Finite Group Databasebeta

← #10844618

Subgroups — #10844618

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
1#83237C211yesmaximal, characteristic
2C2×#4794C211yesmaximal, characteristic
3C2×#4791C211yesmaximal, characteristic, complemented
4#73657C211yesmaximal, characteristic, complemented
5C3×#1981C211yesmaximal, characteristic, complemented
6#71781C211yesmaximal, characteristic, complemented
7#70483C211yesmaximal, characteristic
8#70452C211yesmaximal, characteristic, complemented
9#70261C211yesmaximal, characteristic, complemented
10#70245C211yesmaximal, characteristic
11#70230C211yesmaximal, characteristic
12#70230C211yesmaximal, characteristic
13#70223C211yesmaximal, characteristic, complemented
14#70220C211yesmaximal, characteristic, complemented
15#69463C211yesmaximal, characteristic
16#25287—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C2×#947C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×#888C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×#817C2211yescharacteristic
36C2×#815C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39#4994—21no—
40#4994—21no—
41#4990—21no—
42#4990—21no—
43#4984C2211yescharacteristic
44Dic3×C4⋊C4C2211yescharacteristic
45Dic3×C4⋊C4C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#809C2211yescharacteristic
48C2×#808C2211yescharacteristic
49#4939C2211yescharacteristic
50C4×Dic3⋊C4—21no—
51C4×Dic3⋊C4C2211yescharacteristic
52#4929—21no—
53#4928C2211yescharacteristic
54#4794—24no—
55#4791—24no—
56#4647—21no—
57#4646—21no—
58#4551C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#489—31no—
63C2×#475—31no—
64#2545—31no—
65#2369—31no—
66#2066—31no—
67#2065—31no—
68#2043—31no—
69#2042—31no—
70#1983—31no—
71#1983—31no—
72#1981D311yescharacteristic, complemented
73#1940—31no—
74#1927—31no—
75#1926—31no—
76#1763—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×Dic6—21no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4⋊Q8—22no—
84C3×C4⋊Q8—41no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91#947—22no—
92#947—22no—
93#947—22no—
94#947—41no—
95Q8×Dic3—22no—
96Q8×Dic3—22no—
97C2×(C2×D3)⋊C4—21no—
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×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4D411yescharacteristic
111C2×Dic3⋊C4D411yescharacteristic
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#888—22no—
120#888D414yes—
121C3×C4⋊C8—41no—
122C3×#103—22no—
123C3×#103—22no—
124C3×#103—22no—
125C3×#103—22no—
126C3×#95—21no—
127#835—22no—
128#835—22no—
129#835—22no—
130#835—41no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95D411yescharacteristic
138C3⋊#95D411yescharacteristic
139#821—22no—
140#821—41no—
141#817—22no—
142#815—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—41no—
145#809—22no—
146#809—22no—
147#808—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⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#103C2×D311yescharacteristic
166C2×#103C2×D311yescharacteristic
167C2×#103C2×D311yescharacteristic
168C2×#102—31no—
169C2×#102—31no—
170C2×#102—31no—
171C2×#102—31no—
172C2×#97—31no—
173#396—61no—
174#396—61no—
175#391—31no—
176#391—61no—
177#391—61no—
178#390—31no—
179#389—31no—
180#389—31no—
181#389—61no—
182#388—61no—
183#388—61no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187C4×C4⋊C4—61no—
188C4×C4⋊C4C2×D311yescharacteristic
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C23×D3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8#4112yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×Dic6—24no—
227C2×Dic6—81no—
228C2×Dic6—81no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4C2×D412yes—
240C3×C4⋊C4#4112yes—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—81no—
244(C2×D3)⋊C4—42no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—42no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—42no—
256C4⋊Dic3—81no—
257C4⋊Dic3C2×D412yes—
258Dic3⋊C4—22no—
259Dic3⋊C4—24no—
260Dic3⋊C4—41no—
261Dic3⋊C4—81no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—42no—
265C4×Dic3—42no—
266C4×Dic3—81no—
267C4×Dic3C2×D412yes—
268C4×Dic3#4112yes—
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—61no—
276C22×Q8C22×D311yescharacteristic
277C22×Q8C22×D311yescharacteristic
278C22×D4—31no—
279C22×C8—31no—
280C22×C8—31no—
281C22×C8C22×D311yescharacteristic
282C22×C8C22×D311yescharacteristic
283C4⋊Q8—22no—
284C4⋊Q8—34no—
285C4⋊Q8—41no—
286C4⋊Q8—62no—
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—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
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—31no—
314C2×C42—31no—
315C2×C42—61no—
316C2×C42—61no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319#106—62no—
320#106—62no—
321C4⋊C8—41no—
322C4⋊C8—62no—
323C4⋊C8—121no—
324#103—22no—
325#103—22no—
326#103—22no—
327#103—22no—
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—31no—
339#95—31no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
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—22no—
355C3×Q8—22no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×Q8—41no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
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#13712yes—
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4H(5,𝔽2)11yescharacteristic
385C2×C3×C42−1+411yescharacteristic
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—22no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3#12112yes—
406C2×Dic3#12112yes—
407D12—41no—
408D12—41no—
409D12—41no—
410D12—41no—
411Dic6—84no—
412C3×C8—41no—
413C3×C8—41no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C24—61no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—64no—
429C2×Q8—241no—
430C2×Q8—241no—
431C2×Q8#23512yes—
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—31no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4D3×D411yescharacteristic
450C22×C4D3×D411yescharacteristic
451C22×C4C23×D311yescharacteristic
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—32no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—64no—
470C4⋊C4—62no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—121no—
477C4⋊C4—122no—
478C4⋊C4—241no—
479C4⋊C4—241no—
480C4⋊C4D3×D412yes—
481C4⋊C4#23712yes—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C42—21no—
489C42—21no—
490C42—32no—
491C42—32no—
492C42—61no—
493C42—61no—
494C42—81no—
495C42—122no—
496C42—122no—
497C42—241no—
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—42no—
514C3×C4—42no—
515C3×C4—42no—
516C3×C4—41no—
517C3×C4—81no—
518C3×C4#52911yescharacteristic
519C3×C4#54211yescharacteristic
520C3×C4#54811yescharacteristic
521C3×C4#55011yescharacteristic
522Dic3—22no—
523Dic3—22no—
524Dic3—42no—
525Dic3—41no—
526Dic3—41no—
527Dic3—81no—
528Dic3—81no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23C2×D3×D411yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543Q8—41no—
544Q8—41no—
545Q8—244no—
546D4—121no—
547D4—121no—
548D4—121no—
549D4—121no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—32no—
559C2×C4—32no—
560C2×C4—41no—
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—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
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#91812yes—
589C2×C4C2×#23511yescharacteristic
590C2×C4C2×D3×D411yescharacteristic
591C2×C4C2×#23711yescharacteristic
592C2×C4#100411yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4#101011yescharacteristic
595C8—41no—
596C8—41no—
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—
614C22#549011yescharacteristic
615C22#559811yescharacteristic
616C22D3×#12111yescharacteristic
617C22#576211yescharacteristic
618C22C2×#91811yescharacteristic
619C22#577711yescharacteristic
620C22#577811yescharacteristic
621C4—22no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—41no—
626C4—62no—
627C4—62no—
628C4—81no—
629C4—122no—
630C4—121no—
631C4—121no—
632C4—241no—
633C4—241no—
634C4#553811yescharacteristic
635C4#555211yescharacteristic
636C4#566911yescharacteristic
637C4#571811yescharacteristic
638C3#2528711yescharacteristic, complemented, Sylow
639C2—481no—
640C2#7631211yescharacteristic
641C2#7881111yescharacteristic
642C2#7896911yescharacteristic
643C2#8041911yescharacteristic
644C2#8043011yescharacteristic
645C2#8046411yescharacteristic
646C2#8047611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1981C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1981D3
hypocenterC3#25287
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
3. C2×#4791C2semidirect
4. #73657C2semidirect
5. C3×#1981C2semidirect
6. #71781C2semidirect
8. #70452C2semidirect
9. #70261C2semidirect
13. #70223C2semidirect
14. #70220C2semidirect
72. #1981D3semidirect
638. C3#25287semidirect

Maximal subgroups

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

subgroupindex
#832372
C2×#47942
C2×#47912
#736572
C3×#19812
#717812
#704832
#704522
#702612
#702452
#702302
#702232
#702202
#694632
#252873