Finite Group Databasebeta

← #10840198

Subgroups — #10840198

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
1#82375C211yesmaximal, characteristic
2#82337C211yesmaximal, characteristic
3C3×#1955C211yesmaximal, characteristic
4#70485C211yesmaximal, characteristic, complemented
5#70455C211yesmaximal, characteristic, complemented
6#69940C211yesmaximal, characteristic, complemented
7#69939C211yesmaximal, characteristic, complemented
8#69937C211yesmaximal, characteristic, complemented
9#69936C211yesmaximal, characteristic, complemented
10#69931C211yesmaximal, characteristic, complemented
11#69931C211yesmaximal, characteristic, complemented
12#69926C211yesmaximal, characteristic, complemented
13#69926C211yesmaximal, characteristic, complemented
14#69815C211yesmaximal, characteristic, complemented
15#69813C211yesmaximal, characteristic, complemented
16#24980—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C2×#926C2211yescharacteristic
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#5222—21no—
30#5221—21no—
31#5212C2211yescharacteristic
32#5212C2211yescharacteristic
33#5212C2211yescharacteristic
34#5212C2211yescharacteristic
35C2×#818C2211yescharacteristic
36C2×#817C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41#4983—21no—
42Dic3×C4⋊C4—21no—
43#4951—21no—
44#4951—21no—
45#4944—21no—
46#4944—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#2536—31no—
61#2354—31no—
62#2077—31no—
63#2076—31no—
64#2036—31no—
65#2035—31no—
66#1955D311yescharacteristic
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
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C22⋊C4D411yescharacteristic
87C2×C3×C22⋊C4D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
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—22no—
99#926—41no—
100#926—41no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110#924—22no—
111#924—22no—
112#924—41no—
113#924—41no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×C4×Dic3—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#817—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—
147C2×#117—31no—
148C2×#107—31no—
149C2×#106—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#398—61no—
161#394—61no—
162#390—61no—
163#388—61no—
164#388—61no—
165#388—61no—
166#388—61no—
167#384—31no—
168#382—31no—
169#380—31no—
170#380—31no—
171#380—31no—
172#380—31no—
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×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×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—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6#4112yes—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
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—
233C3×C4⋊C4—22no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C42#4112yes—
237C3×C42#4112yes—
238#217—42no—
239#217—42no—
240#217—42no—
241#217—42no—
242#217—42no—
243#217—42no—
244#217—81no—
245#217—81no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
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—42no—
260C4⋊Dic3—81no—
261C4⋊Dic3#4112yes—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264Dic3⋊C4—42no—
265C4×Dic3—42no—
266C4×Dic3—42no—
267C4×Dic3—42no—
268C4×Dic3—42no—
269C22×Q8—31no—
270C22×D4D1211yescharacteristic
271C23×C4C22×D311yescharacteristic
272C23×C4C22×D311yescharacteristic
273C22×C8C22×D311yescharacteristic
274C22×C8C22×D311yescharacteristic
275C22×C8C22×D311yescharacteristic
276C22×C8C22×D311yescharacteristic
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—62no—
286#117—62no—
287#117—121no—
288#117—121no—
289C2×C4⋊C4—31no—
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—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4D1211yescharacteristic
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4D1211yescharacteristic
311C2×C22⋊C4D1211yescharacteristic
312C2×C42—61no—
313C2×C42—61no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#107—121no—
318#106—121no—
319C4⋊C8—22no—
320#103—62no—
321#103—62no—
322#103—121no—
323#103—121no—
324C22⋊C8—41no—
325C22⋊C8—41no—
326C22⋊C8—41no—
327C22⋊C8—41no—
328#97—22no—
329#95—61no—
330#95—61no—
331#95—61no—
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—21no—
361C2×C3×C4—21no—
362C2×C3×C4—41no—
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#12312yes—
371C2×C3×C4#12312yes—
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4H(5,𝔽2)11yescharacteristic
377C2×C3×C4H(5,𝔽2)11yescharacteristic
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—41no—
381C2×Dic3—41no—
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—
402Dic6—22no—
403Dic6—22no—
404Dic6—41no—
405Dic6—41no—
406C3×C8—41no—
407C3×C8—41no—
408C3×C8—41no—
409C3×C8—41no—
410C24C2×D1211yescharacteristic
411C24C2×D1211yescharacteristic
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—61no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4C2×D1211yescharacteristic
442C22×C4C2×D1211yescharacteristic
443C22×C4C2×D1211yescharacteristic
444C22×C4C2×D1211yescharacteristic
445C22×C4C23×D311yescharacteristic
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C4⋊C4—22no—
459C4⋊C4—32no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—122no—
474C4⋊C4—122no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4—241no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—122no—
481C22⋊C4—122no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—241no—
487C22⋊C4—241no—
488C42—122no—
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#52911yescharacteristic
520C3×C4#54011yescharacteristic
521C3×C4#55311yescharacteristic
522C3×C4#55511yescharacteristic
523Dic3—42no—
524Dic3—42no—
525Dic3—81no—
526Dic3—81no—
527Dic3—81no—
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×D1211yescharacteristic
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—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
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#88612yes—
594C2×C4#88612yes—
595C2×C4C2×#23711yescharacteristic
596C2×C4C2×#23711yescharacteristic
597C2×C4#100411yescharacteristic
598C2×C4#100411yescharacteristic
599C2×C4D3×#4111yescharacteristic
600C2×C4D3×#4111yescharacteristic
601C8—41no—
602C8—41no—
603C8—41no—
604C8—41no—
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×#88611yescharacteristic
629C22#555411yescharacteristic
630C22#555411yescharacteristic
631C22#574811yescharacteristic
632C22#574811yescharacteristic
633C22#575111yescharacteristic
634C22#575311yescharacteristic
635C4—22no—
636C4—22no—
637C4—41no—
638C4—41no—
639C4—122no—
640C4—122no—
641C4—241no—
642C4—241no—
643C4—241no—
644C4—241no—
645C4—241no—
646C4—241no—
647C4#561411yescharacteristic
648C4#561611yescharacteristic
649C4#568011yescharacteristic
650C4#571911yescharacteristic
651C3#2498011yescharacteristic, complemented, Sylow
652C2—81no—
653C2—81no—
654C2#7644011yescharacteristic
655C2#7862311yescharacteristic
656C2#7865211yescharacteristic
657C2#7950411yescharacteristic
658C2#7951211yescharacteristic
659C2#7951411yescharacteristic
660C2#7951611yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1955C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#1955D3
hypocenterC3#24980
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. #70485C2semidirect
5. #70455C2semidirect
6. #69940C2semidirect
7. #69939C2semidirect
8. #69937C2semidirect
9. #69936C2semidirect
10. #69931C2semidirect
11. #69931C2semidirect
12. #69926C2semidirect
13. #69926C2semidirect
14. #69815C2semidirect
15. #69813C2semidirect
651. C3#24980semidirect

Maximal subgroups

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

subgroupindex
#823752
#823372
C3×#19552
#704852
#704552
#699402
#699392
#699372
#699362
#699312
#699262
#698152
#698132
#249803