Finite Group Databasebeta

← #10701931

Subgroups — #10701931

Order 768.

1370 conjugacy classes of subgroups.

The subgroup lattice has 764 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#932C211yesmaximal, characteristic
2C2×C3×#418C211yesmaximal, characteristic, complemented
3C22×#833C211yesmaximal, characteristic, complemented
4#83762C211yesmaximal, characteristic, complemented
5C2×#5005C211yesmaximal, characteristic, complemented
6C2×C3⋊#338C211yesmaximal, characteristic, complemented
7C2×#4533C211yesmaximal, characteristic, complemented
8#11014—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×#235C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C22×#214C412yescomplemented
14C2×#932—22no—
15C2×#932—22no—
16C2×#932—22no—
17C2×#932—22no—
18C2×#932—22no—
19C2×#932—22no—
20C22×#213C412yescomplemented
21C22×#208C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C3×#418—22no—
25C3×#418—22no—
26C2×C3×#102—22no—
27C2×#833—22no—
28C2×#833—22no—
29C2×#833—22no—
30#5129—22no—
31#5126—22no—
32#5110—22no—
33#5005—22no—
34#5005—22no—
35C2×#805—22no—
36C3⋊#338—22no—
37#4533—22no—
38C22×AGL(1,ℤ/8)—31no—
39C2×#421—31no—
40C2×#418D311yescharacteristic, complemented
41C22×#100—31no—
42#2879—31no—
43C2×#344—31no—
44C2×#338—31no—
45C25×C3—21no—
46C22×C3×D4—21no—
47C22×C3×D4—21no—
48C22×C3×D4—21no—
49C22×C3×D4—21no—
50C22×C3×D4—21no—
51C22×C3×D4—21no—
52C22×C3×D4—21no—
53C22×C3×D4—21no—
54C22×C3×D4—21no—
55C22×C3×D4—21no—
56C22×C3×D4—21no—
57C22×C3×D4C2×C412yes—
58C22×C3×D4D412yes—
59C22×C3×D4D412yes—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C2×#235—21no—
63C2×#235—21no—
64C2×#235—21no—
65C2×#235—21no—
66C2×#235—21no—
67C2×#235—21no—
68C2×#235—21no—
69C2×#235—21no—
70C2×#235—21no—
71C2×#235—21no—
72C2×#235D412yes—
73C2×#235D412yes—
74C22×D12C2×C411yescharacteristic
75C22×C4×D3—21no—
76C22×Dic6C2×C411yescharacteristic
77C2×C3×M16—21no—
78C2×C3×M16—21no—
79C2×C3×M16—21no—
80C2×C3×M16—21no—
81C2×C3×M16—21no—
82C2×C3×M16—21no—
83C22×C3×C8D412yes—
84C3×#117—21no—
85C3×#117—21no—
86C3×#117—21no—
87C3×#117D412yes—
88C2×C3×C4⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C42—21no—
91C2×#214—24no—
92C2×#214—24no—
93C2×#214—24no—
94#932—28no—
95#932—28no—
96#932—28no—
97#932—28no—
98C2×#213—24no—
99C2×#213—24no—
100C2×#213—24no—
101C2×#208—21no—
102C2×#208—21no—
103C2×#208—21no—
104C2×#208—21no—
105C2×#208—21no—
106C2×#208—21no—
107C2×#208—21no—
108C2×#208—21no—
109C2×#208—21no—
110C2×#208—21no—
111C2×#208—21no—
112C2×#208—21no—
113C2×#208—21no—
114C2×#208—21no—
115C2×#208D412yes—
116C2×#208D412yes—
117C2×#208D412yes—
118C2×#208D412yes—
119C22×C3⋊C8—21no—
120C22×C3⋊C8—21no—
121C22×C3⋊C8C2×C412yes—
122C3×#102—28no—
123#833—28no—
124(C2×D3)⋊C8—28no—
125#805—28no—
126C22×#41—31no—
127C23×D4C2×D311yescharacteristic
128C2×AGL(1,ℤ/8)—62no—
129C2×AGL(1,ℤ/8)—62no—
130C2×AGL(1,ℤ/8)—62no—
131C2×AGL(1,ℤ/8)—62no—
132C2×AGL(1,ℤ/8)—62no—
133C2×AGL(1,ℤ/8)—62no—
134C22×SD16—32no—
135C22×D8—32no—
136C22×M16—31no—
137C22×M16—31no—
138C22×M16—31no—
139C22×M16C2×D311yescharacteristic
140C2×#117C2×D311yescharacteristic
141#421—62no—
142#421—62no—
143C2×#104—62no—
144#418—22no—
145#418—22no—
146C2×#102—22no—
147C2×#100—62no—
148C2×#100—62no—
149C2×#100—62no—
150#408—62no—
151#408—62no—
152#407—62no—
153#344—62no—
154#338—62no—
155C24×C3—22no—
156C24×C3—22no—
157C24×C3—22no—
158C24×C3—41no—
159C24×C3—41no—
160C24×C3—41no—
161C24×C3—41no—
162C24×C3—84no—
163C24×C3#4111yescharacteristic
164C23×D3—21no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—41no—
181C2×C3×D4—41no—
182C2×C3×D4—41no—
183C2×C3×D4—41no—
184C2×C3×D4—41no—
185C2×C3×D4—41no—
186C2×C3×D4—41no—
187C2×C3×D4—41no—
188C2×C3×D4—41no—
189C2×C3×D4—41no—
190C2×C3×D4—41no—
191C2×C3×D4—41no—
192C2×C3×D4C22⋊C414yes—
193C22×C3×C4—41no—
194C22×C3×C4C22×C411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4#4111yescharacteristic
206C22×C3×C4#4111yescharacteristic
207C22×C3×C4#4111yescharacteristic
208C2×#67—22no—
209C2×#67—22no—
210C2×#67—41no—
211C2×#67—41no—
212C2×#67—41no—
213C2×#67—41no—
214C22×Dic3—21no—
215#235—22no—
216#235—22no—
217#235—22no—
218#235—22no—
219#235—22no—
220#235—22no—
221#235—22no—
222#235—22no—
223#235—41no—
224#235—41no—
225#235—41no—
226#235—41no—
227#235—41no—
228#235—41no—
229#235—41no—
230#235—41no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12C22⋊C412yes—
237C2×C4×D3—22no—
238C2×C4×D3—22no—
239C2×C4×D3—41no—
240C2×C4×D3—41no—
241C2×C4×D3—41no—
242C2×C4×D3—41no—
243C2×Dic6—21no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6C22⋊C412yes—
249C3×M16—41no—
250C3×M16—41no—
251C3×M16—41no—
252C3×M16—41no—
253C2×C3×C8—22no—
254C2×C3×C8—22no—
255C2×C3×C8—22no—
256C3×C4⋊C4—22no—
257C3×C4⋊C4—41no—
258C3×C22⋊C4—22no—
259C3×C22⋊C4—41no—
260C3×C42—22no—
261C3×C42—41no—
262#214—216no—
263#213—216no—
264#208—22no—
265#208—22no—
266#208—22no—
267#208—22no—
268#208—22no—
269#208—22no—
270#208—22no—
271#208—22no—
272#208—41no—
273#208—41no—
274#208—41no—
275#208—41no—
276C2×C3⋊C8—22no—
277C2×C3⋊C8—22no—
278C2×C3⋊C8—22no—
279C2×C3⋊C8—22no—
280C2×C3⋊C8—41no—
281C2×C3⋊C8—41no—
282C2×C3⋊C8—41no—
283C2×C3⋊C8—41no—
284C2×C3⋊C8C22⋊C414yes—
285C25—21no—
286C2×#41—32no—
287C2×#41—32no—
288C2×#41—61no—
289C2×#41—61no—
290C2×#41—61no—
291C2×#41—61no—
292C2×#41—61no—
293C2×#41—61no—
294C2×#41—61no—
295C2×#41—61no—
296C2×#41—61no—
297C2×#41—61no—
298C22×Q8—31no—
299C22×D4—21no—
300C22×D4—21no—
301C22×D4—21no—
302C22×D4—21no—
303C22×D4—21no—
304C22×D4—21no—
305C22×D4—21no—
306C22×D4—21no—
307C22×D4—21no—
308C22×D4—21no—
309C22×D4—21no—
310C22×D4—31no—
311C22×D4—61no—
312C22×D4C4×D312yes—
313C22×D4D1212yes—
314C22×D4#6712yes—
315C23×C4—61no—
316C23×C4C22×D311yescharacteristic
317AGL(1,ℤ/8)—68no—
318AGL(1,ℤ/8)—68no—
319AGL(1,ℤ/8)—68no—
320AGL(1,ℤ/8)—68no—
321C2×SD16—64no—
322C2×SD16—64no—
323C2×SD16—64no—
324C2×D8—64no—
325C2×D8—64no—
326C2×D8—64no—
327C2×M16—21no—
328C2×M16—21no—
329C2×M16—21no—
330C2×M16—21no—
331C2×M16—21no—
332C2×M16—21no—
333C2×M16—32no—
334C2×M16—32no—
335C2×M16—32no—
336C2×M16—32no—
337C2×M16—61no—
338C2×M16—61no—
339C2×M16—61no—
340C2×M16—61no—
341C2×M16—61no—
342C2×M16—61no—
343C2×M16—61no—
344C2×M16—61no—
345C2×M16—61no—
346C2×M16—61no—
347C2×M16—61no—
348C2×M16—61no—
349C2×M16—61no—
350C2×M16—61no—
351C22×C8—32no—
352C22×C8—61no—
353C22×C8—61no—
354C22×C8#6712yes—
355#117—21no—
356#117—21no—
357#117—21no—
358#117#6712yes—
359C2×C4⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C42—21no—
362#104—68no—
363#102—28no—
364#100—68no—
365C22⋊C8—68no—
366C23×C3—24no—
367C23×C3—42no—
368C23×C3—42no—
369C23×C3—42no—
370C23×C3—42no—
371C23×C3—42no—
372C23×C3—42no—
373C23×C3—42no—
374C23×C3—84no—
375C23×C3—84no—
376C23×C3—84no—
377C23×C3—84no—
378C23×C3—84no—
379C23×C3—84no—
380C23×C3—84no—
381C23×C3C4×D411yescharacteristic
382C23×C3#12111yescharacteristic
383C23×C3#12111yescharacteristic
384C23×C3#12111yescharacteristic
385C23×C3#12311yescharacteristic
386C23×C3#12311yescharacteristic
387C23×C3#12411yescharacteristic
388C22×D3—22no—
389C22×D3—41no—
390C22×D3—41no—
391C22×D3—82no—
392C3×D4—24no—
393C3×D4—24no—
394C3×D4—42no—
395C3×D4—42no—
396C3×D4—42no—
397C3×D4—42no—
398C3×D4—42no—
399C3×D4—42no—
400C2×C3×C4—42no—
401C2×C3×C4—82no—
402C2×C3×C4C2×C22⋊C411yescharacteristic
403C2×C3×C4C2×C22⋊C411yescharacteristic
404C2×C3×C4C2×C22⋊C411yescharacteristic
405C2×C3×C4C4×D411yescharacteristic
406C2×C3×C4C4×D411yescharacteristic
407C2×C3×C4C4×D411yescharacteristic
408C2×C3×C4#12011yescharacteristic
409C2×C3×C4#12011yescharacteristic
410C2×C3×C4#12011yescharacteristic
411C2×C3×C4#12011yescharacteristic
412C2×C3×C4#12011yescharacteristic
413C2×C3×C4#12011yescharacteristic
414C2×C3×C4#12011yescharacteristic
415C2×C3×C4#12011yescharacteristic
416C2×C3×C4#12111yescharacteristic
417C2×C3×C4#12111yescharacteristic
418C2×C3×C4#12111yescharacteristic
419C2×C3×C4#12111yescharacteristic
420C2×C3×C4#12111yescharacteristic
421C2×C3×C4#12111yescharacteristic
422C2×C3×C4#12111yescharacteristic
423C2×C3×C4#12111yescharacteristic
424C2×C3×C4#12111yescharacteristic
425C2×C3×C4#12311yescharacteristic
426C2×C3×C4#12311yescharacteristic
427C2×C3×C4#12411yescharacteristic
428C2×C3×C4#12711yescharacteristic
429C2×C3×C4#12711yescharacteristic
430#67—42no—
431#67—42no—
432#67—42no—
433#67—42no—
434C2×Dic3—22no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437D12—22no—
438D12—22no—
439D12—41no—
440D12—41no—
441C4×D3—42no—
442C4×D3—42no—
443C4×D3—42no—
444C4×D3—42no—
445Dic6—22no—
446Dic6—22no—
447Dic6—41no—
448Dic6—41no—
449C3×C8—42no—
450C3⋊C8—24no—
451C3⋊C8—42no—
452C3⋊C8—42no—
453C24—22no—
454C24—22no—
455C24—22no—
456C24—41no—
457C24—41no—
458C24—41no—
459C24—41no—
460C24—61no—
461C24—84no—
462C24#23711yescharacteristic
463#41—62no—
464#41—62no—
465#41—62no—
466#41—62no—
467#41—62no—
468#41—62no—
469#41—62no—
470#41—62no—
471#41—121no—
472#41—121no—
473#41—121no—
474#41—121no—
475#41—121no—
476#41—121no—
477#41—121no—
478#41—121no—
479C2×Q8—32no—
480C2×Q8—61no—
481C2×Q8—61no—
482C2×Q8—61no—
483C2×Q8—61no—
484C2×Q8—61no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—22no—
494C2×D4—22no—
495C2×D4—22no—
496C2×D4—22no—
497C2×D4—22no—
498C2×D4—22no—
499C2×D4—22no—
500C2×D4—32no—
501C2×D4—41no—
502C2×D4—41no—
503C2×D4—41no—
504C2×D4—41no—
505C2×D4—41no—
506C2×D4—41no—
507C2×D4—41no—
508C2×D4—41no—
509C2×D4—41no—
510C2×D4—41no—
511C2×D4—41no—
512C2×D4—41no—
513C2×D4—62no—
514C2×D4—62no—
515C2×D4—61no—
516C2×D4—61no—
517C2×D4—61no—
518C2×D4—61no—
519C2×D4—61no—
520C2×D4—121no—
521C2×D4—121no—
522C2×D4—121no—
523C2×D4—121no—
524C2×D4(C2×D3)⋊C414yes—
525C22×C4—41no—
526C22×C4—61no—
527C22×C4—62no—
528C22×C4—62no—
529C22×C4—121no—
530C22×C4—121no—
531C22×C4—121no—
532C22×C4—121no—
533C22×C4C2×C4×D311yescharacteristic
534C22×C4C2×D1211yescharacteristic
535C22×C4#23511yescharacteristic
536C22×C4#23511yescharacteristic
537C22×C4D3×D411yescharacteristic
538C22×C4D3×D411yescharacteristic
539C22×C4D3×D411yescharacteristic
540C22×C4D3×D411yescharacteristic
541C22×C4D3×D411yescharacteristic
542C22×C4D3×D411yescharacteristic
543C22×C4#23711yescharacteristic
544C22×C4C2×#6711yescharacteristic
545C22×C4C2×#6711yescharacteristic
546C22×C4C2×#6711yescharacteristic
547SD16—616no—
548D8—616no—
549M16—41no—
550M16—41no—
551M16—41no—
552M16—41no—
553M16—62no—
554M16—62no—
555M16—62no—
556M16—62no—
557M16—62no—
558M16—62no—
559M16—62no—
560M16—62no—
561M16—121no—
562M16—121no—
563M16—121no—
564M16—121no—
565C2×C8—22no—
566C2×C8—22no—
567C2×C8—22no—
568C2×C8—34no—
569C2×C8—62no—
570C2×C8—62no—
571C2×C8—62no—
572C2×C8—62no—
573C2×C8—121no—
574C2×C8—121no—
575C2×C8—121no—
576C2×C8—121no—
577C4⋊C4—22no—
578C4⋊C4—41no—
579C22⋊C4—22no—
580C22⋊C4—41no—
581C42—22no—
582C42—41no—
583C22×C3—22no—
584C22×C3—22no—
585C22×C3—22no—
586C22×C3—41no—
587C22×C3—41no—
588C22×C3—41no—
589C22×C3—41no—
590C22×C3—44no—
591C22×C3—84no—
592C22×C3—84no—
593C22×C3—84no—
594C22×C3—84no—
595C22×C3—84no—
596C22×C3—84no—
597C22×C3—84no—
598C22×C3#38611yescharacteristic
599C22×C3#38611yescharacteristic
600C22×C3#39011yescharacteristic
601C22×C3#39211yescharacteristic
602C22×C3#39411yescharacteristic
603C22×C3#39411yescharacteristic
604C22×C3#39411yescharacteristic
605C2×D3—44no—
606C2×D3—42no—
607C2×D3—82no—
608C2×D3—82no—
609C3×C4—82no—
610C3×C4#37911yescharacteristic
611C3×C4#38611yescharacteristic
612C3×C4#38611yescharacteristic
613C3×C4#39011yescharacteristic
614C3×C4#39211yescharacteristic
615C3×C4#39211yescharacteristic
616C3×C4#39211yescharacteristic
617C3×C4#39411yescharacteristic
618Dic3—42no—
619C23—24no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23—42no—
624C23—42no—
625C23—42no—
626C23—42no—
627C23—62no—
628C23—84no—
629C23—84no—
630C23—84no—
631C23—84no—
632C23—84no—
633C23—84no—
634C23—84no—
635C23—121no—
636C23—121no—
637C23—242no—
638C23#67⋊C411yescharacteristic
639C23#88311yescharacteristic
640C23#88511yescharacteristic
641C23#88611yescharacteristic
642C23#93811yescharacteristic
643C23#93911yescharacteristic
644C23#93911yescharacteristic
645Q8—62no—
646Q8—62no—
647Q8—121no—
648Q8—121no—
649D4—24no—
650D4—24no—
651D4—42no—
652D4—42no—
653D4—42no—
654D4—42no—
655D4—42no—
656D4—42no—
657D4—62no—
658D4—62no—
659D4—122no—
660D4—122no—
661D4—122no—
662D4—122no—
663D4—121no—
664D4—121no—
665C2×C4—42no—
666C2×C4—62no—
667C2×C4—82no—
668C2×C4—121no—
669C2×C4—121no—
670C2×C4—122no—
671C2×C4—122no—
672C2×C4—122no—
673C2×C4—122no—
674C2×C4D3×C22⋊C411yescharacteristic
675C2×C4D3×C22⋊C411yescharacteristic
676C2×C4#67⋊C411yescharacteristic
677C2×C4#88211yescharacteristic
678C2×C4#88211yescharacteristic
679C2×C4#88211yescharacteristic
680C2×C4#88311yescharacteristic
681C2×C4#88411yescharacteristic
682C2×C4#88411yescharacteristic
683C2×C4#88411yescharacteristic
684C2×C4#88411yescharacteristic
685C2×C4#88511yescharacteristic
686C2×C4C2×(C2×D3)⋊C411yescharacteristic
687C2×C4C4×#6711yescharacteristic
688C2×C4C4×#6711yescharacteristic
689C2×C4#92911yescharacteristic
690C2×C4#93011yescharacteristic
691C2×C4#93011yescharacteristic
692C2×C4#93711yescharacteristic
693C2×C4#93711yescharacteristic
694C2×C4#93711yescharacteristic
695C2×C4#93711yescharacteristic
696C2×C4#93811yescharacteristic
697C2×C4#93911yescharacteristic
698C2×C4#93911yescharacteristic
699C2×C4#94011yescharacteristic
700C2×C4#94011yescharacteristic
701C2×C4#95311yescharacteristic
702C8—42no—
703C8—64no—
704C8—122no—
705C8—122no—
706C2×C3—21no—
707C2×C3—21no—
708C2×C3—21no—
709C2×C3—21no—
710C2×C3—21no—
711C2×C3—21no—
712C2×C3—84no—
713C2×C3#143811yescharacteristic
714C2×C3#190312yes—
715D3—82no—
716C22—22no—
717C22—22no—
718C22—22no—
719C22—41no—
720C22—41no—
721C22—41no—
722C22—41no—
723C22—44no—
724C22—84no—
725C22—84no—
726C22—84no—
727C22—84no—
728C22—84no—
729C22—84no—
730C22—84no—
731C22—122no—
732C22—124no—
733C22—242no—
734C22—242no—
735C22#495611yescharacteristic
736C22#495711yescharacteristic
737C22#495811yescharacteristic
738C22#495911yescharacteristic
739C22#496111yescharacteristic
740C22#496111yescharacteristic
741C22#522311yescharacteristic
742C4—82no—
743C4—122no—
744C4#495511yescharacteristic
745C4#495611yescharacteristic
746C4#495711yescharacteristic
747C4#495911yescharacteristic
748C4#496011yescharacteristic
749C4#496011yescharacteristic
750C4#521311yescharacteristic
751C4#522311yescharacteristic
752C3#1101411yescharacteristic, complemented, Sylow
753C2—21no—
754C2—21no—
755C2—21no—
756C2—21no—
757C2—21no—
758C2—21no—
759C2—84no—
760C2—242no—
761C2#7032912yes—
762C2#7155011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#418C2
socleC22×C3#386
prime radicalC22×#932C2
cosocleC23×C3×C4C23
hypercenterC2×#418D3
hypocenterC3#11014
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
2. C2×C3×#418C2semidirect
3. C22×#833C2semidirect
4. #83762C2semidirect
5. C2×#5005C2semidirect
6. C2×C3⋊#338C2semidirect
7. C2×#4533C2semidirect
13. C22×#214C4semidirect
20. C22×#213C4semidirect
40. C2×#418D3semidirect
752. C3#11014semidirect

Maximal subgroups

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

subgroupindex
C22×#9322
C2×C3×#4182
C22×#8332
#837622
C2×#50052
C2×C3⋊#3382
C2×#45332
#110143