
622C4—62no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—121no—
627C4—121no—
628C4—122no—
629C4—241no—
630C4—241no—
631C4#553011yescharacteristic
632C4#556811yescharacteristic
633C4#569211yescharacteristic
634C4#570611yescharacteristic
635C3#2704711yescharacteristic, complemented, Sylow
636C2—122no—
637C2—242no—
638C2#7533011yescharacteristic
639C2#7974811yescharacteristic
640C2#7986911yescharacteristic
641C2#8112711yescharacteristic
642C2#8119711yescharacteristic
643C2#8140311yescharacteristic
644C2#8147311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1921C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#1921D3
hypocenterC3#27047
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
2. #73792C2semidirect
3. C3×#1921C2semidirect
5. #70615C2semidirect
6. #69620C2semidirect
7. #69474C2semidirect
8. #68902C2semidirect
9. #68900C2semidirect
10. #68875C2semidirect
11. #68872C2semidirect
13. #68818C2semidirect
14. #68671C2semidirect
15. #68658C2semidirect
69. #1921D3semidirect
635. C3#27047semidirect

Maximal subgroups

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

subgroupindex
#762062
#737922
C3×#19212
#706342
#706152
#696202
#694742
#689022
#689002
#688752
#688722
#688222
#688182
#686712
#686582
#270473