Finite Group Databasebeta

← #10840194

Subgroups — #10840194

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×#2354C211yesmaximal, characteristic
2#82375C211yesmaximal, characteristic
3#71615C211yesmaximal, characteristic
4#71585C211yesmaximal, characteristic, complemented
5#71585C211yesmaximal, characteristic, complemented
6#71575C211yesmaximal, characteristic, complemented
7#71575C211yesmaximal, characteristic, complemented
8#69588C211yesmaximal, characteristic, complemented
9#69575C211yesmaximal, characteristic, complemented
10#69527C211yesmaximal, characteristic, complemented
11#69526C211yesmaximal, characteristic, complemented
12#69520C211yesmaximal, characteristic, complemented
13#69519C211yesmaximal, characteristic, complemented
14#69475C211yesmaximal, characteristic, complemented
15#69468C211yesmaximal, characteristic, complemented
16#24980—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C2×#924C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#390—21no—
24C3×#388—21no—
25C3×#388—21no—
26C3×#384C2211yescharacteristic
27C3×#380C2211yescharacteristic
28C3×#380C2211yescharacteristic
29C3×C4×C4⋊C4—21no—
30C3×C4×C4⋊C4C2211yescharacteristic
31#5222—21no—
32#5212C2211yescharacteristic
33#5212C2211yescharacteristic
34C2×#830C2211yescharacteristic
35C2×#830C2211yescharacteristic
36C2×#830C2211yescharacteristic
37C2×#830C2211yescharacteristic
38#4983—21no—
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4944—21no—
46#4944—21no—
47#4936C2211yescharacteristic
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#2536—31no—
61#2354D311yescharacteristic
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×D4D411yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C23×C3×C4C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C3×C4×D4—22no—
80C3×C4×D4—22no—
81C3×#117—22no—
82C3×#117—22no—
83C3×#117—41no—
84C3×#117—41no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4D411yescharacteristic
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C42—21no—
99C2×C3×C42—21no—
100C2×C3×C42—21no—
101C2×C3×C42C2311yescharacteristic
102C2×#217—21no—
103C2×#217—21no—
104C2×#217—21no—
105C2×#217—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109#924—22no—
110#924—22no—
111#924—41no—
112#924—41no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×C4×Dic3—21no—
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C4×Dic6—22no—
122C4×Dic6—22no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130#830—41no—
131#830—41no—
132#830—41no—
133#830—41no—
134#810—22no—
135#810—22no—
136#810—41no—
137#810—41no—
138#808—41no—
139#807—41no—
140C4⋊(C3⋊C8)—22no—
141#803—22no—
142C2×C22⋊Q8—31no—
143C2×C22⋊Q8—31no—
144C2×C4×Q8—31no—
145C2×C4×D4C2×D311yescharacteristic
146C2×#117C2×D311yescharacteristic
147C2×#117C2×D311yescharacteristic
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—61no—
161#394—61no—
162#390—21no—
163#388—21no—
164#388—21no—
165#388—61no—
166#388—61no—
167#384C2×D311yescharacteristic
168#382—31no—
169#380—31no—
170#380—31no—
171#380C2×D311yescharacteristic
172#380C2×D311yescharacteristic
173C4×C4⋊C4—21no—
174C4×C4⋊C4C2×D311yescharacteristic
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×D4—22no—
188C2×C3×D4—41no—
189C2×C3×D4—41no—
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×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×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2411yescharacteristic
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6#4112yes—
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—21no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4—81no—
231C3×C4⋊C4#4112yes—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—42no—
235C3×C22⋊C4—42no—
236C3×C22⋊C4—81no—
237C3×C22⋊C4—81no—
238C3×C42—42no—
239C3×C42—42no—
240C3×C42—42no—
241C3×C42#4112yes—
242C3×C42#4112yes—
243#217—42no—
244#217—42no—
245#217—42no—
246#217—42no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253Dic3⋊C4—42no—
254Dic3⋊C4—42no—
255Dic3⋊C4—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×Q8—31no—
270C22×D4#6711yescharacteristic
271C23×C4C22×D311yescharacteristic
272C23×C4C22×D311yescharacteristic
273C22×C8—31no—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8—31no—
277C22⋊Q8—62no—
278C22⋊Q8—62no—
279C22⋊Q8—121no—
280C22⋊Q8—121no—
281C4×Q8—62no—
282C4×Q8—62no—
283C4×D4—22no—
284C4×D4—22no—
285#117—22no—
286#117—22no—
287#117—41no—
288#117—41no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4#6711yescharacteristic
298C2×C4⋊C4C22×D311yescharacteristic
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—21no—
314C2×C42—21no—
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—61no—
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×D4—44no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
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#12312yes—
383C2×C3×C4#12312yes—
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4H(5,𝔽2)11yescharacteristic
389C2×C3×C4H(5,𝔽2)11yescharacteristic
390C2×Dic3—22no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402Dic6—22no—
403Dic6—22no—
404Dic6—41no—
405Dic6—41no—
406C3⋊C8—41no—
407C3⋊C8—41no—
408C3⋊C8—41no—
409C3⋊C8—41no—
410C24C2×#6711yescharacteristic
411C24C2×#6711yescharacteristic
412C2×Q8—32no—
413C2×Q8—61no—
414C2×Q8—61no—
415C2×Q8—61no—
416C2×Q8—61no—
417C2×Q8—61no—
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—21no—
466C4⋊C4—42no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—81no—
474C4⋊C4—122no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4#23712yes—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—81no—
483C22⋊C4—81no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C42—42no—
489C42—42no—
490C42—42no—
491C42—122no—
492C42#23512yes—
493C42#23512yes—
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—62no—
547Q8—62no—
548Q8—121no—
549Q8—121no—
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#92912yes—
594C2×C4#92912yes—
595C2×C4C2×#23511yescharacteristic
596C2×C4C2×#23511yescharacteristic
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×#92911yescharacteristic
629C22#580611yescharacteristic
630C22#582211yescharacteristic
631C22#582211yescharacteristic
632C22#582911yescharacteristic
633C22#583611yescharacteristic
634C22#583811yescharacteristic
635C4—22no—
636C4—22no—
637C4—42no—
638C4—41no—
639C4—41no—
640C4—81no—
641C4—81no—
642C4—81no—
643C4—122no—
644C4—241no—
645C4—241no—
646C4—241no—
647C4#555711yescharacteristic
648C4#555811yescharacteristic
649C4#556211yescharacteristic
650C4#556311yescharacteristic
651C3#2498011yescharacteristic, complemented, Sylow
652C2—81no—
653C2—81no—
654C2#8221611yescharacteristic
655C2#8240511yescharacteristic
656C2#8257311yescharacteristic
657C2#8351311yescharacteristic
658C2#8357311yescharacteristic
659C2#8365611yescharacteristic
660C2#8367411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2354C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#2354D3
hypocenterC3#24980
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2354

Complements

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

NH ≅ G/Ntype
4. #71585C2semidirect
5. #71585C2semidirect
6. #71575C2semidirect
7. #71575C2semidirect
8. #69588C2semidirect
9. #69575C2semidirect
10. #69527C2semidirect
11. #69526C2semidirect
12. #69520C2semidirect
13. #69519C2semidirect
14. #69475C2semidirect
15. #69468C2semidirect
651. C3#24980semidirect

Maximal subgroups

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

subgroupindex
C3×#23542
#823752
#716152
#715852
#715752
#695882
#695752
#695272
#695262
#695202
#695192
#694752
#694682
#249803