Finite Group Databasebeta

← #10840196

Subgroups — #10840196

Order 768.

756 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
1C3×#2536C211yesmaximal, characteristic
2#82337C211yesmaximal, characteristic
3#71695C211yesmaximal, characteristic, complemented
4#71694C211yesmaximal, characteristic, complemented
5#71693C211yesmaximal, characteristic, complemented
6#71693C211yesmaximal, characteristic, complemented
7#71631C211yesmaximal, characteristic, complemented
8#71626C211yesmaximal, characteristic, complemented
9#71615C211yesmaximal, characteristic
10#69548C211yesmaximal, characteristic, complemented
11#69540C211yesmaximal, characteristic, complemented
12#69519C211yesmaximal, characteristic, complemented
13#69507C211yesmaximal, characteristic, complemented
14#69503C211yesmaximal, characteristic, complemented
15#69503C211yesmaximal, characteristic, complemented
16#24980—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C3×#398—21no—
24C3×#394—21no—
25C3×#388—21no—
26C3×#388—21no—
27C3×#382C2211yescharacteristic
28C3×#380C2211yescharacteristic
29C3×#380C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34#5221—21no—
35#5212C2211yescharacteristic
36#5212C2211yescharacteristic
37C2×#830C2211yescharacteristic
38C2×#830C2211yescharacteristic
39C2×#830C2211yescharacteristic
40C2×#830C2211yescharacteristic
41Dic3×C4⋊C4—21no—
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4951—21no—
45#4951—21no—
46#4935C2211yescharacteristic
47C4×C4⋊Dic3C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49C2×#803C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2536D311yescharacteristic
61#2354—31no—
62#2077—31no—
63#2076—31no—
64#2036—31no—
65#2035—31no—
66#1955—31no—
67#1949—31no—
68#1873—31no—
69#1873—31no—
70#1836—31no—
71#1825—31no—
72#1825—31no—
73#1803—31no—
74#1763—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4D411yescharacteristic
77C23×C3×C4C2311yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C3×C22⋊Q8—22no—
80C3×C22⋊Q8—22no—
81C3×C22⋊Q8—41no—
82C3×C22⋊Q8—41no—
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C22⋊C4D411yescharacteristic
100C2×C3×C42—21no—
101C2×C3×C42C2311yescharacteristic
102C2×#217—21no—
103C2×#217—21no—
104C2×#217—21no—
105C2×#217—21no—
106#926—22no—
107#926—22no—
108#926—41no—
109#926—41no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127#835—22no—
128#835—22no—
129#835—41no—
130#835—41no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134#830—41no—
135#830—41no—
136#830—41no—
137#830—41no—
138#808—41no—
139#807—41no—
140C4⋊(C3⋊C8)—22no—
141#803—22no—
142C2×C22⋊Q8C2×D311yescharacteristic
143C2×C22⋊Q8C2×D311yescharacteristic
144C2×C4×Q8C2×D311yescharacteristic
145C2×C4×D4C2×D311yescharacteristic
146C2×#117—31no—
147C2×#117—31no—
148C2×#107—31no—
149C2×#106—31no—
150C2×C4⋊C8—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×C22⋊C8—31no—
156C2×C22⋊C8—31no—
157C2×C22⋊C8—31no—
158C2×C22⋊C8—31no—
159C2×#97—31no—
160#398—21no—
161#394—21no—
162#390—61no—
163#388—21no—
164#388—21no—
165#388—61no—
166#388—61no—
167#384—31no—
168#382C2×D311yescharacteristic
169#380—31no—
170#380—31no—
171#380C2×D311yescharacteristic
172#380C2×D311yescharacteristic
173C4×C4⋊C4—31no—
174C4×C4⋊C4—61no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#340—31no—
183#340—31no—
184#340—31no—
185C24×C3C2×D411yescharacteristic
186C24×C3C2×D411yescharacteristic
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8#4112yes—
193C2×C3×D4—22no—
194C2×C3×D4—41no—
195C2×C3×D4—41no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2411yescharacteristic
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—42no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4—42no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—42no—
234C3×C22⋊C4—42no—
235C3×C22⋊C4—42no—
236C3×C22⋊C4—42no—
237C3×C42—42no—
238C3×C42#4112yes—
239C3×C42#4112yes—
240#217—42no—
241#217—42no—
242#217—81no—
243#217—81no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—42no—
252C4⋊Dic3—81no—
253C4⋊Dic3#4112yes—
254C4×Dic3—42no—
255C4×Dic3—42no—
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—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C22×Q8C22×D311yescharacteristic
270C22×D4#6711yescharacteristic
271C23×C4C22×D311yescharacteristic
272C23×C4C22×D311yescharacteristic
273C22×C8—31no—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8—31no—
277C22⋊Q8—22no—
278C22⋊Q8—22no—
279C22⋊Q8—41no—
280C22⋊Q8—41no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×D4—22no—
284C4×D4—22no—
285#117—62no—
286#117—62no—
287#117—121no—
288#117—121no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4#6711yescharacteristic
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4#6711yescharacteristic
311C2×C22⋊C4#6711yescharacteristic
312C2×C42—21no—
313C2×C42—61no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#107—121no—
318#106—121no—
319C4⋊C8—62no—
320#103—62no—
321#103—62no—
322#103—121no—
323#103—121no—
324C22⋊C8—121no—
325C22⋊C8—121no—
326C22⋊C8—121no—
327C22⋊C8—121no—
328#97—62no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—61no—
334#95—61no—
335#95—61no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3C22×D411yescharacteristic
351C3×Q8—22no—
352C3×Q8—22no—
353C3×Q8—41no—
354C3×Q8—41no—
355C3×D4—44no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
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#12312yes—
387C2×C3×C4#12312yes—
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4H(5,𝔽2)11yescharacteristic
393C2×C3×C4H(5,𝔽2)11yescharacteristic
394C2×Dic3—22no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C3⋊C8—41no—
407C3⋊C8—41no—
408C3⋊C8—41no—
409C3⋊C8—41no—
410C24C2×#6711yescharacteristic
411C24C2×#6711yescharacteristic
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8#23712yes—
418C2×D4—22no—
419C2×D4—41no—
420C2×D4—41no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4C2×#6711yescharacteristic
442C22×C4C2×#6711yescharacteristic
443C22×C4C2×#6711yescharacteristic
444C22×C4C2×#6711yescharacteristic
445C22×C4C23×D311yescharacteristic
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—32no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—122no—
477C4⋊C4—241no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—241no—
487C22⋊C4—241no—
488C42—42no—
489C42—122no—
490C42—122no—
491C42—122no—
492C42#23712yes—
493C42#23712yes—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3—41no—
497C22×C3—41no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3—41no—
501C22×C3—41no—
502C22×C3—41no—
503C22×C3—41no—
504C22×C3—41no—
505C22×C3—41no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3C2×#12311yescharacteristic
509C22×C3#54611yescharacteristic
510C22×C3#54611yescharacteristic
511C22×C3#57511yescharacteristic
512C22×C3#57511yescharacteristic
513C22×C3#57711yescharacteristic
514C22×C3#57811yescharacteristic
515C3×C4—22no—
516C3×C4—22no—
517C3×C4—41no—
518C3×C4—41no—
519C3×C4—42no—
520C3×C4—81no—
521C3×C4—81no—
522C3×C4—81no—
523C3×C4#52911yescharacteristic
524C3×C4#54011yescharacteristic
525C3×C4#55311yescharacteristic
526C3×C4#55511yescharacteristic
527Dic3—42no—
528Dic3—81no—
529Dic3—81no—
530Dic3—81no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23C22×#6711yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550D4—44no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—62no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4#93512yes—
594C2×C4#93512yes—
595C2×C4C2×#23711yescharacteristic
596C2×C4C2×#23711yescharacteristic
597C2×C4C2×#23711yescharacteristic
598C2×C4#100411yescharacteristic
599C2×C4D3×#4111yescharacteristic
600C2×C4#100911yescharacteristic
601C8—121no—
602C8—121no—
603C8—121no—
604C8—121no—
605C2×C3—81no—
606C2×C3—81no—
607C2×C3#242011yescharacteristic
608C2×C3#309311yescharacteristic
609C2×C3#309711yescharacteristic
610C2×C3#331311yescharacteristic
611C2×C3#331511yescharacteristic
612C2×C3#331711yescharacteristic
613C2×C3#332111yescharacteristic
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22C2×#93511yescharacteristic
629C22#580611yescharacteristic
630C22#582211yescharacteristic
631C22#582211yescharacteristic
632C22#582911yescharacteristic
633C22#583711yescharacteristic
634C22#583811yescharacteristic
635C4—22no—
636C4—22no—
637C4—41no—
638C4—41no—
639C4—42no—
640C4—81no—
641C4—81no—
642C4—81no—
643C4—122no—
644C4—241no—
645C4—241no—
646C4—241no—
647C4#560011yescharacteristic
648C4#561611yescharacteristic
649C4#561811yescharacteristic
650C4#561911yescharacteristic
651C3#2498011yescharacteristic, complemented, Sylow
652C2—81no—
653C2—81no—
654C2#8264311yescharacteristic
655C2#8291511yescharacteristic
656C2#8339711yescharacteristic
657C2#8353711yescharacteristic
658C2#8353911yescharacteristic
659C2#8360011yescharacteristic
660C2#8360411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2536C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#2536D3
hypocenterC3#24980
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2536

Complements

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

NH ≅ G/Ntype
3. #71695C2semidirect
4. #71694C2semidirect
5. #71693C2semidirect
6. #71693C2semidirect
7. #71631C2semidirect
8. #71626C2semidirect
10. #69548C2semidirect
11. #69540C2semidirect
12. #69519C2semidirect
13. #69507C2semidirect
14. #69503C2semidirect
15. #69503C2semidirect
651. C3#24980semidirect

Maximal subgroups

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

subgroupindex
C3×#25362
#823372
#716952
#716942
#716932
#716312
#716262
#716152
#695482
#695402
#695192
#695072
#695032
#249803