Finite Group Databasebeta

← #10876227

Subgroups — #10876227

Order 768.

842 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2314C211yesmaximal, characteristic, complemented
2#82270C211yesmaximal, characteristic
3#71584C211yesmaximal, characteristic, complemented
4#69589C211yesmaximal, characteristic, complemented
5#69585C211yesmaximal, characteristic, complemented
6#69561C211yesmaximal, characteristic, complemented
7#69558C211yesmaximal, characteristic, complemented
8#69522C211yesmaximal, characteristic
9#69522C211yesmaximal, characteristic
10#69505C211yesmaximal, characteristic, complemented
11#69504C211yesmaximal, characteristic, complemented
12#69465C211yesmaximal, characteristic, complemented
13#69464C211yesmaximal, characteristic, complemented
14#69463C211yesmaximal, characteristic, complemented
15#69462C211yesmaximal, characteristic, complemented
16#27593—31nomaximal, Sylow
17C22×#235C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C3×#388—21no—
21C3×#388—21no—
22C3×#387—21no—
23C3×#387—21no—
24C3×#384C2211yescharacteristic
25C3×#382C2211yescharacteristic
26C3×#380C2211yescharacteristic
27C3×#380C2211yescharacteristic
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C2×#830C2211yescharacteristic
31C2×#830C2211yescharacteristic
32#4987—21no—
33#4987—21no—
34C2×#810C2211yescharacteristic
35C2×#810C2211yescharacteristic
36C2×#810C2211yescharacteristic
37C2×#810C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#807C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4958—21no—
47#4958—21no—
48#4939C2211yescharacteristic
49#4939C2211yescharacteristic
50#4929C2211yescharacteristic
51#4929C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4524C2211yescharacteristic
57#4524C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2314D311yescharacteristic, complemented
61#2311—31no—
62#2097—31no—
63#2092—31no—
64#2072—31no—
65#2071—31no—
66#1928—31no—
67#1927—31no—
68#1926—31no—
69#1925—31no—
70#1853—31no—
71#1852—31no—
72#1818—31no—
73#1788—31no—
74#1788—31no—
75C23×C3×C4C2311yescharacteristic
76C22×#67—21no—
77C23×Dic3—21no—
78C2×#235—22no—
79C2×#235—22no—
80C2×#235—22no—
81C2×#235—22no—
82C2×#235—41no—
83C2×#235—41no—
84C22×D12C2311yescharacteristic
85C22×C4×D3—21no—
86C22×Dic6C2311yescharacteristic
87C3×#117—22no—
88C3×#117—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4D411yescharacteristic
100C2×C3×C22⋊C4D411yescharacteristic
101C2×C3×C42—21no—
102C2×C3×C42—21no—
103C2×C3×C42C2311yescharacteristic
104C2×C3×C42C2311yescharacteristic
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—41no—
111C2×C4⋊Dic3—41no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128#830—22no—
129#830—22no—
130#810—22no—
131#810—22no—
132#810—41no—
133#810—41no—
134#809—22no—
135#809—22no—
136#809—41no—
137#809—41no—
138#808—41no—
139#808—41no—
140#807—41no—
141#807—41no—
142C22×#41—31no—
143C2×#117C2×D311yescharacteristic
144C22×C4⋊C4—31no—
145C2×#107—31no—
146C2×#107—31no—
147C2×#106—31no—
148C2×#106—31no—
149C2×#103—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×C22⋊C8—31no—
158C2×C22⋊C8—31no—
159#391—31no—
160#391—31no—
161#390—31no—
162#390—31no—
163#388—21no—
164#388—21no—
165#387—21no—
166#387—21no—
167#386—61no—
168#386—61no—
169#384C2×D311yescharacteristic
170#382C2×D311yescharacteristic
171#380—61no—
172#380—61no—
173#380C2×D311yescharacteristic
174#380C2×D311yescharacteristic
175C4×C4⋊C4C2×D311yescharacteristic
176C4×C4⋊C4C2×D311yescharacteristic
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336—31no—
182#336—31no—
183#336—31no—
184#336—31no—
185C24×C3C2×D411yescharacteristic
186C23×D3—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
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×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4#4112yes—
201C22×C3×C4#4112yes—
202C22×C3×C4C2411yescharacteristic
203C2×#67—24no—
204C2×#67—42no—
205C2×#67—42no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—42no—
210C22×Dic3—81no—
211C22×Dic3—81no—
212#235—44no—
213#235—44no—
214#235—44no—
215#235—44no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12#4112yes—
222C2×C4×D3—24no—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6#4112yes—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—81no—
242C3×C4⋊C4—81no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—81no—
246C3×C22⋊C4—81no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—81no—
252C3×C42—81no—
253(C2×D3)⋊C4—81no—
254(C2×D3)⋊C4—81no—
255(C2×D3)⋊C4—81no—
256(C2×D3)⋊C4—81no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—42no—
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—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×#41—62no—
275C2×#41—62no—
276C2×#41—62no—
277C2×#41—62no—
278C2×#41—121no—
279C2×#41—121no—
280C22×Q8—31no—
281C22×D4—31no—
282C22×D4—61no—
283C23×C4—61no—
284C23×C4—61no—
285C23×C4C22×D311yescharacteristic
286C22×C8—31no—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8—31no—
290#117—22no—
291#117—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—121no—
298C2×C4⋊C4—121no—
299C2×C4⋊C4#6711yescharacteristic
300C2×C4⋊C4#6711yescharacteristic
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4#6711yescharacteristic
312C2×C22⋊C4#6711yescharacteristic
313C2×C42—21no—
314C2×C42—21no—
315C2×C42C22×D311yescharacteristic
316C2×C42C22×D311yescharacteristic
317#107—121no—
318#107—121no—
319#106—121no—
320#106—121no—
321#103—62no—
322#103—62no—
323#103—121no—
324#103—121no—
325#102—62no—
326#102—62no—
327#102—121no—
328#102—121no—
329C22⋊C8—62no—
330C22⋊C8—62no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3#12312yes—
348C23×C3C22×D411yescharacteristic
349C22×D3—22no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
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×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—48no—
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—84no—
404D12—22no—
405D12—22no—
406D12—41no—
407D12—41no—
408C4×D3—48no—
409Dic6—22no—
410Dic6—22no—
411Dic6—41no—
412Dic6—41no—
413C3⋊C8—41no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C3⋊C8—41no—
417C24—61no—
418C24C2×#6711yescharacteristic
419#41—124no—
420#41—124no—
421#41—124no—
422#41—124no—
423C2×Q8—32no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×D4—32no—
430C2×D4—64no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—122no—
437C2×D4—122no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—64no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—122no—
451C22×C4—122no—
452C22×C4—122no—
453C22×C4—241no—
454C22×C4—241no—
455C22×C4#23512yes—
456C22×C4#23512yes—
457C22×C4C2×#6711yescharacteristic
458C22×C4C2×#6711yescharacteristic
459C22×C4C2×#6711yescharacteristic
460C22×C4C2×#6711yescharacteristic
461C22×C4C2×#6711yescharacteristic
462C22×C4C23×D311yescharacteristic
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—122no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C22⋊C4—241no—
497C22⋊C4—241no—
498C22⋊C4—241no—
499C22⋊C4—241no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—81no—
505C42—81no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3C2×#12311yescharacteristic
514C22×C3#52511yescharacteristic
515C22×C3#53511yescharacteristic
516C22×C3#57611yescharacteristic
517C22×C3#57711yescharacteristic
518C22×C3#57711yescharacteristic
519C22×C3#57811yescharacteristic
520C2×D3—42no—
521C2×D3—42no—
522C2×D3—42no—
523C2×D3—81no—
524C2×D3—81no—
525C2×D3—81no—
526C2×D3—81no—
527C3×C4—22no—
528C3×C4—41no—
529C3×C4—41no—
530C3×C4—41no—
531C3×C4—41no—
532C3×C4—81no—
533C3×C4—81no—
534C3×C4—81no—
535C3×C4—81no—
536C3×C4#55111yescharacteristic
537C3×C4#55111yescharacteristic
538C3×C4#55111yescharacteristic
539C3×C4#55111yescharacteristic
540Dic3—42no—
541Dic3—81no—
542Dic3—81no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—62no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23#92912yes—
557C23C22×#6711yescharacteristic
558Q8—62no—
559Q8—62no—
560Q8—121no—
561Q8—121no—
562D4—62no—
563D4—62no—
564D4—128no—
565D4—121no—
566D4—121no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—62no—
598C2×C4—128no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—244no—
608C2×C4#92912yes—
609C2×C4C2×#23511yescharacteristic
610C2×C4C2×#23511yescharacteristic
611C2×C4D3×#4111yescharacteristic
612C2×C4D3×#4111yescharacteristic
613C2×C4#100911yescharacteristic
614C2×C4#100911yescharacteristic
615C8—121no—
616C8—121no—
617C8—121no—
618C8—121no—
619C2×C3—42no—
620C2×C3#248211yescharacteristic
621C2×C3#309811yescharacteristic
622C2×C3#309911yescharacteristic
623C2×C3#311911yescharacteristic
624C2×C3#312011yescharacteristic
625C2×C3#322411yescharacteristic
626C2×C3#322511yescharacteristic
627D3—82no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—122no—
636C22—122no—
637C22—122no—
638C22—241no—
639C22—241no—
640C22—241no—
641C22—241no—
642C22C2×#92911yescharacteristic
643C22#582811yescharacteristic
644C22#583011yescharacteristic
645C22#583511yescharacteristic
646C22#583611yescharacteristic
647C22#583711yescharacteristic
648C22#583811yescharacteristic
649C4—22no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—81no—
655C4—81no—
656C4—81no—
657C4—81no—
658C4—122no—
659C4—241no—
660C4—241no—
661C4#552711yescharacteristic
662C4#552711yescharacteristic
663C4#552711yescharacteristic
664C4#552711yescharacteristic
665C3#2759311yescharacteristic, complemented, Sylow
666C2—42no—
667C2—242no—
668C2#8228211yescharacteristic
669C2#8256711yescharacteristic
670C2#8257411yescharacteristic
671C2#8343411yescharacteristic
672C2#8344611yescharacteristic
673C2#8346311yescharacteristic
674C2#8346411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2314C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#2314D3
hypocenterC3#27593
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2314

Complements

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

NH ≅ G/Ntype
1. C3×#2314C2semidirect
3. #71584C2semidirect
4. #69589C2semidirect
5. #69585C2semidirect
6. #69561C2semidirect
7. #69558C2semidirect
10. #69505C2semidirect
11. #69504C2semidirect
12. #69465C2semidirect
13. #69464C2semidirect
14. #69463C2semidirect
15. #69462C2semidirect
60. #2314D3semidirect
665. C3#27593semidirect

Maximal subgroups

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

subgroupindex
C3×#23142
#822702
#715842
#695892
#695852
#695612
#695582
#695222
#695052
#695042
#694652
#694642
#694632
#694622
#275933