Finite Group Databasebeta

← #10835962

Subgroups — #10835962

Order 768.

912 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5153C211yesmaximal, characteristic, complemented
2#82677C211yesmaximal, characteristic, complemented
3#82633C211yesmaximal, characteristic, complemented
4C2×#4720C211yesmaximal, characteristic, complemented
5C3×#1999C211yesmaximal, characteristic
6#72039C211yesmaximal, characteristic, complemented
7#71685C211yesmaximal, characteristic, complemented
8#71680C211yesmaximal, characteristic, complemented
9#70189C211yesmaximal, characteristic, complemented
10#70185C211yesmaximal, characteristic, complemented
11#70162C211yesmaximal, characteristic, complemented
12#70153C211yesmaximal, characteristic, complemented
13#69936C211yesmaximal, characteristic, complemented
14#68855C211yesmaximal, characteristic, complemented
15#68796C211yesmaximal, characteristic, complemented
16#24667—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C2×C3×#121C2211yescharacteristic
20C2×#936C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C22×#214C2211yescharacteristic
23C2×#926C2211yescharacteristic
24C2×#924C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28#5222—21no—
29#5222—21no—
30#5221—21no—
31#5221C2211yescharacteristic
32#5221C2211yescharacteristic
33#5220—21no—
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37#5212C2211yescharacteristic
38C2×#830C2211yescharacteristic
39#5153—24no—
40C2×#817C2211yescharacteristic
41C2×#816C2211yescharacteristic
42C2×#816C2211yescharacteristic
43C2×#815C2211yescharacteristic
44C2×Dic3⋊C8C2211yescharacteristic
45C2×Dic3⋊C8C2211yescharacteristic
46C2×C8×Dic3C2211yescharacteristic
47#4984C2211yescharacteristic
48#4975C2211yescharacteristic
49Dic3×C4⋊C4C2211yescharacteristic
50#4973C2211yescharacteristic
51C2×#810C2211yescharacteristic
52C2×#808C2211yescharacteristic
53#4951—21no—
54#4950—21no—
55#4948—21no—
56#4948—21no—
57#4720—24no—
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#463—31no—
63C2×#433—31no—
64#2620—31no—
65#2324—31no—
66#2035—31no—
67#2034—31no—
68#2000—31no—
69#1999D311yescharacteristic
70#1983—31no—
71#1972—31no—
72#1968—31no—
73#1948—31no—
74#1935—31no—
75#1934—31no—
76#1802—31no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4C2311yescharacteristic
82C23×Dic3—21no—
83C22×Dic6C2311yescharacteristic
84C2×C3×D8—22no—
85C2×C3×D8—22no—
86C2×C3×D8—41no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×#121—22no—
90C3×#121—22no—
91C3×#121—41no—
92C3×#121—41no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104C2×#217—21no—
105C2×#217—21no—
106C2×#217D411yescharacteristic
107C2×#217D411yescharacteristic
108#936—22no—
109#936—22no—
110D4×Dic3—22no—
111D4×Dic3—22no—
112C2×#214—22no—
113C2×#214—22no—
114C2×#214—41no—
115#926—22no—
116#926—22no—
117C2×C4⋊Dic3—21no—
118C2×C4⋊Dic3D411yescharacteristic
119C2×C4⋊Dic3D411yescharacteristic
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2311yescharacteristic
122C2×C4⋊Dic3C2311yescharacteristic
123#924—41no—
124#924—41no—
125C2×Dic3⋊C4—21no—
126C2×Dic3⋊C4—21no—
127C2×Dic3⋊C4—21no—
128C2×C4×Dic3—21no—
129C2×C4×Dic3C2311yescharacteristic
130C2×C4×Dic3C2311yescharacteristic
131C22×C3⋊C8C2311yescharacteristic
132C22×C3⋊C8C2311yescharacteristic
133C3×#102—22no—
134C3×#102—41no—
135C3×C22⋊C8—22no—
136#832—22no—
137#832—41no—
138#832—41no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144#830—22no—
145#817—41no—
146#816—22no—
147#816—41no—
148#815—22no—
149Dic3⋊C8—22no—
150Dic3⋊C8D414yescomplemented
151C8×Dic3—22no—
152#810—22no—
153#808—41no—
154C22×SD16—31no—
155C22×D8C2×D311yescharacteristic
156C2×#124—31no—
157C2×C22⋊Q8—31no—
158C2×#121C2×D311yescharacteristic
159C2×#121C2×D311yescharacteristic
160C2×C4×D4—31no—
161C2×#117—31no—
162#463—64no—
163#433—64no—
164C2×#106—31no—
165C2×#106—31no—
166C2×C4⋊C8—31no—
167C2×C4⋊C8—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#102—31no—
172C2×#102—31no—
173C2×#102—31no—
174C2×#102C2×D311yescharacteristic
175C2×#102C2×D311yescharacteristic
176C2×C22⋊C8—31no—
177C2×C22⋊C8C2×D311yescharacteristic
178C2×#97—31no—
179C2×C4×C8—31no—
180#396—61no—
181#396—61no—
182#394—61no—
183#394—61no—
184#391—31no—
185#390—31no—
186#390—31no—
187#390—61no—
188#389—31no—
189#388—61no—
190#386—61no—
191#385—61no—
192#384—31no—
193#380—31no—
194C4×C4⋊C4—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340C2×D311yescharacteristic
199C24×C3—21no—
200C24×C3—21no—
201C24×C3C2×D411yescharacteristic
202C2×C3×D4—24no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—42no—
215C2×C3×D4—81no—
216C2×C3×D4—81no—
217C2×C3×D4—81no—
218C2×C3×D4—81no—
219C2×C3×D4#4112yes—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2411yescharacteristic
230C22×Dic3—21no—
231C22×Dic3—21no—
232C22×Dic3—21no—
233C22×Dic3—21no—
234C22×Dic3—42no—
235C22×Dic3—81no—
236C22×Dic3—81no—
237C22×Dic3C2×D411yescharacteristic
238C22×Dic3C2×D411yescharacteristic
239C22×Dic3C2×D411yescharacteristic
240C22×Dic3C2×D411yescharacteristic
241C2×Dic6—21no—
242C2×Dic6—21no—
243C2×Dic6—21no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C3×D8—44no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8C2×D412yes—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4#4112yes—
258C3×C22⋊C4—24no—
259C3×C22⋊C4—42no—
260#217—22no—
261#217—22no—
262#217—42no—
263#217—42no—
264#217—42no—
265#217—42no—
266#217—81no—
267#217—81no—
268#217—81no—
269#217—81no—
270#217—81no—
271#214—44no—
272C4⋊Dic3—22no—
273C4⋊Dic3—22no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3—81no—
281Dic3⋊C4—42no—
282Dic3⋊C4—81no—
283Dic3⋊C4—81no—
284C4×Dic3—21no—
285C4×Dic3—21no—
286C4×Dic3—81no—
287C4×Dic3C2×D412yes—
288C4×Dic3#4112yes—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8C2×D412yes—
295C22×Q8—31no—
296C22×D4—21no—
297C22×D4—21no—
298C22×D4C22×D311yescharacteristic
299C22×D4C22×D311yescharacteristic
300C23×C4—61no—
301C23×C4C22×D311yescharacteristic
302C2×SD16—62no—
303C2×SD16—62no—
304C2×SD16—121no—
305C2×D8—22no—
306C2×D8—22no—
307C2×D8—41no—
308C22×C8—31no—
309C22×C8—31no—
310C22×C8C22×D311yescharacteristic
311C22×C8C22×D311yescharacteristic
312#124—62no—
313#124—62no—
314C22⋊Q8—121no—
315C22⋊Q8—121no—
316#121—22no—
317#121—22no—
318#121—41no—
319#121—41no—
320C4×D4—62no—
321C4×D4—62no—
322#117—62no—
323#117—62no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4C22×D311yescharacteristic
334C2×C4⋊C4C22×D311yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—31no—
338C2×C22⋊C4—31no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C22⋊C4—61no—
342C2×C22⋊C4—61no—
343C2×C22⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C22⋊C4—61no—
346C2×C22⋊C4—61no—
347C2×C22⋊C4—61no—
348C2×C42—31no—
349C2×C42—31no—
350C2×C42—61no—
351#106—121no—
352#106—121no—
353C4⋊C8—34no—
354C4⋊C8—62no—
355#103—62no—
356#103—62no—
357#103—121no—
358#102—22no—
359#102—41no—
360#102—62no—
361#102—121no—
362#102—121no—
363C22⋊C8—22no—
364C22⋊C8—62no—
365#97—62no—
366C4×C8—62no—
367#95—61no—
368#95—61no—
369#95—61no—
370#95—61no—
371#95—61no—
372C23×C3—21no—
373C23×C3—22no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—21no—
378C23×C3—21no—
379C23×C3—21no—
380C23×C3—41no—
381C23×C3—41no—
382C23×C3—41no—
383C23×C3—41no—
384C23×C3—41no—
385C23×C3—41no—
386C23×C3—41no—
387C23×C3—41no—
388C23×C3—41no—
389C23×C3—41no—
390C23×C3—41no—
391C23×C3—41no—
392C23×C3—41no—
393C23×C3C22×D411yescharacteristic
394C3×D4—22no—
395C3×D4—22no—
396C3×D4—41no—
397C3×D4—41no—
398C3×D4—41no—
399C3×D4—41no—
400C3×D4—41no—
401C3×D4—41no—
402C3×D4—84no—
403C3×D4—84no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—22no—
407C2×C3×C4—21no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4C22×D411yescharacteristic
418C2×C3×C4C2×#4111yescharacteristic
419C2×C3×C4C2×#4111yescharacteristic
420C2×C3×C4C2×#4111yescharacteristic
421C2×C3×C4H(5,𝔽2)11yescharacteristic
422C2×C3×C4H(5,𝔽2)11yescharacteristic
423C2×Dic3—21no—
424C2×Dic3—21no—
425C2×Dic3—21no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439C2×Dic3—41no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—41no—
445C2×Dic3—84no—
446C2×Dic3#12112yes—
447C2×Dic3#12112yes—
448Dic6—41no—
449Dic6—41no—
450Dic6—41no—
451Dic6—41no—
452C3×C8—22no—
453C3×C8—41no—
454C3⋊C8—22no—
455C3⋊C8—41no—
456C24—21no—
457C24—21no—
458C24D3×D411yescharacteristic
459C2×Q8—61no—
460C2×Q8—61no—
461C2×Q8—61no—
462C2×Q8—61no—
463C2×Q8—61no—
464C2×Q8—61no—
465C2×D4—24no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—42no—
478C2×D4—81no—
479C2×D4—81no—
480C2×D4—81no—
481C2×D4—81no—
482C2×D4#23712yes—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—31no—
492C22×C4—31no—
493C22×C4—31no—
494C22×C4—31no—
495C22×C4—61no—
496C22×C4—61no—
497C22×C4—61no—
498C22×C4—61no—
499C22×C4—122no—
500C22×C4—241no—
501C22×C4—241no—
502C22×C4D3×D411yescharacteristic
503C22×C4C23×D311yescharacteristic
504SD16—124no—
505D8—44no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—32no—
512C2×C8—61no—
513C2×C8—61no—
514C2×C8—61no—
515C2×C8—61no—
516C2×C8—61no—
517C2×C8D3×D412yes—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—62no—
522C4⋊C4—62no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C4⋊C4—122no—
530C4⋊C4—241no—
531C4⋊C4—241no—
532C4⋊C4—241no—
533C4⋊C4#23712yes—
534C22⋊C4—24no—
535C22⋊C4—42no—
536C22⋊C4—62no—
537C22⋊C4—62no—
538C22⋊C4—122no—
539C22⋊C4—122no—
540C22⋊C4—122no—
541C22⋊C4—122no—
542C22⋊C4—241no—
543C22⋊C4—241no—
544C22⋊C4—241no—
545C22⋊C4—241no—
546C22⋊C4—241no—
547C42—32no—
548C42—32no—
549C42—61no—
550C42—61no—
551C42—241no—
552C22×C3—41no—
553C22×C3—42no—
554C22×C3—41no—
555C22×C3—42no—
556C22×C3—41no—
557C22×C3—42no—
558C22×C3—41no—
559C22×C3—41no—
560C22×C3—41no—
561C22×C3—41no—
562C22×C3—81no—
563C22×C3—81no—
564C22×C3—81no—
565C22×C3—81no—
566C22×C3—81no—
567C22×C3—81no—
568C22×C3—81no—
569C22×C3—81no—
570C22×C3—81no—
571C22×C3—81no—
572C22×C3—81no—
573C22×C3C2×#12111yescharacteristic
574C22×C3#52511yescharacteristic
575C22×C3#53511yescharacteristic
576C22×C3#57611yescharacteristic
577C22×C3#57711yescharacteristic
578C22×C3#57711yescharacteristic
579C22×C3#57711yescharacteristic
580C3×C4—42no—
581C3×C4—41no—
582C3×C4—81no—
583C3×C4#54611yescharacteristic
584C3×C4#55011yescharacteristic
585C3×C4#55311yescharacteristic
586C3×C4#55511yescharacteristic
587Dic3—22no—
588Dic3—22no—
589Dic3—41no—
590Dic3—41no—
591Dic3—81no—
592Dic3—81no—
593Dic3—81no—
594Dic3—81no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—22no—
600C23—21no—
601C23—21no—
602C23—21no—
603C23—41no—
604C23—41no—
605C23—41no—
606C23—41no—
607C23—41no—
608C23—41no—
609C23—41no—
610C23—41no—
611C23—41no—
612C23—41no—
613C23—41no—
614C23—41no—
615C23—41no—
616C23C2×D3×D411yescharacteristic
617Q8—121no—
618Q8—121no—
619Q8—121no—
620Q8—121no—
621D4—22no—
622D4—22no—
623D4—41no—
624D4—41no—
625D4—41no—
626D4—41no—
627D4—41no—
628D4—41no—
629D4—84no—
630D4—84no—
631C2×C4—21no—
632C2×C4—22no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—32no—
636C2×C4—32no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—61no—
647C2×C4—61no—
648C2×C4—61no—
649C2×C4—61no—
650C2×C4—61no—
651C2×C4—61no—
652C2×C4—61no—
653C2×C4—61no—
654C2×C4—61no—
655C2×C4—61no—
656C2×C4—121no—
657C2×C4—121no—
658C2×C4—121no—
659C2×C4—121no—
660C2×C4—121no—
661C2×C4—121no—
662C2×C4—121no—
663C2×C4—121no—
664C2×C4—121no—
665C2×C4—121no—
666C2×C4—121no—
667C2×C4—121no—
668C2×C4—244no—
669C2×C4C2×D3×D411yescharacteristic
670C2×C4C2×#23711yescharacteristic
671C2×C4C2×#23711yescharacteristic
672C2×C4#100411yescharacteristic
673C2×C4#100411yescharacteristic
674C2×C4D3×#4111yescharacteristic
675C8—22no—
676C8—41no—
677C8—62no—
678C8—121no—
679C2×C3—81no—
680C2×C3—82no—
681C2×C3—161no—
682C2×C3#241411yescharacteristic
683C2×C3#306011yescharacteristic
684C2×C3#306211yescharacteristic
685C2×C3#311811yescharacteristic
686C2×C3#312511yescharacteristic
687C2×C3#322111yescharacteristic
688C2×C3#322411yescharacteristic
689C22—41no—
690C22—41no—
691C22—42no—
692C22—41no—
693C22—42no—
694C22—41no—
695C22—41no—
696C22—42no—
697C22—41no—
698C22—41no—
699C22—81no—
700C22—81no—
701C22—81no—
702C22—81no—
703C22—81no—
704C22—81no—
705C22—81no—
706C22—81no—
707C22—81no—
708C22—81no—
709C22—81no—
710C22#558611yescharacteristic
711C22#559211yescharacteristic
712C22D3×#12111yescharacteristic
713C22#575711yescharacteristic
714C22#577011yescharacteristic
715C22#577511yescharacteristic
716C22#577511yescharacteristic
717C4—42no—
718C4—41no—
719C4—62no—
720C4—62no—
721C4—81no—
722C4—121no—
723C4—121no—
724C4—241no—
725C4—241no—
726C4—241no—
727C4—241no—
728C4#559111yescharacteristic
729C4#560111yescharacteristic
730C4#561611yescharacteristic
731C4#571811yescharacteristic
732C3#2466711yescharacteristic, complemented, Sylow
733C2—81no—
734C2—82no—
735C2—161no—
736C2#7667011yescharacteristic
737C2#7999211yescharacteristic
738C2#7999911yescharacteristic
739C2#8007511yescharacteristic
740C2#8012711yescharacteristic
741C2#8071811yescharacteristic
742C2#8086611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#1999C2
socleC23×C3C22×D4
hypercenter#1999D3
hypocenterC3#24667
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1999

Complements

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

NH ≅ G/Ntype
1. C2×#5153C2semidirect
2. #82677C2semidirect
3. #82633C2semidirect
4. C2×#4720C2semidirect
6. #72039C2semidirect
7. #71685C2semidirect
8. #71680C2semidirect
9. #70189C2semidirect
10. #70185C2semidirect
11. #70162C2semidirect
12. #70153C2semidirect
13. #69936C2semidirect
14. #68855C2semidirect
15. #68796C2semidirect
150. Dic3⋊C8D4semidirect
732. C3#24667semidirect

Maximal subgroups

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

subgroupindex
C2×#51532
#826772
#826332
C2×#47202
C3×#19992
#720392
#716852
#716802
#701892
#701852
#701622
#701532
#699362
#688552
#687962
#246673