Finite Group Databasebeta

← #10800518

Subgroups — #10800518

Order 768.

1122 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86019C211yesmaximal, characteristic
2Dic3×#117C211yesmaximal, characteristic, complemented
3#78620C211yesmaximal, characteristic, complemented
4C3×#1897C211yesmaximal, characteristic, complemented
5#71711C211yesmaximal, characteristic, complemented
6#71591C211yesmaximal, characteristic, complemented
7#70357C211yesmaximal, characteristic, complemented
8#70356C211yesmaximal, characteristic, complemented
9#70354C211yesmaximal, characteristic, complemented
10Dic3×#104C211yesmaximal, characteristic, complemented
11#69965C211yesmaximal, characteristic, complemented
12#68732C211yesmaximal, characteristic, complemented
13#68731C211yesmaximal, characteristic, complemented
14#68729C211yesmaximal, characteristic, complemented
15#68728C211yesmaximal, characteristic, complemented
16#21419—31nomaximal, Sylow
17C3×#525C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19#5832C2211yescharacteristic
20Dic3×#41C2211yescharacteristic
21#5791C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24#5665—22no—
25#5643C412yescomplemented
26#5625C412yescomplemented
27#5619C412yescomplemented
28#5603C412yescomplemented
29#5593—22no—
30#5557—22no—
31#5538—22no—
32#5522C2211yescharacteristic
33#5517C2211yescharacteristic
34C2×C3×#104C2211yescharacteristic
35C2×C3×#104C2211yescharacteristic
36C3×#407C2211yescharacteristic
37C2×#837C2211yescharacteristic
38C2×#837C2211yescharacteristic
39#5207C2211yescharacteristic
40C2×#825C2211yescharacteristic
41C2×#825C2211yescharacteristic
42#5118C2211yescharacteristic
43#5118C2211yescharacteristic
44Dic3×M16C2211yescharacteristic
45Dic3×M16C2211yescharacteristic
46Dic3×C4⋊C4—22no—
47Dic3×C22⋊C4—22no—
48#4932C2211yescharacteristic
49#4932C2211yescharacteristic
50C42×Dic3C2211yescharacteristic
51C42×Dic3C2211yescharacteristic
52C2×#805C2211yescharacteristic
53C2×#805C2211yescharacteristic
54#4720—22no—
55#4720—22no—
56C3×#339C2211yescharacteristic
57C3×#339C2211yescharacteristic
58#4555C2211yescharacteristic
59#4555C2211yescharacteristic
60#4532C2211yescharacteristic
61#4532C2211yescharacteristic
62#4523C2211yescharacteristic
63#4523C2211yescharacteristic
64#3455—31no—
65#2921—31no—
66C4×#117—31no—
67#1995—31no—
68#1994—31no—
69#1963—31no—
70#1962—31no—
71#1897—31no—
72#1897D311yescharacteristic, complemented
73#1896—31no—
74#1896—31no—
75#1759—31no—
76#1759—31no—
77C4×#104—31no—
78C4×#104—31no—
79C2×C3×#41C2311yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C23×Dic3—21no—
82C2×#235C2311yescharacteristic
83C2×C3×M16C2311yescharacteristic
84C2×C3×M16C2311yescharacteristic
85C3×#123—21no—
86C3×C22⋊Q8C2×C412yes—
87C3×#121C2×C412yes—
88C3×#120—21no—
89C3×C4×D4—22no—
90C3×#117—22no—
91C3×#117—22no—
92C3×#117C2311yescharacteristic
93C2×C3×C4⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97#953—21no—
98C2×#217D412yes—
99#947C2×C411yescharacteristic, complemented
100Q8×Dic3C2×C412yescomplemented
101#944C2×C411yescharacteristic, complemented
102#940C2×C411yescharacteristic, complemented
103#939—22no—
104#939—22no—
105#938—22no—
106#936—22no—
107#936C2×C411yescharacteristic, complemented
108#935—22no—
109D4×Dic3—22no—
110D4×Dic3C2×C412yescomplemented
111#930C2×C412yes—
112#929—21no—
113C4×#67—22no—
114C4×#67—22no—
115#926—21no—
116#926C2×C412yes—
117#926C2×C412yes—
118#926D412yes—
119#926D412yes—
120#926C2311yescharacteristic
121C2×C4⋊Dic3D412yes—
122#924C2×C412yes—
123C2×C4×Dic3—21no—
124C2×C4×Dic3—21no—
125C2×C4×Dic3—22no—
126C2×C4×Dic3—22no—
127C2×C4×Dic3—22no—
128C2×C4×Dic3—22no—
129C2×C4×Dic3C2311yescharacteristic
130C2×C4×Dic3C2311yescharacteristic
131C2×C4×Dic3C2311yescharacteristic
132C2×C4×Dic3C2311yescharacteristic
133C2×#208C2311yescharacteristic
134C2×#208C2311yescharacteristic
135#898—22no—
136#897—22no—
137#893C2×C412yescomplemented
138#890—22no—
139#890C2×C412yescomplemented
140#889—22no—
141#888C2×C412yescomplemented
142#887C2×C412yescomplemented
143#886—22no—
144#885—22no—
145#883—22no—
146#883—22no—
147#67⋊C4—22no—
148#879—22no—
149#879—22no—
150#879—22no—
151#878—22no—
152#878—22no—
153#872—21no—
154C42×D3—21no—
155C4×Dic6—22no—
156C3×#104—22no—
157C3×#104—22no—
158C3×C22⋊C8—22no—
159#837—22no—
160#837—22no—
161C3⋊#95—22no—
162C3⋊#95—22no—
163#830—22no—
164#825—22no—
165#825—22no—
166#815—21no—
167#815—21no—
168Dic3⋊C8—22no—
169Dic3⋊C8—22no—
170C8×Dic3—21no—
171C8×Dic3—21no—
172#805—22no—
173#805—22no—
174#542—32no—
175#542—32no—
176#541—32no—
177#540—32no—
178#539—62no—
179#538—62no—
180#533—31no—
181#532—31no—
182#529—62no—
183#529—62no—
184#525—31no—
185#525C2×D311yescharacteristic
186C4×#41—31no—
187C4×#41—31no—
188C2×#117—31no—
189C2×#117C2×D311yescharacteristic
190C22×C42—31no—
191#433—62no—
192#433—62no—
193#423—31no—
194#423—31no—
195C2×#104—31no—
196C2×#104—31no—
197C2×#104—31no—
198C2×#104—31no—
199C2×#104—31no—
200C2×#104—31no—
201C2×#104C2×D311yescharacteristic
202C2×#104C2×D311yescharacteristic
203#407—31no—
204#407C2×D311yescharacteristic
205C4×M16—31no—
206C4×M16—31no—
207C4×C4⋊C4—62no—
208C4×C22⋊C4—62no—
209#376—31no—
210#376—31no—
211C43—31no—
212C43—31no—
213#339—31no—
214#339—31no—
215#339—31no—
216#339—31no—
217#339—31no—
218#339—31no—
219#339C2×D311yescharacteristic
220#339C2×D311yescharacteristic
221C24×C3C2×D411yescharacteristic
222C3×#41—21no—
223C3×#41—21no—
224C3×#41—21no—
225C3×#41—21no—
226C2×C3×Q8C22×C411yescharacteristic
227C2×C3×D4—21no—
228C2×C3×D4—22no—
229C2×C3×D4C22×C411yescharacteristic
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4—21no—
234C22×C3×C4—21no—
235C22×C3×C4—21no—
236C22×C3×C4—21no—
237C22×C3×C4C22×C412yes—
238C22×C3×C4C2×D411yescharacteristic
239C22×C3×C4C2×D412yes—
240C22×C3×C4C2411yescharacteristic
241C2×#67—21no—
242C2×#67—22no—
243C2×#67—22no—
244C22×Dic3—21no—
245C22×Dic3—24no—
246C22×Dic3—21no—
247C22×Dic3—22no—
248C22×Dic3—44no—
249C22×Dic3C2×D412yes—
250C22×Dic3C2×D412yes—
251#235—21no—
252#235—21no—
253#235—21no—
254#235—21no—
255C2×D12C22×C411yescharacteristic
256C2×C4×D3—21no—
257C2×C4×D3—22no—
258C2×Dic6—22no—
259C2×Dic6C22×C411yescharacteristic
260C3×M16—21no—
261C3×M16—21no—
262C3×M16—21no—
263C3×M16—21no—
264C2×C3×C8—21no—
265C2×C3×C8—21no—
266C3×C4⋊C4—24no—
267C3×C4⋊C4—22no—
268C3×C4⋊C4C22×C412yes—
269C3×C22⋊C4—24no—
270C3×C22⋊C4—22no—
271C3×C22⋊C4—21no—
272C3×C22⋊C4—22no—
273C3×C42—21no—
274C3×C42—21no—
275C3×C42—21no—
276C3×C42#4112yes—
277C3×C42#4112yes—
278#217—22no—
279#217—21no—
280#217—22no—
281#217—22no—
282#217C22⋊C412yes—
283#217C22⋊C412yes—
284#217C22⋊C412yes—
285#217C22⋊C412yes—
286(C2×D3)⋊C4—22no—
287(C2×D3)⋊C4—22no—
288(C2×D3)⋊C4—22no—
289C4⋊Dic3—22no—
290C4⋊Dic3—22no—
291C4⋊Dic3C22⋊C412yes—
292C4⋊Dic3C22⋊C412yes—
293C4⋊Dic3C22⋊C412yes—
294C4⋊Dic3C22⋊C412yes—
295C4⋊Dic3C22×C412yes—
296Dic3⋊C4—22no—
297Dic3⋊C4—22no—
298Dic3⋊C4—22no—
299Dic3⋊C4—22no—
300Dic3⋊C4—22no—
301C4×Dic3—24no—
302C4×Dic3—21no—
303C4×Dic3—21no—
304C4×Dic3—21no—
305C4×Dic3—22no—
306C4×Dic3—22no—
307C4×Dic3—44no—
308C4×Dic3—44no—
309C4×Dic3C22×C412yes—
310C4×Dic3C22×C411yescharacteristic
311C4×Dic3C22×C411yescharacteristic
312C4×Dic3C2×D411yescharacteristic
313C4×Dic3C2×D411yescharacteristic
314C4×Dic3C2×D412yes—
315C4×Dic3#4112yes—
316C4×Dic3#4112yes—
317#208—21no—
318#208—21no—
319#208—21no—
320#208—21no—
321C2×C3⋊C8—21no—
322C2×C3⋊C8—21no—
323C2×#41—31no—
324C2×#41C22×D311yescharacteristic
325C23×C4—61no—
326C23×C4C22×D311yescharacteristic
327C2×M16—31no—
328C2×M16—31no—
329C2×M16C22×D311yescharacteristic
330C2×M16C22×D311yescharacteristic
331C4⋊Q8—32no—
332C4⋊Q8—31no—
333#127—31no—
334#126—62no—
335#126—62no—
336#126—62no—
337#126—62no—
338#125—32no—
339#125—62no—
340#125—62no—
341#124—31no—
342#124—31no—
343#124—62no—
344#124—62no—
345#123—21no—
346#123—62no—
347#123—62no—
348#123—62no—
349#123—62no—
350#123—61no—
351C22⋊Q8—32no—
352C22⋊Q8—62no—
353C22⋊Q8—62no—
354C22⋊Q8—62no—
355C22⋊Q8C4×D312yes—
356#121—32no—
357#121—62no—
358#121—62no—
359#121—62no—
360#121C4×D312yes—
361#120—21no—
362#120—61no—
363C4×Q8—32no—
364C4×Q8—32no—
365C4×Q8—62no—
366C4×D4—22no—
367C4×D4—32no—
368C4×D4—32no—
369C4×D4—62no—
370C4×D4—62no—
371C4×D4—62no—
372C4×D4—62no—
373#117—22no—
374#117—22no—
375#117—32no—
376#117—32no—
377#117—32no—
378#117—32no—
379#117—31no—
380#117—61no—
381#117—61no—
382#117C22×D311yescharacteristic
383C2×C4⋊C4—21no—
384C2×C4⋊C4—32no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—32no—
387C2×C42—31no—
388C2×C42—31no—
389C2×C42—31no—
390C2×C42—31no—
391C2×C42—61no—
392C2×C42—61no—
393C2×C42—62no—
394C2×C42—62no—
395C2×C42—61no—
396C2×C42—62no—
397C2×C42—62no—
398C2×C42C22×D311yescharacteristic
399C2×C42C22×D311yescharacteristic
400C4⋊C8—62no—
401C4⋊C8—62no—
402#104—22no—
403#104—22no—
404#104—62no—
405#104—62no—
406#104—62no—
407#104—62no—
408#104—62no—
409#104—62no—
410C22⋊C8—22no—
411C22⋊C8—62no—
412#97—61no—
413#97—61no—
414C4×C8—61no—
415C4×C8—61no—
416#95—62no—
417#95—62no—
418C23×C3—21no—
419C23×C3—21no—
420C23×C3—21no—
421C23×C3—42no—
422C23×C3C2×C22⋊C412yes—
423C23×C3C22×D411yescharacteristic
424C22×D3—21no—
425C3×Q8—21no—
426C3×Q8—21no—
427C3×D4—21no—
428C3×D4—21no—
429C3×D4—41no—
430C3×D4—41no—
431C3×D4—82no—
432C2×C3×C4—21no—
433C2×C3×C4—21no—
434C2×C3×C4—22no—
435C2×C3×C4—22no—
436C2×C3×C4—22no—
437C2×C3×C4—22no—
438C2×C3×C4—21no—
439C2×C3×C4—22no—
440C2×C3×C4—22no—
441C2×C3×C4—22no—
442C2×C3×C4—42no—
443C2×C3×C4—42no—
444C2×C3×C4—41no—
445C2×C3×C4—41no—
446C2×C3×C4C2×C22⋊C412yes—
447C2×C3×C4C23×C411yescharacteristic
448C2×C3×C4C22×D411yescharacteristic
449C2×C3×C4C2×#4111yescharacteristic
450C2×C3×C4C2×#4111yescharacteristic
451C2×C3×C4C2×#4111yescharacteristic
452C2×C3×C4C2×#4111yescharacteristic
453#67—41no—
454#67—41no—
455#67—82no—
456#67—82no—
457C2×Dic3—24no—
458C2×Dic3—22no—
459C2×Dic3—21no—
460C2×Dic3—22no—
461C2×Dic3—22no—
462C2×Dic3—42no—
463C2×Dic3—42no—
464C2×Dic3—48no—
465C2×Dic3—42no—
466C2×Dic3—82no—
467C2×Dic3C2×C22⋊C412yes—
468C2×Dic3C2×C22⋊C412yes—
469C2×Dic3C2×C22⋊C412yes—
470C2×Dic3C2×C22⋊C412yes—
471D12—21no—
472D12—21no—
473C4×D3—41no—
474C4×D3—41no—
475C4×D3—82no—
476Dic6—21no—
477Dic6—21no—
478Dic6—82no—
479C3×C8—41no—
480C3×C8—41no—
481C3⋊C8—41no—
482C3⋊C8—41no—
483C24D3×D411yescharacteristic
484#41—21no—
485#41—21no—
486#41—21no—
487#41—21no—
488#41—61no—
489#41—61no—
490#41—61no—
491#41—61no—
492C2×Q8—31no—
493C2×Q8—62no—
494C2×Q8C2×C4×D311yescharacteristic
495C2×D4—21no—
496C2×D4—22no—
497C2×D4—31no—
498C2×D4—62no—
499C2×D4—61no—
500C2×D4—62no—
501C2×D4C2×C4×D311yescharacteristic
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—32no—
510C22×C4—32no—
511C22×C4—62no—
512C22×C4—62no—
513C22×C4—61no—
514C22×C4—64no—
515C22×C4—61no—
516C22×C4—61no—
517C22×C4—124no—
518C22×C4C2×C4×D312yes—
519C22×C4D3×D411yescharacteristic
520C22×C4D3×D412yes—
521C22×C4C23×D311yescharacteristic
522M16—21no—
523M16—21no—
524M16—21no—
525M16—21no—
526M16—61no—
527M16—61no—
528M16—61no—
529M16—61no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—61no—
533C2×C8—61no—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—32no—
537C4⋊C4—32no—
538C4⋊C4—32no—
539C4⋊C4—32no—
540C4⋊C4—32no—
541C4⋊C4—62no—
542C4⋊C4—62no—
543C4⋊C4—62no—
544C4⋊C4—62no—
545C4⋊C4—62no—
546C4⋊C4—62no—
547C4⋊C4—62no—
548C4⋊C4C2×C4×D312yes—
549C22⋊C4—24no—
550C22⋊C4—22no—
551C22⋊C4—21no—
552C22⋊C4—22no—
553C22⋊C4—32no—
554C22⋊C4—32no—
555C22⋊C4—32no—
556C22⋊C4—32no—
557C22⋊C4—62no—
558C22⋊C4—62no—
559C22⋊C4—62no—
560C22⋊C4—62no—
561C22⋊C4—62no—
562C22⋊C4—61no—
563C22⋊C4—62no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—32no—
568C42—31no—
569C42—31no—
570C42—31no—
571C42—31no—
572C42—32no—
573C42—32no—
574C42—32no—
575C42—62no—
576C42—62no—
577C42—64no—
578C42—61no—
579C42—61no—
580C42—61no—
581C42—124no—
582C42—124no—
583C42#23712yes—
584C42#23712yes—
585C22×C3—22no—
586C22×C3—22no—
587C22×C3—22no—
588C22×C3—22no—
589C22×C3—42no—
590C22×C3—42no—
591C22×C3—42no—
592C22×C3—41no—
593C22×C3C22×C22⋊C411yescharacteristic
594C22×C3#52511yescharacteristic
595C22×C3#52511yescharacteristic
596C2×D3—42no—
597C2×D3—41no—
598C3×C4—22no—
599C3×C4—42no—
600C3×C4—42no—
601C3×C4—42no—
602C3×C4—41no—
603C3×C4C4×#4111yescharacteristic
604C3×C4C4×#4111yescharacteristic
605C3×C4#53211yescharacteristic
606C3×C4#53211yescharacteristic
607Dic3—24no—
608Dic3—22no—
609Dic3—22no—
610Dic3—22no—
611Dic3—22no—
612Dic3—42no—
613Dic3—42no—
614Dic3—41no—
615C23—21no—
616C23—21no—
617C23—21no—
618C23—42no—
619C23—61no—
620C23D3×C22⋊C412yes—
621C23C2×D3×D411yescharacteristic
622Q8—21no—
623Q8—21no—
624Q8—61no—
625Q8—61no—
626Q8—242no—
627D4—21no—
628D4—21no—
629D4—41no—
630D4—41no—
631D4—61no—
632D4—61no—
633D4—82no—
634D4—121no—
635D4—121no—
636D4—242no—
637D4—242no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—21no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—32no—
649C2×C4—32no—
650C2×C4—32no—
651C2×C4—32no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—64no—
657C2×C4—62no—
658C2×C4—62no—
659C2×C4—62no—
660C2×C4—61no—
661C2×C4—122no—
662C2×C4—122no—
663C2×C4—128no—
664C2×C4—122no—
665C2×C4—121no—
666C2×C4—121no—
667C2×C4—242no—
668C2×C4—242no—
669C2×C4D3×C22⋊C412yes—
670C2×C4C22×C4×D311yescharacteristic
671C2×C4C2×D3×D411yescharacteristic
672C2×C4C2×#23711yescharacteristic
673C2×C4C2×#23711yescharacteristic
674C2×C4D3×#4111yescharacteristic
675C2×C4D3×#4111yescharacteristic
676C8—41no—
677C8—41no—
678C8—121no—
679C8—121no—
680C2×C3—21no—
681C2×C3—21no—
682C2×C3—42no—
683C2×C3—81no—
684C2×C3#229711yescharacteristic
685C2×C3#290112yes—
686D3—81no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—41no—
695C22—122no—
696C22—121no—
697C22C2×D3×C22⋊C411yescharacteristic
698C22#558611yescharacteristic
699C22#565611yescharacteristic
700C4—22no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—41no—
705C4—64no—
706C4—62no—
707C4—62no—
708C4—62no—
709C4—62no—
710C4—122no—
711C4—122no—
712C4—121no—
713C4C4×#23711yescharacteristic
714C4C4×#23711yescharacteristic
715C4#559611yescharacteristic
716C4#559611yescharacteristic
717C3#2141911yescharacteristic, complemented, Sylow
718C2—21no—
719C2—21no—
720C2—42no—
721C2—81no—
722C2—241no—
723C2#7465811yescharacteristic
724C2#7899212yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1897C2
socleC22×C3C22×C22⋊C4
prime radical#86019C2
cosocleC22×C3×C4C24
hypercenter#1897D3
hypocenterC3#21419
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1897

Complements

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

NH ≅ G/Ntype
2. Dic3×#117C2semidirect
3. #78620C2semidirect
4. C3×#1897C2semidirect
5. #71711C2semidirect
6. #71591C2semidirect
7. #70357C2semidirect
8. #70356C2semidirect
9. #70354C2semidirect
10. Dic3×#104C2semidirect
11. #69965C2semidirect
12. #68732C2semidirect
13. #68731C2semidirect
14. #68729C2semidirect
15. #68728C2semidirect
25. #5643C4semidirect
26. #5625C4semidirect
27. #5619C4semidirect
28. #5603C4semidirect
72. #1897D3semidirect
99. #947C2×C4semidirect
100. Q8×Dic3C2×C4semidirect
101. #944C2×C4semidirect
102. #940C2×C4semidirect
107. #936C2×C4semidirect
110. D4×Dic3C2×C4semidirect
137. #893C2×C4semidirect
139. #890C2×C4semidirect
141. #888C2×C4semidirect
142. #887C2×C4semidirect
717. C3#21419semidirect

Maximal subgroups

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

subgroupindex
#860192
Dic3×#1172
#786202
C3×#18972
#717112
#715912
#703572
#703562
#703542
Dic3×#1042
#699652
#687322
#687312
#687292
#687282
#214193