Finite Group Databasebeta

← #10833998

Subgroups — #10833998

Order 768.

896 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#451C211yesmaximal, characteristic
2#84062C211yesmaximal, characteristic
3#82787C211yesmaximal, characteristic
4C2×#4720C211yesmaximal, characteristic
5#72132C211yesmaximal, characteristic
6#72068C211yesmaximal, characteristic, complemented
7#72068C211yesmaximal, characteristic, complemented
8#72067C211yesmaximal, characteristic, complemented
9#72066C211yesmaximal, characteristic, complemented
10#71693C211yesmaximal, characteristic
11#70256C211yesmaximal, characteristic, complemented
12#70254C211yesmaximal, characteristic, complemented
13#70248C211yesmaximal, characteristic, complemented
14#70247C211yesmaximal, characteristic, complemented
15#69929C211yesmaximal, characteristic
16#24514—31nomaximal, Sylow
17C23×C3×Q8C2211yescharacteristic
18C22×C3×Q16C2211yescharacteristic
19C22×C3×Q16C2211yescharacteristic
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C3×#451—24no—
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27#5229—21no—
28#5229—21no—
29#5229C2211yescharacteristic
30#5229C2211yescharacteristic
31#5229C2211yescharacteristic
32#5229C2211yescharacteristic
33#5229C2211yescharacteristic
34#5229C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#835C2211yescharacteristic
39C2×#835C2211yescharacteristic
40C2×#835C2211yescharacteristic
41#5212C2211yescharacteristic
42#5211—21no—
43#5211—21no—
44C2×#830C2211yescharacteristic
45C2×#818C2211yescharacteristic
46C2×#817C2211yescharacteristic
47C2×#815C2211yescharacteristic
48C2×Dic3⋊C8C2211yescharacteristic
49C2×Dic3⋊C8C2211yescharacteristic
50C2×C8×Dic3C2211yescharacteristic
51#4986—21no—
52#4981—21no—
53#4972C2211yescharacteristic
54#4972C2211yescharacteristic
55#4952—21no—
56#4952—21no—
57#4720—24no—
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#451D311yescharacteristic
63C2×#433—31no—
64#2625—31no—
65#2310—31no—
66#2068—31no—
67#2066—31no—
68#2056—31no—
69#2056—31no—
70#1971—31no—
71#1969—31no—
72#1935—31no—
73#1933—31no—
74#1873—31no—
75#1854—31no—
76#1787—31no—
77C22×C3×Q8—22no—
78C22×C3×Q8—22no—
79C22×C3×Q8—21no—
80C22×C3×Q8—22no—
81C22×C3×Q8—22no—
82C22×C3×Q8—41no—
83C22×C3×Q8—41no—
84C22×C3×Q8C2311yescharacteristic
85C22×C3×Q8C2311yescharacteristic
86C22×C3×Q8C2311yescharacteristic
87C23×C3×C4—21no—
88C23×C3×C4C2311yescharacteristic
89C2×C3×Q16—22no—
90C2×C3×Q16—22no—
91C2×C3×Q16—22no—
92C2×C3×Q16—22no—
93C2×C3×Q16—41no—
94C2×C3×Q16—41no—
95C22×C3×C8C2311yescharacteristic
96C22×C3×C8C2311yescharacteristic
97C3×C22⋊Q8—22no—
98C3×C22⋊Q8—22no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×#217—21no—
103C2×#217—21no—
104C2×#217D411yescharacteristic
105C2×#217D411yescharacteristic
106#926—22no—
107#926—22no—
108#926—41no—
109#926—41no—
110C2×C4⋊Dic3D411yescharacteristic
111C2×C4⋊Dic3D411yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C3×#103—22no—
124C3×#103—22no—
125C3×C22⋊C8D414yes—
126#835—22no—
127#835—22no—
128#835—41no—
129#835—41no—
130#835—41no—
131#835—41no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147#830—22no—
148#818—41no—
149#817—41no—
150#815—22no—
151Dic3⋊C8—22no—
152Dic3⋊C8—22no—
153C8×Dic3—22no—
154C23×Q8C2×D311yescharacteristic
155C22×Q16C2×D311yescharacteristic
156C22×Q16C2×D311yescharacteristic
157C2×C22⋊Q8C2×D311yescharacteristic
158C2×#117—31no—
159C2×#117—31no—
160#451—24no—
161#433—64no—
162C2×#107—31no—
163C2×#106—31no—
164C2×C4⋊C8—31no—
165C2×C4⋊C8—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#103C2×D311yescharacteristic
173C2×#103C2×D311yescharacteristic
174C2×C22⋊C8—31no—
175C2×C22⋊C8C2×D311yescharacteristic
176C2×#97—31no—
177C2×C4×C8—31no—
178#400—61no—
179#398—61no—
180#397—61no—
181#397—61no—
182#391—31no—
183#391—31no—
184#391—31no—
185#391—31no—
186#391—31no—
187#391—31no—
188#391—61no—
189#391—61no—
190#389—31no—
191#389—31no—
192#381—61no—
193#381—61no—
194#380—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340—31no—
199C24×C3C2×D411yescharacteristic
200C2×C3×Q8—24no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—44no—
218C2×C3×Q8—44no—
219C2×C3×Q8—44no—
220C2×C3×Q8—44no—
221C2×C3×Q8—42no—
222C2×C3×Q8—42no—
223C2×C3×Q8#4112yes—
224C2×C3×Q8#4112yes—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—24no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4—21no—
234C22×C3×C4—42no—
235C22×C3×C4—42no—
236C22×C3×C4C2×D411yescharacteristic
237C22×C3×C4C2×D412yes—
238C22×C3×C4#4112yes—
239C22×C3×C4C2411yescharacteristic
240C22×Dic3—21no—
241C22×Dic3—21no—
242C22×Dic3—21no—
243C22×Dic3—21no—
244C22×Dic3C2×D411yescharacteristic
245C22×Dic3C2×D411yescharacteristic
246C22×Dic3C2×D411yescharacteristic
247C22×Dic3C2×D411yescharacteristic
248C3×Q16—44no—
249C3×Q16—44no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8C2×D412yes—
255C2×C3×C8C2×D412yes—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—42no—
259C3×C22⋊C4—42no—
260#217—22no—
261#217—22no—
262#217—42no—
263#217—81no—
264C4⋊Dic3—22no—
265C4⋊Dic3—22no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274Dic3⋊C4—81no—
275Dic3⋊C4—81no—
276C4×Dic3—21no—
277C4×Dic3—21no—
278C4×Dic3—21no—
279C4×Dic3—21no—
280C4×Dic3—42no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C22×Q8—22no—
288C22×Q8—22no—
289C22×Q8—21no—
290C22×Q8—22no—
291C22×Q8—22no—
292C22×Q8—41no—
293C22×Q8—41no—
294C22×Q8C22×D311yescharacteristic
295C22×Q8C22×D311yescharacteristic
296C22×Q8C22×D311yescharacteristic
297C23×C4—21no—
298C23×C4C22×D311yescharacteristic
299C2×Q16—22no—
300C2×Q16—22no—
301C2×Q16—22no—
302C2×Q16—22no—
303C2×Q16—41no—
304C2×Q16—41no—
305C22×C8—31no—
306C22×C8—31no—
307C22×C8C22×D311yescharacteristic
308C22×C8C22×D311yescharacteristic
309C22⋊Q8—22no—
310C22⋊Q8—22no—
311#117—62no—
312#117—62no—
313#117—121no—
314#117—121no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C22⋊C4—21no—
326C2×C22⋊C4—31no—
327C2×C22⋊C4—31no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C42—31no—
331C2×C42—31no—
332C2×C42—61no—
333#107—121no—
334#106—121no—
335C4⋊C8—62no—
336C4⋊C8—62no—
337#103—22no—
338#103—22no—
339#103—62no—
340#103—62no—
341#103—121no—
342#103—121no—
343#103—121no—
344#103—121no—
345C22⋊C8—62no—
346C22⋊C8#6714yes—
347#97—62no—
348C4×C8—62no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360#95—61no—
361#95—61no—
362#95—61no—
363#95—61no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3#12112yes—
371C23×C3C22×D411yescharacteristic
372C3×Q8—22no—
373C3×Q8—22no—
374C3×Q8—22no—
375C3×Q8—22no—
376C3×Q8—48no—
377C3×Q8—41no—
378C3×Q8—41no—
379C3×Q8—41no—
380C3×Q8—41no—
381C3×Q8—41no—
382C3×Q8—41no—
383C3×Q8—41no—
384C3×Q8—41no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—48no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4#12112yes—
405C2×C3×C4C22×D411yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4H(5,𝔽2)11yescharacteristic
410C2×C3×C4H(5,𝔽2)11yescharacteristic
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C3×C8—22no—
436C3×C8—22no—
437C3⋊C8—41no—
438C3⋊C8—41no—
439C24D3×D411yescharacteristic
440C2×Q8—24no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—44no—
458C2×Q8—44no—
459C2×Q8—44no—
460C2×Q8—44no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8#23712yes—
464C2×Q8#23712yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—24no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—31no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4D3×D411yescharacteristic
485C22×C4#23712yes—
486C22×C4C2×#6712yes—
487C22×C4C23×D311yescharacteristic
488Q16—44no—
489Q16—44no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C2×C8C2×#6712yes—
501C2×C8C2×#6712yes—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—42no—
505C4⋊C4—62no—
506C4⋊C4—62no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—241no—
516C4⋊C4—241no—
517C22⋊C4—42no—
518C22⋊C4—62no—
519C22⋊C4—62no—
520C22⋊C4—122no—
521C22⋊C4—241no—
522C42—61no—
523C42—61no—
524C42—61no—
525C42—61no—
526C42—122no—
527C22×C3—22no—
528C22×C3—22no—
529C22×C3—22no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3—41no—
533C22×C3—41no—
534C22×C3C2×#12111yescharacteristic
535C22×C3#52511yescharacteristic
536C22×C3#53511yescharacteristic
537C22×C3#57711yescharacteristic
538C22×C3#57811yescharacteristic
539C22×C3#57811yescharacteristic
540C22×C3#57811yescharacteristic
541C3×C4—22no—
542C3×C4—42no—
543C3×C4—42no—
544C3×C4—81no—
545C3×C4—81no—
546C3×C4#54611yescharacteristic
547C3×C4#54611yescharacteristic
548C3×C4#55111yescharacteristic
549C3×C4#55111yescharacteristic
550Dic3—41no—
551Dic3—41no—
552Dic3—41no—
553Dic3—41no—
554Dic3—81no—
555Dic3—81no—
556Dic3—81no—
557Dic3—81no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23#93812yes—
565C23C2×D3×D411yescharacteristic
566Q8—22no—
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—48no—
571Q8—41no—
572Q8—41no—
573Q8—41no—
574Q8—41no—
575Q8—41no—
576Q8—41no—
577Q8—41no—
578Q8—41no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—48no—
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—41no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4#93812yes—
623C2×C4C2×#23711yescharacteristic
624C2×C4C2×#23711yescharacteristic
625C2×C4C2×#23711yescharacteristic
626C2×C4#100411yescharacteristic
627C2×C4#100411yescharacteristic
628C2×C4C22×#6711yescharacteristic
629C8—22no—
630C8—22no—
631C8—121no—
632C8—121no—
633C2×C3—42no—
634C2×C3#241111yescharacteristic
635C2×C3#305911yescharacteristic
636C2×C3#306311yescharacteristic
637C2×C3#312111yescharacteristic
638C2×C3#312711yescharacteristic
639C2×C3#322311yescharacteristic
640C2×C3#322511yescharacteristic
641C22—22no—
642C22—22no—
643C22—22no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22#558611yescharacteristic
649C22#559211yescharacteristic
650C22#576611yescharacteristic
651C22#576611yescharacteristic
652C22#577811yescharacteristic
653C22#577911yescharacteristic
654C22C2×#93811yescharacteristic
655C4—22no—
656C4—42no—
657C4—42no—
658C4—81no—
659C4—81no—
660C4—121no—
661C4—121no—
662C4—121no—
663C4—121no—
664C4—241no—
665C4—241no—
666C4—241no—
667C4—241no—
668C4#558711yescharacteristic
669C4#558711yescharacteristic
670C4#580611yescharacteristic
671C4#580611yescharacteristic
672C3#2451411yescharacteristic, complemented, Sylow
673C2—42no—
674C2#8017611yescharacteristic
675C2#8017711yescharacteristic
676C2#8018011yescharacteristic
677C2#8020311yescharacteristic
678C2#8258311yescharacteristic
679C2#8388711yescharacteristic
680C2#8398611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#451C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#451D3
hypocenterC3#24514
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
6. #72068C2semidirect
7. #72068C2semidirect
8. #72067C2semidirect
9. #72066C2semidirect
11. #70256C2semidirect
12. #70254C2semidirect
13. #70248C2semidirect
14. #70247C2semidirect
672. C3#24514semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4512
#840622
#827872
C2×#47202
#721322
#720682
#720672
#720662
#716932
#702562
#702542
#702482
#702472
#699292
#245143