Finite Group Databasebeta

← #10763573

Subgroups — #10763573

Order 768.

996 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86219C211yesmaximal, characteristic
2D3×#431C211yesmaximal, characteristic, complemented
3C2×#4835C211yesmaximal, characteristic, complemented
4C3×#1540C211yesmaximal, characteristic, complemented
5#71804C211yesmaximal, characteristic, complemented
6#70776C211yesmaximal, characteristic, complemented
7#70437C211yesmaximal, characteristic, complemented
8#68427C212yesmaximal, complemented
9#68424C212yesmaximal, complemented
10#68386C212yesmaximal, complemented
11#68385C212yesmaximal, complemented
12#17713—31nomaximal, Sylow
13C3×#532C2211yescharacteristic
14#5828—21no—
15Dic3×#41—21no—
16#5728C2211yescharacteristic
17#5728C2211yescharacteristic
18#5716C2211yescharacteristic
19#5716C2211yescharacteristic
20#5680—21no—
21#5680—21no—
22#5606—21no—
23#5606C412yes—
24#5600—21no—
25#5600C412yes—
26#5557—21no—
27#5557C412yes—
28C4×#237—21no—
29C4×#237C412yes—
30C2×C42×D3C2211yescharacteristic
31#5470C2211yescharacteristic
32C3×#431C2211yescharacteristic
33C2×C3×C4⋊C8C2211yescharacteristic
34C2×(C2×D3)⋊C8—21no—
35C2×#815—21no—
36#4925C2211yescharacteristic
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38#4835—21no—
39#4835—21no—
40#4835C2212yes—
41#4835C2212yes—
42#4833C2212yes—
43D3×C4⋊C8C2212yes—
44D3×C22⋊C8—21no—
45#4718—21no—
46#4685—21no—
47C4×D3×C8—21no—
48C3×#332C2212yescomplemented
49C3×#331C2212yescomplemented
50#4547C2212yescomplemented
51#4544C2212yescomplemented
52#4486C2212yescomplemented
53#4482C2212yescomplemented
54#4481C2212yescomplemented
55#4481C2212yescomplemented
56#3444—31no—
57C2×#432—31no—
58C2×#431—31no—
59#1554—32no—
60#1553—32no—
61#1552—32no—
62#1551—32no—
63#1542—31no—
64#1542—31no—
65#1540—31no—
66#1540D311yescharacteristic, complemented
67C2×C3×#41—21no—
68C2×#237—21no—
69C2×#237C2×C412yes—
70C22×C4×D3C2×C411yescharacteristic
71C22×C4×D3D411yescharacteristic
72C22×C4×D3D411yescharacteristic
73C22×C3×C8—21no—
74C3×C4⋊Q8C2311yescharacteristic
75C3×#127C2311yescharacteristic
76C3×#124—21no—
77C3×#121—21no—
78C3×#121C2×C412yes—
79C3×C4×D4—21no—
80C3×C4×D4C2×C412yes—
81C2×C3×C42C2311yescharacteristic
82#946—21no—
83#946—21no—
84Q8×Dic3—21no—
85Q8×Dic3—21no—
86#938—21no—
87#938—21no—
88#938—21no—
89#938C2×C412yes—
90#937—22no—
91#936—21no—
92#936C2×C412yes—
93#935—22no—
94D4×Dic3—22no—
95D4×Dic3—22no—
96D4×Dic3—21no—
97D4×Dic3—21no—
98D4×Dic3—21no—
99D4×Dic3C2×C412yes—
100C4×#67—22no—
101C4×#67—22no—
102#926—21no—
103#926—22no—
104#924—21no—
105#924C2×C412yes—
106C2×C4×Dic3—22no—
107C2×C4×Dic3C2×C411yescharacteristic
108C2×C4×Dic3D411yescharacteristic
109C2×C4×Dic3D411yescharacteristic
110C22×C3⋊C8—21no—
111C2×D3×C8—22no—
112#897—21no—
113#897C2×C412yes—
114#892—21no—
115#892C2×C412yes—
116#890—21no—
117#890C2×C412yes—
118#887—21no—
119#887C2×C412yes—
120#883—22no—
121#883—22no—
122#67⋊C4—22no—
123#67⋊C4—22no—
124#879—22no—
125#879—22no—
126#877—22no—
127#877—22no—
128C42×D3C2×C411yescharacteristic
129C42×D3C2×C411yescharacteristic
130C42×D3D411yescharacteristic
131C42×D3D411yescharacteristic
132C42×D3D411yescharacteristic
133C42×D3D411yescharacteristic
134C42×D3C2311yescharacteristic
135C42×D3C2311yescharacteristic
136#870—21no—
137#869C2311yescharacteristic
138#869C2311yescharacteristic
139C4×Dic6—21no—
140C4×Dic6C2×C412yes—
141C3×C4⋊C8—21no—
142C3×C4⋊C8C2312yes—
143C3×C4⋊C8C2312yes—
144C3×C22⋊C8—21no—
145C3×C4×C8—21no—
146#830—21no—
147(C2×D3)⋊C8—22no—
148(C2×D3)⋊C8—22no—
149(C2×D3)⋊C8—22no—
150#815—22no—
151#815—22no—
152Dic3⋊C8—22no—
153C8×Dic3—22no—
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)C2312yes—
156C4⋊(C3⋊C8)C2312yes—
157C4×C3⋊C8—21no—
158#533—31no—
159#533—31no—
160#533—31no—
161#532—31no—
162#532—31no—
163#532C2×D311yescharacteristic
164#529—32no—
165#529—32no—
166#529—61no—
167#529—61no—
168#529—61no—
169#529—61no—
170#525—32no—
171#525—61no—
172#525—61no—
173C4×#41—32no—
174C4×#41—61no—
175C4×#41—61no—
176C22×C42—31no—
177#432—32no—
178#432—32no—
179#432—61no—
180#432—61no—
181#431—31no—
182#431—32no—
183#431—61no—
184#431—61no—
185#431C2×D311yescharacteristic
186C2×C4⋊C8—31no—
187C2×C4⋊C8—32no—
188C2×C4⋊C8C2×D311yescharacteristic
189C2×C22⋊C8—61no—
190C2×C22⋊C8—61no—
191C2×#97—61no—
192C2×C4×C8—61no—
193#333—32no—
194#333—32no—
195#332—32no—
196#332—32no—
197#332—32no—
198#332C2×D312yescomplemented
199#331—32no—
200#331C2×D312yescomplemented
201C23×D3C22⋊C411yescharacteristic
202C3×#41—42no—
203C3×#41—42no—
204C2×C3×Q8—21no—
205C2×C3×D4—22no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4C22×C412yes—
209C22×C3×C4—22no—
210C22×C3×C4C22×C411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C2×#67—22no—
214C2×#67—22no—
215C22×Dic3—22no—
216C22×Dic3—22no—
217C22×Dic3C22⋊C411yescharacteristic
218C22×Dic3C22⋊C411yescharacteristic
219C22×Dic3C22⋊C411yescharacteristic
220#237—44no—
221#237—44no—
222C2×C4×D3—22no—
223C2×C4×D3—22no—
224C2×C4×D3—42no—
225C2×C4×D3—42no—
226C2×C4×D3C22⋊C411yescharacteristic
227C2×C4×D3C22⋊C411yescharacteristic
228C2×C4×D3C22⋊C411yescharacteristic
229C2×C4×D3C22⋊C411yescharacteristic
230C2×C4×D3C22⋊C411yescharacteristic
231C2×C4×D3C22⋊C411yescharacteristic
232C2×C4×D3C22×C411yescharacteristic
233C2×C4×D3C22×C411yescharacteristic
234C2×C4×D3C2×D411yescharacteristic
235C2×C4×D3C2×D411yescharacteristic
236C2×C4×D3C2×D411yescharacteristic
237C2×C4×D3C2×D411yescharacteristic
238C2×Dic6—21no—
239C2×Dic6C22×C412yes—
240C2×C3×C8—22no—
241C2×C3×C8—22no—
242C2×C3×C8—42no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4C22×C412yes—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—22no—
247C3×C42C22×C411yescharacteristic
248C3×C42C2×D411yescharacteristic
249C3×C42C2×D411yescharacteristic
250C3×C42C2411yescharacteristic
251#217—22no—
252#217—22no—
253#217—22no—
254#217—22no—
255#217—22no—
256(C2×D3)⋊C4—22no—
257(C2×D3)⋊C4—22no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—22no—
262C4⋊Dic3—22no—
263C4⋊Dic3C22×C412yes—
264Dic3⋊C4—22no—
265Dic3⋊C4—22no—
266Dic3⋊C4—22no—
267Dic3⋊C4—22no—
268C4×Dic3—22no—
269C4×Dic3—22no—
270C4×Dic3—22no—
271C4×Dic3C22⋊C411yescharacteristic
272C4×Dic3C22⋊C411yescharacteristic
273C4×Dic3C22⋊C411yescharacteristic
274C4×Dic3C22⋊C411yescharacteristic
275C4×Dic3C22⋊C411yescharacteristic
276C4×Dic3C22⋊C411yescharacteristic
277C4×Dic3C22×C411yescharacteristic
278C4×Dic3C22×C411yescharacteristic
279C4×Dic3C2×D411yescharacteristic
280C4×Dic3C2×D411yescharacteristic
281C4×Dic3C2×D411yescharacteristic
282C4×Dic3C2×D411yescharacteristic
283C2×C3⋊C8—22no—
284C2×C3⋊C8—22no—
285C2×C3⋊C8—42no—
286D3×C8—44no—
287C2×#41—21no—
288C2×#41—32no—
289C2×#41—61no—
290C23×C4—31no—
291C23×C4—31no—
292C23×C4—31no—
293C22×C8—21no—
294C22×C8—62no—
295C22×C8—61no—
296C4⋊Q8—31no—
297C4⋊Q8—31no—
298C4⋊Q8C22×D311yescharacteristic
299#127C22×D311yescharacteristic
300#126—62no—
301#126—62no—
302#125—32no—
303#125—61no—
304#125—61no—
305#124—21no—
306#124—32no—
307#124—61no—
308#123—62no—
309#123—62no—
310#123—62no—
311C22⋊Q8—32no—
312C22⋊Q8—32no—
313C22⋊Q8—61no—
314C22⋊Q8—61no—
315C22⋊Q8—61no—
316C22⋊Q8—61no—
317#121—21no—
318#121—32no—
319#121—61no—
320#121—61no—
321#121—61no—
322#121C4×D312yes—
323#120—62no—
324C4×Q8—32no—
325C4×Q8—32no—
326C4×Q8—61no—
327C4×Q8—61no—
328C4×Q8—61no—
329C4×Q8—61no—
330C4×D4—21no—
331C4×D4—32no—
332C4×D4—61no—
333C4×D4—61no—
334C4×D4—61no—
335C4×D4—62no—
336C4×D4—62no—
337C4×D4—62no—
338C4×D4—62no—
339C4×D4—62no—
340C4×D4—62no—
341C4×D4C4×D312yes—
342#117—32no—
343#117—62no—
344#117—62no—
345#117—62no—
346#117—61no—
347#117—61no—
348C2×C42—31no—
349C2×C42—31no—
350C2×C42—31no—
351C2×C42—31no—
352C2×C42—31no—
353C2×C42—31no—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—31no—
357C2×C42—31no—
358C2×C42—31no—
359C2×C42—62no—
360C2×C42C22×D311yescharacteristic
361C4⋊C8—21no—
362C4⋊C8—32no—
363C4⋊C8—32no—
364C4⋊C8—62no—
365C4⋊C8—61no—
366C4⋊C8C22×D312yes—
367C4⋊C8C22×D312yes—
368C22⋊C8—21no—
369C22⋊C8—62no—
370C22⋊C8—62no—
371C22⋊C8—62no—
372C22⋊C8—61no—
373#97—62no—
374#97—62no—
375C4×C8—21no—
376C4×C8—62no—
377C4×C8—61no—
378C23×C3—22no—
379C23×C3C2×C22⋊C411yescharacteristic
380C22×D3—22no—
381C22×D3—22no—
382C22×D3—41no—
383C22×D3C2×C22⋊C411yescharacteristic
384C22×D3C2×C22⋊C411yescharacteristic
385C3×Q8—41no—
386C3×Q8—41no—
387C3×D4—42no—
388C3×D4—44no—
389C3×D4—41no—
390C3×D4—41no—
391C3×D4—42no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—41no—
395C2×C3×C4—44no—
396C2×C3×C4C2×C22⋊C411yescharacteristic
397C2×C3×C4C2×C22⋊C411yescharacteristic
398C2×C3×C4C2×C22⋊C411yescharacteristic
399C2×C3×C4#13512yes—
400C2×C3×C4#13512yes—
401C2×C3×C4C23×C411yescharacteristic
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C22×D411yescharacteristic
404#67—44no—
405#67—44no—
406C2×Dic3—22no—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—44no—
412C2×Dic3—44no—
413C2×Dic3C2×C22⋊C411yescharacteristic
414C2×Dic3C2×C22⋊C411yescharacteristic
415C2×Dic3C2×C22⋊C411yescharacteristic
416C2×Dic3C2×C22⋊C411yescharacteristic
417C2×Dic3C2×C22⋊C411yescharacteristic
418C2×Dic3C2×C22⋊C411yescharacteristic
419C4×D3—22no—
420C4×D3—22no—
421C4×D3—21no—
422C4×D3—21no—
423C4×D3—21no—
424C4×D3—21no—
425C4×D3—22no—
426C4×D3—22no—
427C4×D3—22no—
428C4×D3—22no—
429C4×D3—41no—
430C4×D3—41no—
431C4×D3—41no—
432C4×D3—41no—
433Dic6—42no—
434Dic6—42no—
435C3×C8—42no—
436C3×C8—42no—
437C3⋊C8—42no—
438C3⋊C8—42no—
439C24—31no—
440#41—42no—
441#41—42no—
442#41—124no—
443#41—124no—
444C2×Q8—21no—
445C2×Q8—32no—
446C2×Q8—61no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—62no—
451C2×D4—62no—
452C2×D4C2×C4×D312yes—
453C22×C4—22no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—31no—
460C22×C4—31no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—31no—
467C22×C4—31no—
468C22×C4—31no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4—122no—
474C22×C4—122no—
475C22×C4C2×C4×D311yescharacteristic
476C22×C4D3×D411yescharacteristic
477C22×C4D3×D411yescharacteristic
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—42no—
481C2×C8—62no—
482C2×C8—62no—
483C2×C8—124no—
484C2×C8—122no—
485C4⋊C4—21no—
486C4⋊C4—32no—
487C4⋊C4—62no—
488C4⋊C4—62no—
489C4⋊C4—62no—
490C4⋊C4—62no—
491C4⋊C4—62no—
492C4⋊C4—62no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4C2×C4×D312yes—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—62no—
500C22⋊C4—62no—
501C22⋊C4—62no—
502C22⋊C4—62no—
503C22⋊C4—62no—
504C22⋊C4—62no—
505C22⋊C4—62no—
506C42—31no—
507C42—31no—
508C42—31no—
509C42—31no—
510C42—31no—
511C42—31no—
512C42—31no—
513C42—31no—
514C42—31no—
515C42—31no—
516C42—31no—
517C42—31no—
518C42—62no—
519C42—62no—
520C42—62no—
521C42C2×C4×D311yescharacteristic
522C42D3×D411yescharacteristic
523C42D3×D411yescharacteristic
524C42C23×D311yescharacteristic
525C22×C3—21no—
526C22×C3—21no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3#41612yes—
531C22×C3C22×C22⋊C411yescharacteristic
532C2×D3—21no—
533C2×D3—21no—
534C2×D3—21no—
535C2×D3—21no—
536C2×D3—21no—
537C2×D3—44no—
538C2×D3#41312yes—
539C3×C4—21no—
540C3×C4—21no—
541C3×C4—21no—
542C3×C4—42no—
543C3×C4#41612yes—
544C3×C4C2×#13511yescharacteristic
545C3×C4C2×#13511yescharacteristic
546C3×C4#57512yes—
547Dic3—21no—
548Dic3—21no—
549Dic3—21no—
550Dic3—21no—
551Dic3—21no—
552Dic3—21no—
553Dic3—42no—
554Dic3—42no—
555C23—22no—
556C23—31no—
557C23—31no—
558C23—62no—
559C23—62no—
560C23—121no—
561C23D3×C22⋊C411yescharacteristic
562Q8—41no—
563Q8—41no—
564Q8—122no—
565Q8—122no—
566D4—42no—
567D4—44no—
568D4—41no—
569D4—41no—
570D4—42no—
571D4—124no—
572D4—124no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—31no—
576C2×C4—31no—
577C2×C4—31no—
578C2×C4—31no—
579C2×C4—31no—
580C2×C4—31no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—124no—
603C2×C4—124no—
604C2×C4D3×C22⋊C411yescharacteristic
605C2×C4D3×C22⋊C411yescharacteristic
606C2×C4D3×C22⋊C411yescharacteristic
607C2×C4#91212yes—
608C2×C4#91612yes—
609C2×C4C22×C4×D311yescharacteristic
610C2×C4C2×D3×D411yescharacteristic
611C2×C4C2×D3×D411yescharacteristic
612C8—42no—
613C8—42no—
614C8—122no—
615C8—122no—
616C2×C3—22no—
617C2×C3—82no—
618C2×C3C2×#41311yescharacteristic
619C2×C3C2×#41611yescharacteristic
620C2×C3#289611yescharacteristic
621D3—22no—
622D3—41no—
623C22—21no—
624C22—21no—
625C22—32no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—61no—
630C22—61no—
631C22—61no—
632C22—61no—
633C22—61no—
634C22—124no—
635C22#477212yes—
636C22C2×D3×C22⋊C411yescharacteristic
637C4—21no—
638C4—21no—
639C4—21no—
640C4—42no—
641C4—61no—
642C4—61no—
643C4—61no—
644C4—61no—
645C4—61no—
646C4—61no—
647C4—122no—
648C4—122no—
649C4#477212yes—
650C4C2×#91211yescharacteristic
651C4C2×#91611yescharacteristic
652C4#576812yes—
653C3#1771311yescharacteristic, complemented, Sylow
654C2—22no—
655C2—62no—
656C2—82no—
657C2—121no—
658C2D3×#41311yescharacteristic
659C2C2×#477211yescharacteristic
660C2#7883011yescharacteristic
subgroupG/H
centerC2×C4D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1540C2
socleC22×C3C22×C22⋊C4
prime radical#86219C2
cosocleC3×C42C24
hypercenter#1540D3
hypocenterC3#17713
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1540

Complements

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

NH ≅ G/Ntype
2. D3×#431C2semidirect
3. C2×#4835C2semidirect
4. C3×#1540C2semidirect
5. #71804C2semidirect
6. #70776C2semidirect
7. #70437C2semidirect
8. #68427C2semidirect
9. #68424C2semidirect
10. #68386C2semidirect
11. #68385C2semidirect
48. C3×#332C22semidirect
49. C3×#331C22semidirect
50. #4547C22semidirect
51. #4544C22semidirect
52. #4486C22semidirect
53. #4482C22semidirect
54. #4481C22semidirect
55. #4481C22semidirect
66. #1540D3semidirect
198. #332C2×D3semidirect
200. #331C2×D3semidirect
653. C3#17713semidirect

Maximal subgroups

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

subgroupindex
#862192
D3×#4312
C2×#48352
C3×#15402
#718042
#707762
#704372
#684272
#684242
#683862
#683852
#177133