Finite Group Databasebeta

← #10844798

Subgroups — #10844798

Order 768.

830 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4828C211yesmaximal, characteristic, complemented
2C2×#4818C211yesmaximal, characteristic
3#76545C211yesmaximal, characteristic, complemented
4#76059C211yesmaximal, characteristic
5C3×#2068C211yesmaximal, characteristic, complemented
6#70629C211yesmaximal, characteristic
7#70615C211yesmaximal, characteristic, complemented
8#70482C211yesmaximal, characteristic
9#70280C211yesmaximal, characteristic
10#70256C211yesmaximal, characteristic, complemented
11#70225C211yesmaximal, characteristic, complemented
12#69679C211yesmaximal, characteristic
13#69616C211yesmaximal, characteristic, complemented
14#68913C211yesmaximal, characteristic, complemented
15#68780C211yesmaximal, characteristic
16#25301—31nomaximal, Sylow
17C2×#944C2211yescharacteristic
18C22×#215C2211yescharacteristic
19C22×#204C2211yescharacteristic
20C2×#896—21no—
21C2×#893C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#389C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#815C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37#4994—21no—
38#4988C2211yescharacteristic
39#4975C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41#4973C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4939C2211yescharacteristic
48#4929C2211yescharacteristic
49C2×#803C2211yescharacteristic
50#4828—24no—
51#4818—24no—
52#4676—21no—
53#4667—21no—
54#4660—21no—
55#4658—21no—
56#4650—21no—
57#4646—21no—
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#461—31no—
63C2×#446—31no—
64#2568—31no—
65#2538—31no—
66#2068—31no—
67#2068D311yescharacteristic, complemented
68#2055—31no—
69#2055—31no—
70#1973—31no—
71#1973—31no—
72#1956—31no—
73#1947—31no—
74#1935—31no—
75#1935—31no—
76C4×#102—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×Dic6D411yescharacteristic
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89#944—22no—
90#944—22no—
91C2×#215—22no—
92C2×#215—22no—
93C2×#215—41no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×C4⋊Dic3D411yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4D411yescharacteristic
108C2×Dic3⋊C4D411yescharacteristic
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C2×#204—22no—
115C2×#204—22no—
116C2×#204—41no—
117#896—44no—
118#893—22no—
119#893—22no—
120C4×Dic6—22no—
121C4×Dic6—22no—
122C3×#107—41no—
123C3×C4⋊C8—22no—
124C3×#103—22no—
125C3×#103—41no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129#835—22no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138#821—22no—
139#816—41no—
140#815—22no—
141Dic3⋊C8—22no—
142Dic3⋊C8D414yes—
143C8×Dic3—22no—
144#810—22no—
145#810—22no—
146#809—22no—
147#807—41no—
148#803—22no—
149C22×SD16—31no—
150C22×SD16—31no—
151C2×C4⋊Q8—31no—
152C2×C22⋊Q8—61no—
153C2×C4×Q8—31no—
154C2×C4×D4—31no—
155#461—64no—
156#446—64no—
157C2×#107—31no—
158C2×#107C2×D311yescharacteristic
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8C2×D311yescharacteristic
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103C2×D311yescharacteristic
167C2×#103C2×D311yescharacteristic
168C2×#102—31no—
169C2×#102—31no—
170C2×#97—31no—
171C2×#97—31no—
172C2×C4×C8—31no—
173#400—61no—
174#400—61no—
175#396—61no—
176#396—61no—
177#391—31no—
178#391—31no—
179#391—31no—
180#391—61no—
181#391C2×D311yescharacteristic
182#390—31no—
183#390—31no—
184#389—31no—
185#389C2×D311yescharacteristic
186#384—31no—
187#382—61no—
188#380—61no—
189C4×C4⋊C4—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340C2×D311yescharacteristic
194C23×D3—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—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12#4112yes—
221C2×C4×D3—42no—
222C2×C4×D3—42no—
223C2×C4×D3—42no—
224C2×Dic6—22no—
225C2×Dic6—22no—
226C2×Dic6—22no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8C2×D412yes—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—42no—
246C3×C42—21no—
247C3×C42—21no—
248#215—44no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—81no—
253(C2×D3)⋊C4—81no—
254C4⋊Dic3—22no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259Dic3⋊C4—22no—
260Dic3⋊C4—22no—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264Dic3⋊C4—42no—
265Dic3⋊C4—81no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—42no—
269C4×Dic3C2×D412yes—
270C4×Dic3#4112yes—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8C2×D412yes—
276C2×C3⋊C8#4112yes—
277#204—44no—
278C22×Q8—31no—
279C22×Q8—31no—
280C22×Q8C22×D311yescharacteristic
281C22×D4—31no—
282C23×C4—61no—
283C2×SD16—62no—
284C2×SD16—62no—
285C2×SD16—62no—
286C2×SD16—62no—
287C2×SD16—121no—
288C2×SD16—121no—
289C22×C8—31no—
290C22×C8—31no—
291C22×C8C22×D311yescharacteristic
292C22×C8C22×D311yescharacteristic
293C4⋊Q8—62no—
294C4⋊Q8—62no—
295C22⋊Q8—124no—
296C4×Q8—62no—
297C4×Q8—62no—
298C4×D4—62no—
299C4×D4—62no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C42—31no—
320C2×C42—31no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323#107—41no—
324#107—121no—
325C4⋊C8—22no—
326C4⋊C8—34no—
327C4⋊C8—62no—
328#103—22no—
329#103—41no—
330#103—62no—
331#103—62no—
332#103—62no—
333#103—121no—
334#102—62no—
335#102—62no—
336#97—62no—
337#97—62no—
338C4×C8—62no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3C22×D411yescharacteristic
351C22×D3—22no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3—41no—
357C22×D3—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×Q8—41no—
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—41no—
369C2×C3×C4—41no—
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—41no—
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—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—41no—
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#12112yes—
404C2×Dic3#12112yes—
405C2×Dic3AGL(1,ℤ/8)12yes—
406C2×Dic3#13712yes—
407D12—22no—
408D12—22no—
409D12—41no—
410D12—41no—
411C4×D3—84no—
412Dic6—44no—
413Dic6—41no—
414Dic6—41no—
415Dic6—41no—
416Dic6—41no—
417C3×C8—22no—
418C3×C8—41no—
419C3⋊C8—22no—
420C3⋊C8—22no—
421C24—61no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—62no—
429C2×Q8—62no—
430C2×Q8—62no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×D4—32no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—122no—
455C22×C4—122no—
456C22×C4—122no—
457C22×C4D3×D411yescharacteristic
458C22×C4D3×D411yescharacteristic
459C22×C4C23×D311yescharacteristic
460SD16—124no—
461SD16—124no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—32no—
468C2×C8—32no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8D3×D412yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—62no—
482C4⋊C4—62no—
483C4⋊C4—62no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—122no—
489C4⋊C4—122no—
490C4⋊C4—122no—
491C4⋊C4—122no—
492C4⋊C4—241no—
493C22⋊C4—122no—
494C22⋊C4—122no—
495C22⋊C4—122no—
496C22⋊C4—241no—
497C22⋊C4—241no—
498C42—21no—
499C42—21no—
500C42—32no—
501C42—32no—
502C42—61no—
503C42—61no—
504C42—122no—
505C22×C3C2×#12111yescharacteristic
506C22×C3#54811yescharacteristic
507C22×C3#55011yescharacteristic
508C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
509C22×C3C2×#13711yescharacteristic
510C22×C3#57711yescharacteristic
511C22×C3#57811yescharacteristic
512C2×D3—42no—
513C2×D3—42no—
514C2×D3—42no—
515C2×D3—81no—
516C2×D3—81no—
517C2×D3—81no—
518C2×D3—81no—
519C3×C4—41no—
520C3×C4—41no—
521C3×C4—81no—
522C3×C4—81no—
523C3×C4—81no—
524C3×C4—81no—
525C3×C4#52911yescharacteristic
526C3×C4#54211yescharacteristic
527C3×C4#54611yescharacteristic
528C3×C4#54811yescharacteristic
529Dic3—22no—
530Dic3—41no—
531Dic3—41no—
532Dic3—81no—
533Dic3—81no—
534Dic3—81no—
535Dic3#46814yes—
536C23—62no—
537C23—121no—
538C23—121no—
539C23—121no—
540C23—121no—
541C23—121no—
542C23—121no—
543C23C2×D3×D411yescharacteristic
544Q8—41no—
545Q8—41no—
546Q8—41no—
547Q8—41no—
548Q8—124no—
549Q8—121no—
550Q8—121no—
551Q8—121no—
552Q8—121no—
553D4—62no—
554D4—62no—
555D4—121no—
556D4—121no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—32no—
564C2×C4—32no—
565C2×C4—32no—
566C2×C4—32no—
567C2×C4—41no—
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—41no—
578C2×C4—41no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—244no—
597C2×C4C2×D3×D411yescharacteristic
598C2×C4#100411yescharacteristic
599C2×C4#100711yescharacteristic
600C2×C4D3×#4111yescharacteristic
601C2×C4D3×#4111yescharacteristic
602C2×C4D3×#4111yescharacteristic
603C8—22no—
604C8—41no—
605C8—62no—
606C8—62no—
607C2×C3#242611yescharacteristic
608C2×C3C2×#46811yescharacteristic
609C2×C3#306311yescharacteristic
610C2×C3#336711yescharacteristic
611C2×C3#337211yescharacteristic
612C2×C3#339711yescharacteristic
613C2×C3#339811yescharacteristic
614D3—82no—
615C22—122no—
616C22—122no—
617C22—122no—
618C22—241no—
619C22—241no—
620C22—241no—
621C22—241no—
622C22#559911yescharacteristic
623C22D3×#12111yescharacteristic
624C22#563311yescharacteristic
625C22#576611yescharacteristic
626C22#577011yescharacteristic
627C22D3×AGL(1,ℤ/8)11yescharacteristic
628C22D3×#13711yescharacteristic
629C4—34no—
630C4—41no—
631C4—41no—
632C4—62no—
633C4—81no—
634C4—81no—
635C4—81no—
636C4—81no—
637C4—121no—
638C4—121no—
639C4—241no—
640C4—241no—
641C4—241no—
642C4#561011yescharacteristic
643C4#563311yescharacteristic
644C4#564011yescharacteristic
645C4#570611yescharacteristic
646C3#2530111yescharacteristic, complemented, Sylow
647C2—242no—
648C2#7728611yescharacteristic
649C2#8046911yescharacteristic
650C2#8052611yescharacteristic
651C2#8076211yescharacteristic
652C2D3×#46811yescharacteristic
653C2#8110011yescharacteristic
654C2#8121911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2068C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2068D3
hypocenterC3#25301
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2068

Complements

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

NH ≅ G/Ntype
1. C2×#4828C2semidirect
3. #76545C2semidirect
5. C3×#2068C2semidirect
7. #70615C2semidirect
10. #70256C2semidirect
11. #70225C2semidirect
13. #69616C2semidirect
14. #68913C2semidirect
67. #2068D3semidirect
646. C3#25301semidirect

Maximal subgroups

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

subgroupindex
C2×#48282
C2×#48182
#765452
#760592
C3×#20682
#706292
#706152
#704822
#702802
#702562
#702252
#696792
#696162
#689132
#687802
#253013