Finite Group Databasebeta

← #10844592

Subgroups — #10844592

Order 768.

822 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2365C211yesmaximal, characteristic, complemented
2C2×#5091C211yesmaximal, characteristic, complemented
3C2×#5089C211yesmaximal, characteristic
4#73710C211yesmaximal, characteristic
5#69635C211yesmaximal, characteristic, complemented
6#69633C211yesmaximal, characteristic, complemented
7#69630C211yesmaximal, characteristic
8#69628C211yesmaximal, characteristic
9#69488C211yesmaximal, characteristic
10#69488C211yesmaximal, characteristic
11#69486C211yesmaximal, characteristic, complemented
12#69485C211yesmaximal, characteristic, complemented
13#69475C211yesmaximal, characteristic, complemented
14#69465C211yesmaximal, characteristic, complemented
15#69462C211yesmaximal, characteristic
16#25286—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8—21no—
19C2×#875C2211yescharacteristic
20C2×#875C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#391—21no—
24C3×#389C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×#387—21no—
27C3×#387—21no—
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×C4×C4⋊C4C2211yescharacteristic
32#5091—24no—
33#5089—24no—
34#4994—21no—
35#4994—21no—
36C2×#810C2211yescharacteristic
37C2×#810C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#809C2211yescharacteristic
48C2×#807C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4939C2211yescharacteristic
51#4929C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C2×#803C2211yescharacteristic
56#4670—21no—
57#4670—21no—
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#489—31no—
63C2×#477—31no—
64#2545—31no—
65#2365D311yescharacteristic, complemented
66#2068—31no—
67#2067—31no—
68#2043—31no—
69#2042—31no—
70#1983—31no—
71#1983—31no—
72#1981—31no—
73#1940—31no—
74#1928—31no—
75#1925—31no—
76#1763—31no—
77C22×C3×Q8—21no—
78C22×D12C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C3×C4⋊Q8—22no—
82C3×C4⋊Q8D414yes—
83C3×C4×Q8—44no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C42—21no—
94C2×C3×C42—21no—
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114#875—22no—
115#875—22no—
116#875—22no—
117#875—41no—
118#869—22no—
119#869—41no—
120C4×Dic6—22no—
121C4×Dic6—22no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95D411yescharacteristic
128C3×#95D411yescharacteristic
129C3⋊#95—21no—
130C3⋊#95—21no—
131#810—22no—
132#810—22no—
133#810—22no—
134#810—22no—
135#810—22no—
136#810—22no—
137#810—22no—
138#810—41no—
139#809—22no—
140#809—22no—
141#809—22no—
142#809—41no—
143#807—22no—
144#807—22no—
145C4⋊(C3⋊C8)—22no—
146C4⋊(C3⋊C8)—41no—
147C4⋊(C3⋊C8)—41no—
148#803—22no—
149C2×C4⋊Q8—31no—
150C2×C4⋊Q8C2×D311yescharacteristic
151C2×#124—31no—
152C2×#124—31no—
153C2×C4×Q8—21no—
154C2×C4×Q8—31no—
155#489—64no—
156#477—64no—
157C2×#107—31no—
158C2×#107—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#102—31no—
171C2×#102—31no—
172C2×#102—31no—
173C2×#102—31no—
174C2×#97—31no—
175#396—61no—
176#396—61no—
177#391—21no—
178#391—31no—
179#390—31no—
180#389C2×D311yescharacteristic
181#389C2×D311yescharacteristic
182#388—61no—
183#388—61no—
184#387—21no—
185#387—21no—
186C4×C4⋊C4C2×D311yescharacteristic
187C4×C4⋊C4C2×D311yescharacteristic
188C4×C4⋊C4C2×D311yescharacteristic
189C4×C4⋊C4C2×D311yescharacteristic
190#340—31no—
191#340—31no—
192#336—31no—
193#336—31no—
194C23×D3—21no—
195C2×C3×Q8—24no—
196C2×C3×Q8—42no—
197C2×C3×Q8—42no—
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×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6#4112yes—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—41no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4C2×D412yes—
241C3×C4⋊C4C2×D412yes—
242C3×C4⋊C4#4112yes—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—42no—
248C3×C42—42no—
249C3×C42—42no—
250C3×C42C2×D412yes—
251C3×C42#4112yes—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—42no—
261C4⋊Dic3—42no—
262Dic3⋊C4—42no—
263C4×Dic3—42no—
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×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C22×Q8—21no—
277C22×Q8—31no—
278C22×Q8—31no—
279C22×D4—31no—
280C22×C8—31no—
281C22×C8—31no—
282C22×C8—31no—
283C22×C8—31no—
284C4⋊Q8—22no—
285C4⋊Q8—62no—
286C4⋊Q8—121no—
287C4⋊Q8D1214yes—
288#124—62no—
289#124—62no—
290#124—62no—
291#124—121no—
292C4×Q8—44no—
293C4×Q8—62no—
294C4×Q8—62no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4#6711yescharacteristic
303C2×C4⋊C4#6711yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319C2×C42C22×D311yescharacteristic
320C2×C42C22×D311yescharacteristic
321#107—62no—
322#107—62no—
323C4⋊C8—62no—
324C4⋊C8—121no—
325C4⋊C8—121no—
326#103—62no—
327#103—62no—
328#103—62no—
329#103—62no—
330#103—62no—
331#103—62no—
332#103—62no—
333#103—121no—
334#102—62no—
335#102—62no—
336#102—62no—
337#102—121no—
338#97—62no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—61no—
345#95—61no—
346#95#6711yescharacteristic
347#95#6711yescharacteristic
348C23×C3C22×D411yescharacteristic
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C3×Q8—48no—
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—22no—
364C2×C3×C4—22no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
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#12112yes—
385C2×C3×C4#12112yes—
386C2×C3×C4AGL(1,ℤ/8)12yes—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4H(5,𝔽2)11yescharacteristic
392C2×C3×C42−1+411yescharacteristic
393C2×Dic3—22no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402D12—41no—
403D12—41no—
404D12—41no—
405D12—41no—
406Dic6—22no—
407Dic6—22no—
408Dic6—41no—
409Dic6—41no—
410Dic6—41no—
411Dic6—41no—
412Dic6—41no—
413Dic6—41no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C3⋊C8—41no—
417C3⋊C8—41no—
418C24—61no—
419C2×Q8—24no—
420C2×Q8—32no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—61no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4C2×#6711yescharacteristic
448C22×C4C2×#6711yescharacteristic
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6711yescharacteristic
453C22×C4C23×D311yescharacteristic
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—41no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—122no—
479C4⋊C4—122no—
480C4⋊C4—122no—
481C4⋊C4C2×D1212yes—
482C4⋊C4C2×D1212yes—
483C4⋊C4#23912yes—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C22⋊C4—122no—
489C22⋊C4—122no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—42no—
495C42—42no—
496C42—42no—
497C42—122no—
498C42C2×D1212yes—
499C42#23512yes—
500C22×C3C2×#12111yescharacteristic
501C22×C3#54611yescharacteristic
502C22×C3#54811yescharacteristic
503C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
504C22×C3#57511yescharacteristic
505C22×C3#57811yescharacteristic
506C22×C3#57811yescharacteristic
507C2×D3—81no—
508C2×D3—81no—
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C2×D3—81no—
513C2×D3—81no—
514C3×C4—22no—
515C3×C4—22no—
516C3×C4—22no—
517C3×C4—42no—
518C3×C4—42no—
519C3×C4—42no—
520C3×C4—41no—
521C3×C4—41no—
522C3×C4—41no—
523C3×C4—81no—
524C3×C4#52911yescharacteristic
525C3×C4#54211yescharacteristic
526C3×C4#54811yescharacteristic
527C3×C4#55011yescharacteristic
528Dic3—42no—
529Dic3—81no—
530Dic3—81no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23—121no—
538C23C22×#6711yescharacteristic
539Q8—48no—
540Q8—62no—
541Q8—62no—
542Q8—121no—
543Q8—121no—
544Q8—121no—
545Q8—121no—
546Q8—121no—
547Q8—121no—
548D4—121no—
549D4—121no—
550D4—121no—
551D4—121no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
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—62no—
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#93012yes—
589C2×C4#93012yes—
590C2×C4#93212yes—
591C2×C4C22×D1211yescharacteristic
592C2×C4C2×#23511yescharacteristic
593C2×C4C2×#23911yescharacteristic
594C2×C4#100711yescharacteristic
595C2×C4D3×#4111yescharacteristic
596C2×C4#100911yescharacteristic
597C8—121no—
598C8—121no—
599C8—121no—
600C8—121no—
601C2×C3#242611yescharacteristic
602C2×C3#303611yescharacteristic
603C2×C3#304411yescharacteristic
604C2×C3#330811yescharacteristic
605C2×C3#331311yescharacteristic
606C2×C3#336711yescharacteristic
607C2×C3#337711yescharacteristic
608D3—161no—
609C22—241no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22—241no—
615C22—241no—
616C22C2×#93011yescharacteristic
617C22C2×#93211yescharacteristic
618C22#581711yescharacteristic
619C22#582211yescharacteristic
620C22#582911yescharacteristic
621C22#583811yescharacteristic
622C22#583811yescharacteristic
623C4—22no—
624C4—22no—
625C4—22no—
626C4—41no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—41no—
631C4—41no—
632C4—81no—
633C4—122no—
634C4—241no—
635C4—241no—
636C4#557611yescharacteristic
637C4#557711yescharacteristic
638C4#558011yescharacteristic
639C4#558211yescharacteristic
640C3#2528611yescharacteristic, complemented, Sylow
641C2—481no—
642C2#8230711yescharacteristic
643C2#8249511yescharacteristic
644C2#8255411yescharacteristic
645C2#8353211yescharacteristic
646C2#8356311yescharacteristic
647C2#8367411yescharacteristic
648C2#8370011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2365C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2365D3
hypocenterC3#25286
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2365

Complements

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

NH ≅ G/Ntype
1. C3×#2365C2semidirect
2. C2×#5091C2semidirect
5. #69635C2semidirect
6. #69633C2semidirect
11. #69486C2semidirect
12. #69485C2semidirect
13. #69475C2semidirect
14. #69465C2semidirect
65. #2365D3semidirect
640. C3#25286semidirect

Maximal subgroups

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

subgroupindex
C3×#23652
C2×#50912
C2×#50892
#737102
#696352
#696332
#696302
#696282
#694882
#694862
#694852
#694752
#694652
#694622
#252863