Finite Group Databasebeta

← #10840258

Subgroups — #10840258

Order 768.

778 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82376C211yesmaximal, characteristic
2#82335C211yesmaximal, characteristic
3C3×#1955C211yesmaximal, characteristic
4#70486C211yesmaximal, characteristic, complemented
5#70453C211yesmaximal, characteristic, complemented
6#69939C211yesmaximal, characteristic, complemented
7#69938C211yesmaximal, characteristic, complemented
8#69937C211yesmaximal, characteristic, complemented
9#69937C211yesmaximal, characteristic, complemented
10#69931C211yesmaximal, characteristic, complemented
11#69930C211yesmaximal, characteristic, complemented
12#69926C211yesmaximal, characteristic, complemented
13#69924C211yesmaximal, characteristic, complemented
14#69815C211yesmaximal, characteristic, complemented
15#69813C211yesmaximal, characteristic, complemented
16#24984—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C2×#924C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29#5221—21no—
30#5220—21no—
31#5212C2211yescharacteristic
32#5212C2211yescharacteristic
33#5212C2211yescharacteristic
34#5212C2211yescharacteristic
35C2×#818C2211yescharacteristic
36C2×#818C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41#4980—21no—
42#4972—21no—
43#4951—21no—
44#4948—21no—
45#4948—21no—
46Dic3×C22⋊C4—21no—
47#4936C2211yescharacteristic
48#4935C2211yescharacteristic
49C4×C4⋊Dic3C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2538—31no—
61#2352—31no—
62#2076—31no—
63#2075—31no—
64#2036—31no—
65#2036—31no—
66#1955D311yescharacteristic
67#1947—31no—
68#1873—31no—
69#1872—31no—
70#1836—31no—
71#1825—31no—
72#1823—31no—
73#1805—31no—
74#1763—31no—
75C22×C3×D4D411yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C23×C3×C4C2311yescharacteristic
78C23×Dic3—21no—
79C22×Dic6C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C22⋊C4D411yescharacteristic
88C2×C3×C22⋊C4D411yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97#926—22no—
98#926—41no—
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—41no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111#924—22no—
112#924—22no—
113#924—41no—
114#924—41no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C4×Dic6—22no—
122C4×Dic6—22no—
123C3×C4⋊C8—22no—
124C3×C22⋊C8—41no—
125C3×C22⋊C8—41no—
126C3×C22⋊C8—41no—
127C3×C22⋊C8—41no—
128C3×#97—22no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#818—41no—
137#818—41no—
138#816—22no—
139#816—22no—
140#816—41no—
141#816—41no—
142C2×C22⋊Q8—31no—
143C2×C22⋊Q8—31no—
144C2×C4×Q8—31no—
145C2×C4×D4C2×D311yescharacteristic
146C2×#117—31no—
147C22×C4⋊C4—31no—
148C2×#107—31no—
149C2×#107—31no—
150C2×C4⋊C8C2×D311yescharacteristic
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×C22⋊C8C2×D311yescharacteristic
156C2×C22⋊C8C2×D311yescharacteristic
157C2×C22⋊C8C2×D311yescharacteristic
158C2×C22⋊C8C2×D311yescharacteristic
159C2×#97C2×D311yescharacteristic
160#396—61no—
161#396—61no—
162#390—61no—
163#389—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—31no—
173C4×C4⋊C4—31no—
174C4×C22⋊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×D4—22no—
188C2×C3×D4—22no—
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×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—42no—
216C22×Dic3—42no—
217C22×Dic3—81no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6#4112yes—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C3×C4⋊C4—22no—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—22no—
239C3×C42#4112yes—
240C3×C42#4112yes—
241#217—42no—
242#217—42no—
243#217—42no—
244#217—42no—
245#217—81no—
246#217—81no—
247#217—81no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—42no—
262C4⋊Dic3—42no—
263C4⋊Dic3—81no—
264C4⋊Dic3#4112yes—
265Dic3⋊C4—42no—
266Dic3⋊C4—42no—
267Dic3⋊C4—42no—
268C4×Dic3—42no—
269C4×Dic3—42no—
270C4×Dic3—81no—
271C22×Q8—31no—
272C22×D4D1211yescharacteristic
273C23×C4—61no—
274C23×C4C22×D311yescharacteristic
275C23×C4C22×D311yescharacteristic
276C22×C8C22×D311yescharacteristic
277C22×C8C22×D311yescharacteristic
278C22×C8C22×D311yescharacteristic
279C22×C8C22×D311yescharacteristic
280C22⋊Q8—62no—
281C22⋊Q8—62no—
282C22⋊Q8—121no—
283C22⋊Q8—121no—
284C4×Q8—62no—
285C4×Q8—62no—
286C4×D4—22no—
287C4×D4—22no—
288#117—62no—
289#117—121no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—62no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—121no—
305C2×C4⋊C4D1211yescharacteristic
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4D1211yescharacteristic
314C2×C22⋊C4D1211yescharacteristic
315C2×C42—61no—
316C2×C42—61no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319#107—121no—
320#107—121no—
321C4⋊C8—22no—
322#103—62no—
323#103—62no—
324#103—121no—
325#103—121no—
326C22⋊C8—41no—
327C22⋊C8—41no—
328C22⋊C8—41no—
329C22⋊C8—41no—
330#97—22no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
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×C3—21no—
351C23×C3#13712yes—
352C23×C3C22×D411yescharacteristic
353C3×D4—44no—
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—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
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#12312yes—
373C2×C3×C4#12312yes—
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4H(5,𝔽2)11yescharacteristic
379C2×C3×C4H(5,𝔽2)11yescharacteristic
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
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—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—84no—
405Dic6—22no—
406Dic6—22no—
407Dic6—41no—
408Dic6—41no—
409C3×C8—41no—
410C3×C8—41no—
411C3×C8—41no—
412C3×C8—41no—
413C24C2×D1211yescharacteristic
414C24C2×D1211yescharacteristic
415C2×Q8—32no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—41no—
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—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4—241no—
447C22×C4C2×D1211yescharacteristic
448C22×C4C2×D1211yescharacteristic
449C22×C4C2×D1211yescharacteristic
450C22×C4C2×D1211yescharacteristic
451C22×C4C23×D311yescharacteristic
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C4⋊C4—22no—
465C4⋊C4—32no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—122no—
480C4⋊C4—122no—
481C4⋊C4—122no—
482C4⋊C4—122no—
483C4⋊C4—122no—
484C4⋊C4—241no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—122no—
488C22⋊C4—122no—
489C22⋊C4—122no—
490C22⋊C4—122no—
491C22⋊C4—241no—
492C22⋊C4—241no—
493C22⋊C4—241no—
494C42—122no—
495C42—122no—
496C42—241no—
497C42#23712yes—
498C42#23712yes—
499C22×C3—22no—
500C22×C3—22no—
501C22×C3—22no—
502C22×C3—41no—
503C22×C3—41no—
504C22×C3—41no—
505C22×C3—41no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3C2×#12311yescharacteristic
514C22×C3#54611yescharacteristic
515C22×C3#54611yescharacteristic
516C22×C3C2×#13711yescharacteristic
517C22×C3#57511yescharacteristic
518C22×C3#57611yescharacteristic
519C22×C3#57811yescharacteristic
520C3×C4—22no—
521C3×C4—22no—
522C3×C4—41no—
523C3×C4—41no—
524C3×C4#52911yescharacteristic
525C3×C4#54011yescharacteristic
526C3×C4#55311yescharacteristic
527C3×C4#55511yescharacteristic
528Dic3—42no—
529Dic3—42no—
530Dic3—81no—
531Dic3—81no—
532Dic3—81no—
533Dic3—81no—
534Dic3—81no—
535Dic3—81no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23#90912yes—
550C23C22×D1211yescharacteristic
551Q8—62no—
552Q8—62no—
553Q8—121no—
554Q8—121no—
555D4—44no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
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—62no—
575C2×C4—62no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
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—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—244no—
599C2×C4#88612yes—
600C2×C4#88612yes—
601C2×C4C2×#23711yescharacteristic
602C2×C4C2×#23711yescharacteristic
603C2×C4#100411yescharacteristic
604C2×C4#100411yescharacteristic
605C2×C4D3×#4111yescharacteristic
606C2×C4D3×#4111yescharacteristic
607C8—41no—
608C8—41no—
609C8—41no—
610C8—41no—
611C2×C3—42no—
612C2×C3—81no—
613C2×C3#242011yescharacteristic
614C2×C3#309511yescharacteristic
615C2×C3#309811yescharacteristic
616C2×C3#330911yescharacteristic
617C2×C3#331211yescharacteristic
618C2×C3#331311yescharacteristic
619C2×C3#331411yescharacteristic
620C22—22no—
621C22—22no—
622C22—22no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22C2×#88611yescharacteristic
635C22#555411yescharacteristic
636C22#555411yescharacteristic
637C22C2×#90911yescharacteristic
638C22#574811yescharacteristic
639C22#575211yescharacteristic
640C22#575311yescharacteristic
641C4—22no—
642C4—22no—
643C4—41no—
644C4—41no—
645C4—122no—
646C4—122no—
647C4—241no—
648C4—241no—
649C4—241no—
650C4—241no—
651C4—241no—
652C4—241no—
653C4#561411yescharacteristic
654C4#561611yescharacteristic
655C4#568011yescharacteristic
656C4#571911yescharacteristic
657C3#2498411yescharacteristic, complemented, Sylow
658C2—42no—
659C2—81no—
660C2#7644011yescharacteristic
661C2#7862511yescharacteristic
662C2#7865311yescharacteristic
663C2#7950311yescharacteristic
664C2#7951011yescharacteristic
665C2#7951111yescharacteristic
666C2#7951211yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1955C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#1955D3
hypocenterC3#24984
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1955

Complements

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

NH ≅ G/Ntype
4. #70486C2semidirect
5. #70453C2semidirect
6. #69939C2semidirect
7. #69938C2semidirect
8. #69937C2semidirect
9. #69937C2semidirect
10. #69931C2semidirect
11. #69930C2semidirect
12. #69926C2semidirect
13. #69924C2semidirect
14. #69815C2semidirect
15. #69813C2semidirect
657. C3#24984semidirect

Maximal subgroups

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

subgroupindex
#823762
#823352
C3×#19552
#704862
#704532
#699392
#699382
#699372
#699312
#699302
#699262
#699242
#698152
#698132
#249843