Finite Group Databasebeta

← #10840229

Subgroups — #10840229

Order 768.

862 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82272C211yesmaximal, characteristic, complemented
2#73935C211yesmaximal, characteristic
3C3×#1825C211yesmaximal, characteristic, complemented
4#71586C211yesmaximal, characteristic, complemented
5#69935C211yesmaximal, characteristic, complemented
6#69917C211yesmaximal, characteristic, complemented
7#69507C211yesmaximal, characteristic, complemented
8#68893C211yesmaximal, characteristic, complemented
9#68880C211yesmaximal, characteristic, complemented
10#68871C211yesmaximal, characteristic, complemented
11#68849C211yesmaximal, characteristic, complemented
12#68832C211yesmaximal, characteristic, complemented
13#68823C211yesmaximal, characteristic, complemented
14#68796C211yesmaximal, characteristic, complemented
15#68782C211yesmaximal, characteristic, complemented
16#24982—31nomaximal, Sylow
17C2×C3×#117C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C2×C4×#67C2211yescharacteristic
20C2×#895C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×D3×C4⋊C4C2211yescharacteristic
23C2×#884—21no—
24C2×C3×C22⋊C8C2211yescharacteristic
25C3×#380C2211yescharacteristic
26#5212C2211yescharacteristic
27C2×#830C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#815C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35#4987C2211yescharacteristic
36#4987C2211yescharacteristic
37#4977C2211yescharacteristic
38#4975C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#808C2211yescharacteristic
43#4959—21no—
44#4956—21no—
45#4945—21no—
46C4×(C2×D3)⋊C4—21no—
47#4933—21no—
48#4672—21no—
49#4667—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54#4551C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2537—31no—
61#2352—31no—
62#2077—31no—
63#2077—31no—
64#2034—31no—
65#2034—31no—
66#1955—31no—
67#1948—31no—
68#1875—31no—
69#1874—31no—
70#1836—31no—
71#1825D311yescharacteristic, complemented
72#1823—31no—
73#1804—31no—
74#1762—31no—
75C23×C3×C4C2311yescharacteristic
76C22×#67—21no—
77C22×#67D411yescharacteristic
78C22×D12—21no—
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×C4×D3C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×#117—22no—
86C3×#117—41no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C42—21no—
94C2×#217—21no—
95C2×#217D411yescharacteristic
96#930—22no—
97#930—41no—
98C4×#67—22no—
99C4×#67—22no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4D411yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4D411yescharacteristic
113C2×C4×Dic3—21no—
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117#895—22no—
118#895—41no—
119#893—22no—
120#893—22no—
121D3×C4⋊C4—22no—
122D3×C4⋊C4—41no—
123#884—44no—
124C3×C22⋊C8—41no—
125C3×#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132#830—41no—
133#821—22no—
134#821—41no—
135(C2×D3)⋊C8—41no—
136(C2×D3)⋊C8—41no—
137#817—41no—
138#815—22no—
139Dic3⋊C8—22no—
140#809—22no—
141#809—41no—
142#808—41no—
143C2×#121—31no—
144C2×#121—31no—
145C2×#121—61no—
146C2×C4×D4—31no—
147C2×C4×D4—31no—
148C2×#117C2×D311yescharacteristic
149C22×C4⋊C4—31no—
150C2×#106—31no—
151C2×#106—31no—
152C2×C4⋊C8—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×C22⋊C8—31no—
158C2×C22⋊C8—31no—
159C2×C22⋊C8—31no—
160C2×C22⋊C8C2×D311yescharacteristic
161C2×#97—31no—
162#394—61no—
163#394—61no—
164#388—61no—
165#387—61no—
166#386—61no—
167#384—31no—
168#382—31no—
169#380—31no—
170#380—31no—
171#380—31no—
172#380—61no—
173#380C2×D311yescharacteristic
174C4×C4⋊C4—31no—
175C4×C22⋊C4—61no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#340C2×D311yescharacteristic
183#340C2×D311yescharacteristic
184#340C2×D311yescharacteristic
185#340C2×D311yescharacteristic
186C24×C3C2×D411yescharacteristic
187C23×D3—21no—
188C23×D3C2×D411yescharacteristic
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2411yescharacteristic
201C2×#67—22no—
202C2×#67—22no—
203C2×#67—41no—
204C2×#67—42no—
205C2×#67—42no—
206C2×#67—81no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—42no—
219C2×D12—42no—
220C2×D12—81no—
221C2×D12#4112yes—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×C4×D3—42no—
231C2×C4×D3—42no—
232C2×C4×D3—42no—
233C2×C4×D3—81no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4#4112yes—
248C3×C22⋊C4—42no—
249C3×C22⋊C4—42no—
250C3×C42—42no—
251#217—22no—
252#217—81no—
253(C2×D3)⋊C4—22no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265Dic3⋊C4—22no—
266Dic3⋊C4—42no—
267Dic3⋊C4—42no—
268Dic3⋊C4—81no—
269C4×Dic3—81no—
270C4×Dic3#4112yes—
271C4×Dic3#4112yes—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C22×D4—31no—
279C22×D4—31no—
280C22×D4—61no—
281C22×D4—61no—
282C23×C4—31no—
283C23×C4—61no—
284C23×C4—61no—
285C23×C4C22×D311yescharacteristic
286C22×C8—31no—
287C22×C8—31no—
288C22×C8C22×D311yescharacteristic
289C22×C8C22×D311yescharacteristic
290#121—62no—
291#121—62no—
292#121—124no—
293#121—121no—
294#121—121no—
295C4×D4—62no—
296C4×D4—62no—
297C4×D4—62no—
298C4×D4—62no—
299#117—22no—
300#117—41no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—62no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—121no—
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—31no—
317C2×C22⋊C4—31no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C42—21no—
325C2×C42—31no—
326C2×C42—61no—
327#106—121no—
328#106—121no—
329C4⋊C8—62no—
330#102—62no—
331#102—62no—
332#102—121no—
333#102—121no—
334C22⋊C8—41no—
335C22⋊C8—121no—
336C22⋊C8—121no—
337C22⋊C8—121no—
338#97—62no—
339#95—21no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346C23×C3—21no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3C22×D411yescharacteristic
354C22×D3—21no—
355C22×D3—21no—
356C22×D3—21no—
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—22no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3AGL(1,ℤ/8)12yes—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4H(5,𝔽2)11yescharacteristic
392C2×C3×C4H(5,𝔽2)11yescharacteristic
393#67—44no—
394#67—84no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3#12312yes—
412C2×Dic3#12312yes—
413D12—22no—
414D12—22no—
415D12—41no—
416D12—41no—
417D12—84no—
418C4×D3—41no—
419C4×D3—41no—
420C4×D3—41no—
421C4×D3—41no—
422C4×D3—84no—
423C4×D3—84no—
424C3×C8—41no—
425C3×C8—41no—
426C3⋊C8—41no—
427C3⋊C8—41no—
428C24—31no—
429C24—61no—
430C24D3×D411yescharacteristic
431C2×D4—32no—
432C2×D4—62no—
433C2×D4—62no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—122no—
440C2×D4—122no—
441C2×D4—121no—
442C2×D4—122no—
443C2×D4—122no—
444C2×D4—241no—
445C2×D4—241no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—122no—
469C22×C4—122no—
470C22×C4—122no—
471C22×C4—122no—
472C22×C4—122no—
473C22×C4—241no—
474C22×C4D3×D411yescharacteristic
475C22×C4C23×D311yescharacteristic
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—62no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—122no—
503C4⋊C4—122no—
504C4⋊C4—241no—
505C4⋊C4#23512yes—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—62no—
509C22⋊C4—62no—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—122no—
514C22⋊C4—122no—
515C22⋊C4—241no—
516C42—32no—
517C42—32no—
518C42—42no—
519C42—241no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3—41no—
524C22×C3—41no—
525C22×C3—41no—
526C22×C3—41no—
527C22×C3C2×#12311yescharacteristic
528C22×C3#54611yescharacteristic
529C22×C3#54611yescharacteristic
530C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
531C22×C3#57511yescharacteristic
532C22×C3#57711yescharacteristic
533C22×C3#57711yescharacteristic
534C2×D3—22no—
535C2×D3—22no—
536C2×D3—22no—
537C2×D3—41no—
538C2×D3—41no—
539C2×D3—41no—
540C2×D3—41no—
541C2×D3—42no—
542C2×D3—42no—
543C2×D3—42no—
544C2×D3—81no—
545C2×D3—81no—
546C2×D3—81no—
547C2×D3—81no—
548C3×C4—41no—
549C3×C4—42no—
550C3×C4—81no—
551C3×C4—81no—
552C3×C4—81no—
553C3×C4#52911yescharacteristic
554C3×C4#54011yescharacteristic
555C3×C4#55311yescharacteristic
556C3×C4#55511yescharacteristic
557Dic3—22no—
558Dic3—22no—
559Dic3—41no—
560Dic3—81no—
561Dic3—81no—
562Dic3—81no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—32no—
571C23—61no—
572C23—61no—
573C23—61no—
574C23—61no—
575C23—61no—
576C23—61no—
577C23—62no—
578C23—121no—
579C23—121no—
580C23—121no—
581C23—121no—
582C23—121no—
583C23—121no—
584C23C2×D3×D411yescharacteristic
585D4—62no—
586D4—62no—
587D4—124no—
588D4—121no—
589D4—121no—
590D4—244no—
591D4—244no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—32no—
597C2×C4—32no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—244no—
634C2×C4—244no—
635C2×C4C2×#23511yescharacteristic
636C2×C4#100411yescharacteristic
637C2×C4D3×#4111yescharacteristic
638C2×C4D3×#4111yescharacteristic
639C2×C4D3×#4111yescharacteristic
640C2×C4#100911yescharacteristic
641C8—41no—
642C8—41no—
643C8—121no—
644C8—121no—
645C2×C3—81no—
646C2×C3#242011yescharacteristic
647C2×C3#309411yescharacteristic
648C2×C3#309911yescharacteristic
649C2×C3#330611yescharacteristic
650C2×C3#330711yescharacteristic
651C2×C3#331011yescharacteristic
652C2×C3#332111yescharacteristic
653D3—42no—
654D3—82no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—62no—
663C22—62no—
664C22—62no—
665C22—121no—
666C22—121no—
667C22—121no—
668C22—121no—
669C22—122no—
670C22—122no—
671C22—122no—
672C22—241no—
673C22—241no—
674C22—241no—
675C22—241no—
676C22#549011yescharacteristic
677C22D3×#12311yescharacteristic
678C22#565411yescharacteristic
679C22#576811yescharacteristic
680C22#577011yescharacteristic
681C22D3×AGL(1,ℤ/8)11yescharacteristic
682C22#577811yescharacteristic
683C4—42no—
684C4—41no—
685C4—62no—
686C4—62no—
687C4—81no—
688C4—81no—
689C4—81no—
690C4—121no—
691C4—241no—
692C4—241no—
693C4—241no—
694C4#552811yescharacteristic
695C4#553211yescharacteristic
696C4#566111yescharacteristic
697C4#566311yescharacteristic
698C3#2498211yescharacteristic, complemented, Sylow
699C2—81no—
700C2—122no—
701C2—242no—
702C2#7406611yescharacteristic
703C2#7886011yescharacteristic
704C2#7889411yescharacteristic
705C2#8137311yescharacteristic
706C2#8138311yescharacteristic
707C2#8152111yescharacteristic
708C2#8154611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1825C2
socleC23×C3C22×D4
prime radical#73935C2
hypercenter#1825D3
hypocenterC3#24982
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
1. #82272C2semidirect
3. C3×#1825C2semidirect
4. #71586C2semidirect
5. #69935C2semidirect
6. #69917C2semidirect
7. #69507C2semidirect
8. #68893C2semidirect
9. #68880C2semidirect
10. #68871C2semidirect
11. #68849C2semidirect
12. #68832C2semidirect
13. #68823C2semidirect
14. #68796C2semidirect
15. #68782C2semidirect
71. #1825D3semidirect
698. C3#24982semidirect

Maximal subgroups

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

subgroupindex
#822722
#739352
C3×#18252
#715862
#699352
#699172
#695072
#688932
#688802
#688712
#688492
#688322
#688232
#687962
#687822
#249823