Finite Group Databasebeta

← #10844607

Subgroups — #10844607

Order 768.

794 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2369C211yesmaximal, characteristic, complemented
2C2×#5094C211yesmaximal, characteristic, complemented
3C2×#5089C211yesmaximal, characteristic
4#73710C211yesmaximal, characteristic
5#69634C211yesmaximal, characteristic, complemented
6#69633C211yesmaximal, characteristic, complemented
7#69629C211yesmaximal, characteristic
8#69628C211yesmaximal, characteristic
9#69488C211yesmaximal, characteristic
10#69488C211yesmaximal, characteristic
11#69486C211yesmaximal, characteristic, complemented
12#69485C211yesmaximal, characteristic, complemented
13#69475C211yesmaximal, characteristic, complemented
14#69464C211yesmaximal, characteristic, complemented
15#69463C211yesmaximal, characteristic
16#25287—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×#875C2211yescharacteristic
19C2×#875C2211yescharacteristic
20C2×#869C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C3×#391—21no—
23C3×#391—21no—
24C3×#389—21no—
25C3×#389C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×C4×C4⋊C4—21no—
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×C4×C4⋊C4C2211yescharacteristic
32#5094—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×#808C2211yescharacteristic
49C2×#808C2211yescharacteristic
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×#475—31no—
64#2545—31no—
65#2369D311yescharacteristic, complemented
66#2066—31no—
67#2065—31no—
68#2043—31no—
69#2042—31no—
70#1983—31no—
71#1983—31no—
72#1981—31no—
73#1940—31no—
74#1927—31no—
75#1926—31no—
76#1763—31no—
77C22×C3×Q8—21no—
78C22×D12C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C3×C4⋊Q8—22no—
82C3×C4⋊Q8D414yes—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C42—21no—
93C2×C3×C42—21no—
94C2×C3×C42—21no—
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×(C2×D3)⋊C4—21no—
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×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113#875—22no—
114#875—22no—
115#875—22no—
116#875—41no—
117#869—22no—
118#869—41no—
119C4×Dic6—22no—
120C4×Dic6—22no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95D411yescharacteristic
127C3×#95D411yescharacteristic
128C3⋊#95—21no—
129C3⋊#95—21no—
130#810—22no—
131#810—22no—
132#810—22no—
133#810—22no—
134#810—22no—
135#810—22no—
136#810—22no—
137#810—41no—
138#809—22no—
139#809—22no—
140#809—22no—
141#809—41no—
142#808—22no—
143#808—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—41no—
146C4⋊(C3⋊C8)—41no—
147#803—22no—
148C2×C4⋊Q8—31no—
149C2×C4⋊Q8C2×D311yescharacteristic
150C2×#124—31no—
151C2×#124—31no—
152C2×C4×Q8—31no—
153#489—64no—
154#475—64no—
155C2×#106—31no—
156C2×#106—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102—31no—
171C2×#102—31no—
172C2×#97—31no—
173#396—61no—
174#396—61no—
175#391—21no—
176#391—21no—
177#391—31no—
178#390—31no—
179#389—21no—
180#389C2×D311yescharacteristic
181#389C2×D311yescharacteristic
182#388—61no—
183#388—61no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4C2×D311yescharacteristic
186C4×C4⋊C4C2×D311yescharacteristic
187C4×C4⋊C4C2×D311yescharacteristic
188C4×C4⋊C4C2×D311yescharacteristic
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C23×D3—21no—
194C2×C3×Q8—24no—
195C2×C3×Q8—81no—
196C2×C3×Q8—81no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×Dic6—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#4112yes—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—41no—
237C3×C4⋊C4—81no—
238C3×C4⋊C4—81no—
239C3×C4⋊C4C2×D412yes—
240C3×C4⋊C4C2×D412yes—
241C3×C4⋊C4#4112yes—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—42no—
247C3×C42—81no—
248C3×C42—81no—
249C3×C42C2×D412yes—
250C3×C42#4112yes—
251(C2×D3)⋊C4—42no—
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—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—42no—
260C4⋊Dic3—42no—
261Dic3⋊C4—42no—
262C4×Dic3—42no—
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×C3⋊C8—21no—
275C22×Q8—21no—
276C22×Q8—31no—
277C22×Q8—31no—
278C22×D4—31no—
279C22×C8—31no—
280C22×C8—31no—
281C22×C8—31no—
282C22×C8—31no—
283C4⋊Q8—22no—
284C4⋊Q8—62no—
285C4⋊Q8—121no—
286C4⋊Q8D1214yes—
287#124—62no—
288#124—62no—
289#124—62no—
290#124—121no—
291C4×Q8—62no—
292C4×Q8—62no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4#6711yescharacteristic
301C2×C4⋊C4#6711yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
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×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317C2×C42C22×D311yescharacteristic
318C2×C42C22×D311yescharacteristic
319#106—62no—
320#106—62no—
321C4⋊C8—62no—
322C4⋊C8—121no—
323C4⋊C8—121no—
324#103—62no—
325#103—62no—
326#103—62no—
327#103—62no—
328#103—62no—
329#103—62no—
330#103—62no—
331#103—121no—
332#102—62no—
333#102—62no—
334#102—62no—
335#102—121no—
336#97—62no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—61no—
343#95—61no—
344#95#6711yescharacteristic
345#95#6711yescharacteristic
346C23×C3C22×D411yescharacteristic
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C3×Q8—84no—
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—22no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
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—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#12112yes—
383C2×C3×C4#12112yes—
384C2×C3×C4#13712yes—
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4H(5,𝔽2)11yescharacteristic
390C2×C3×C42−1+411yescharacteristic
391C2×Dic3—22no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400D12—41no—
401D12—41no—
402D12—41no—
403D12—41no—
404Dic6—22no—
405Dic6—22no—
406Dic6—41no—
407Dic6—41no—
408Dic6—41no—
409Dic6—41no—
410Dic6—41no—
411Dic6—41no—
412C3⋊C8—41no—
413C3⋊C8—41no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C24—61no—
417C2×Q8—24no—
418C2×Q8—32no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—81no—
431C2×Q8—81no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4C2×#6711yescharacteristic
446C22×C4C2×#6711yescharacteristic
447C22×C4C2×#6711yescharacteristic
448C22×C4C2×#6711yescharacteristic
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C23×D311yescharacteristic
452C2×C8—61no—
453C2×C8—61no—
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—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—41no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4—122no—
479C4⋊C4C2×D1212yes—
480C4⋊C4C2×D1212yes—
481C4⋊C4#23912yes—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—42no—
493C42—81no—
494C42—81no—
495C42—122no—
496C42C2×D1212yes—
497C42#23512yes—
498C22×C3C2×#12111yescharacteristic
499C22×C3#54611yescharacteristic
500C22×C3#54811yescharacteristic
501C22×C3C2×#13711yescharacteristic
502C22×C3#57511yescharacteristic
503C22×C3#57711yescharacteristic
504C22×C3#57711yescharacteristic
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C2×D3—81no—
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C3×C4—22no—
513C3×C4—22no—
514C3×C4—22no—
515C3×C4—42no—
516C3×C4—42no—
517C3×C4—42no—
518C3×C4—41no—
519C3×C4—41no—
520C3×C4—41no—
521C3×C4—81no—
522C3×C4#52911yescharacteristic
523C3×C4#54211yescharacteristic
524C3×C4#54811yescharacteristic
525C3×C4#55011yescharacteristic
526Dic3—42no—
527Dic3—81no—
528Dic3—81no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23C22×#6711yescharacteristic
537Q8—62no—
538Q8—62no—
539Q8—84no—
540Q8—121no—
541Q8—121no—
542Q8—121no—
543Q8—121no—
544Q8—121no—
545Q8—121no—
546D4—121no—
547D4—121no—
548D4—121no—
549D4—121no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
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—62no—
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#93012yes—
587C2×C4#93012yes—
588C2×C4#94212yes—
589C2×C4C22×D1211yescharacteristic
590C2×C4C2×#23511yescharacteristic
591C2×C4C2×#23911yescharacteristic
592C2×C4#100711yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4#100911yescharacteristic
595C8—121no—
596C8—121no—
597C8—121no—
598C8—121no—
599C2×C3#242611yescharacteristic
600C2×C3#303711yescharacteristic
601C2×C3#304211yescharacteristic
602C2×C3#331611yescharacteristic
603C2×C3#332111yescharacteristic
604C2×C3#336911yescharacteristic
605C2×C3#337611yescharacteristic
606D3—161no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22C2×#93011yescharacteristic
615C22C2×#94211yescharacteristic
616C22#581711yescharacteristic
617C22#582211yescharacteristic
618C22#582911yescharacteristic
619C22#583611yescharacteristic
620C22#583611yescharacteristic
621C4—22no—
622C4—22no—
623C4—22no—
624C4—42no—
625C4—42no—
626C4—41no—
627C4—42no—
628C4—41no—
629C4—41no—
630C4—81no—
631C4—122no—
632C4—241no—
633C4—241no—
634C4#557611yescharacteristic
635C4#557711yescharacteristic
636C4#558011yescharacteristic
637C4#558211yescharacteristic
638C3#2528711yescharacteristic, complemented, Sylow
639C2—481no—
640C2#8244111yescharacteristic
641C2#8249511yescharacteristic
642C2#8254411yescharacteristic
643C2#8364111yescharacteristic
644C2#8364511yescharacteristic
645C2#8365611yescharacteristic
646C2#8370411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2369C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2369D3
hypocenterC3#25287
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2369

Complements

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

NH ≅ G/Ntype
1. C3×#2369C2semidirect
2. C2×#5094C2semidirect
5. #69634C2semidirect
6. #69633C2semidirect
11. #69486C2semidirect
12. #69485C2semidirect
13. #69475C2semidirect
14. #69464C2semidirect
65. #2369D3semidirect
638. C3#25287semidirect

Maximal subgroups

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

subgroupindex
C3×#23692
C2×#50942
C2×#50892
#737102
#696342
#696332
#696292
#696282
#694882
#694862
#694852
#694752
#694642
#694632
#252873