Finite Group Databasebeta

← #10804134

Subgroups — #10804134

Order 768.

1554 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5656C211yesmaximal, characteristic
2#82267C211yesmaximal, characteristic, complemented
3D3×#381C211yesmaximal, characteristic, complemented
4C3×#1899C211yesmaximal, characteristic, complemented
5#71705C211yesmaximal, characteristic, complemented
6#71595C211yesmaximal, characteristic, complemented
7#70036C212yesmaximal, complemented
8#70027C212yesmaximal, complemented
9#69535C211yesmaximal, characteristic, complemented
10#68970C212yesmaximal, complemented
11#68953C212yesmaximal, complemented
12#21748—31nomaximal, Sylow
13C22×#237C2211yescharacteristic
14C23×C4×D3C2211yescharacteristic
15C2×C3×#123C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C2×#937C2211yescharacteristic
18C2×#929C2211yescharacteristic
19C2×#926C2211yescharacteristic
20#5656—22no—
21#5656—22no—
22#5656—22no—
23#5656—22no—
24C2×#896C412yescomplemented
25C2×#893C412yescomplemented
26C2×#884C412yescomplemented
27C2×#882C2211yescharacteristic
28C2×#67⋊C4C412yescomplemented
29C2×D3×C22⋊C4C2212yes—
30C2×#877C2211yescharacteristic
31C2×C3×#99C2212yes—
32C3×#381C2211yescharacteristic
33C2×#834C2212yes—
34#5211C2211yescharacteristic
35#4987—22no—
36#4955C2212yes—
37C2×#806C2212yes—
38C2×#806C2212yes—
39#4943C2212yes—
40#4943C2212yes—
41C4×(C2×D3)⋊C4—22no—
42#4664—22no—
43D3×#95—22no—
44C3×#342C2212yes—
45#4527C2212yes—
46#4527C2212yes—
47#4526C2212yes—
48C2×#525—31no—
49#2280—31no—
50C2×#381—31no—
51#1899—31no—
52#1899—31no—
53#1899—31no—
54#1899D311yescharacteristic, complemented
55#1898—32no—
56#1898—32no—
57#1820—32no—
58#1782—32no—
59C24×D3C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67C2×C412yes—
63C22×#67C2×C411yescharacteristic, complemented
64C22×#67C2311yescharacteristic
65C23×Dic3C2×C411yescharacteristic, complemented
66C23×Dic3C2311yescharacteristic
67C2×#237—22no—
68C2×#237—22no—
69C2×#237—42no—
70C2×#237—42no—
71C22×D12C2×C412yes—
72C22×C4×D3—22no—
73C22×C4×D3—42no—
74C22×C4×D3—42no—
75C22×C4×D3C2×C412yes—
76C22×C4×D3C2×C412yes—
77C22×C4×D3C2×C411yescharacteristic, complemented
78C22×C4×D3D412yes—
79C22×C4×D3D412yes—
80C22×C4×D3D412yes—
81C22×C4×D3D412yes—
82C22×Dic6C2×C411yescharacteristic, complemented
83C3×#123—22no—
84C3×#123—21no—
85C3×#123—21no—
86C3×#117—42no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2×C412yescomplemented
89C2×C3×C22⋊C4C2×C412yescomplemented
90C2×C3×C22⋊C4C2312yes—
91C2×C3×C22⋊C4C2312yes—
92C2×C3×C22⋊C4C2311yescharacteristic
93C2×C3×C42—21no—
94C2×#217C2312yes—
95C2×#217C2312yes—
96C2×#217C2311yescharacteristic
97#937—21no—
98#937—22no—
99#937—21no—
100#929—22no—
101#929—21no—
102#929—21no—
103C2×(C2×D3)⋊C4—22no—
104C2×(C2×D3)⋊C4—24no—
105C2×(C2×D3)⋊C4C2×C412yes—
106C2×(C2×D3)⋊C4C2×C412yes—
107C2×(C2×D3)⋊C4C2×C412yescomplemented
108#926—42no—
109C2×C4⋊Dic3—21no—
110C2×Dic3⋊C4C2×C412yescomplemented
111C2×Dic3⋊C4C2×C412yescomplemented
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2×C412yescomplemented
114#896—22no—
115#896—22no—
116#896—22no—
117#896—22no—
118#893—22no—
119#893—22no—
120#893—22no—
121#893—22no—
122#884—22no—
123#884—22no—
124#884—22no—
125#884—22no—
126#882—21no—
127#882—22no—
128#882—21no—
129#67⋊C4—22no—
130#67⋊C4—22no—
131#67⋊C4—22no—
132#67⋊C4—22no—
133D3×C22⋊C4—42no—
134D3×C22⋊C4—42no—
135#877—22no—
136#877—22no—
137C3×#99—42no—
138C3×#95—22no—
139#834—42no—
140C3⋊#95—22no—
141C3⋊#95—24no—
142C3⋊#95—24no—
143#806—42no—
144#806—42no—
145C22×#41—31no—
146C24×C4—31no—
147#525—62no—
148#525—62no—
149#525—62no—
150#525—62no—
151C2×#123—31no—
152C2×#123C2×D311yescharacteristic
153C2×C22⋊Q8—32no—
154C2×#121—32no—
155C2×#120—31no—
156C2×#120—31no—
157C2×C4×D4—32no—
158C2×C4×D4—32no—
159C2×#117—31no—
160C2×#117—31no—
161C2×#117C2×D311yescharacteristic
162C22×C22⋊C4—32no—
163C2×#99—32no—
164C2×#99—32no—
165C2×#99—32no—
166C2×#99C2×D312yes—
167#381—31no—
168#381—32no—
169#381—32no—
170#381—62no—
171#381C2×D311yescharacteristic
172#380—62no—
173#379—32no—
174C4×C22⋊C4—62no—
175C2×#95—62no—
176#342—32no—
177#342—32no—
178#342—32no—
179#342C2×D312yes—
180C24×C3C22×C411yescharacteristic
181C24×C3C2411yescharacteristic
182C23×D3—22no—
183C23×D3—22no—
184C23×D3—44no—
185C23×D3C22×C412yes—
186C23×D3C22×C411yescharacteristic
187C23×D3C2×D411yescharacteristic
188C23×D3C2×D412yes—
189C23×D3C2×D411yescharacteristic
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4C22×C411yescharacteristic
199C22×C3×C4C22×C412yes—
200C22×C3×C4C22×C412yes—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C2×#67—21no—
205C2×#67—21no—
206C2×#67—22no—
207C2×#67—22no—
208C2×#67—22no—
209C2×#67—22no—
210C2×#67—22no—
211C2×#67—42no—
212C2×#67—44no—
213C22×Dic3—22no—
214C22×Dic3—22no—
215C22×Dic3—22no—
216C22×Dic3—22no—
217C22×Dic3—22no—
218C22×Dic3—42no—
219C22×Dic3C22×C411yescharacteristic
220C22×Dic3C22×C411yescharacteristic
221C22×Dic3C22×C411yescharacteristic
222C22×Dic3C22×C412yes—
223C22×Dic3C2×D412yes—
224C22×Dic3C2×D412yes—
225#237—44no—
226#237—44no—
227#237—48no—
228C2×D12—22no—
229C2×D12—22no—
230C2×D12—44no—
231C2×C4×D3—22no—
232C2×C4×D3—44no—
233C2×C4×D3—44no—
234C2×C4×D3—42no—
235C2×C4×D3—42no—
236C2×C4×D3—42no—
237C2×C4×D3—42no—
238C2×C4×D3—44no—
239C2×C4×D3—44no—
240C2×C4×D3—42no—
241C2×C4×D3—44no—
242C2×C4×D3—48no—
243C2×C4×D3C22⋊C412yes—
244C2×C4×D3C22⋊C412yes—
245C2×C4×D3C22⋊C412yes—
246C2×C4×D3C22⋊C412yes—
247C2×C4×D3C22⋊C412yes—
248C2×C4×D3C22⋊C412yes—
249C2×C4×D3C22⋊C412yes—
250C2×C4×D3C22⋊C412yes—
251C2×Dic6—22no—
252C2×Dic6—42no—
253C3×C4⋊C4—22no—
254C3×C4⋊C4—22no—
255C3×C4⋊C4—42no—
256C3×C22⋊C4—21no—
257C3×C22⋊C4—21no—
258C3×C22⋊C4—22no—
259C3×C22⋊C4—22no—
260C3×C22⋊C4—22no—
261C3×C22⋊C4—22no—
262C3×C22⋊C4—22no—
263C3×C22⋊C4—22no—
264C3×C42—42no—
265#217—21no—
266#217—21no—
267#217—22no—
268#217—22no—
269#217—22no—
270#217—22no—
271(C2×D3)⋊C4—22no—
272(C2×D3)⋊C4—22no—
273(C2×D3)⋊C4—22no—
274(C2×D3)⋊C4—22no—
275(C2×D3)⋊C4—22no—
276(C2×D3)⋊C4—22no—
277(C2×D3)⋊C4—44no—
278(C2×D3)⋊C4—48no—
279C4⋊Dic3—42no—
280Dic3⋊C4—22no—
281Dic3⋊C4—22no—
282Dic3⋊C4—22no—
283Dic3⋊C4—22no—
284C4×Dic3—22no—
285C4×Dic3—22no—
286C4×Dic3—42no—
287C25—31no—
288C2×#41—62no—
289C2×#41—62no—
290C2×#41—122no—
291C2×#41—122no—
292C22×Q8—31no—
293C22×D4—31no—
294C22×D4—32no—
295C22×D4—31no—
296C22×D4—32no—
297C22×D4C22×D311yescharacteristic
298C23×C4—31no—
299C23×C4—32no—
300C23×C4—32no—
301C23×C4—32no—
302C23×C4—32no—
303C23×C4—32no—
304C23×C4—32no—
305C23×C4—31no—
306C23×C4—31no—
307C23×C4—62no—
308C23×C4—122no—
309C23×C4—122no—
310C23×C4C22×D311yescharacteristic
311#123—22no—
312#123—21no—
313#123—21no—
314#123—62no—
315#123—61no—
316#123—61no—
317C22⋊Q8—62no—
318C22⋊Q8—62no—
319C22⋊Q8—62no—
320C22⋊Q8—62no—
321#121—62no—
322#121—62no—
323#121—62no—
324#121—62no—
325#120—61no—
326#120—62no—
327#120—61no—
328#120—61no—
329#120—62no—
330#120—61no—
331C4×D4—62no—
332C4×D4—62no—
333C4×D4—62no—
334C4×D4—62no—
335C4×D4—62no—
336C4×D4—62no—
337C4×D4—62no—
338C4×D4—62no—
339#117—42no—
340#117—62no—
341#117—62no—
342#117—122no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—32no—
345C2×C4⋊C4—32no—
346C2×C4⋊C4—61no—
347C2×C4⋊C4C4×D312yescomplemented
348C2×C22⋊C4—31no—
349C2×C22⋊C4—32no—
350C2×C22⋊C4—32no—
351C2×C22⋊C4—32no—
352C2×C22⋊C4—32no—
353C2×C22⋊C4—32no—
354C2×C22⋊C4—62no—
355C2×C22⋊C4—64no—
356C2×C22⋊C4—122no—
357C2×C22⋊C4—122no—
358C2×C22⋊C4C4×D312yescomplemented
359C2×C22⋊C4C22×D312yes—
360C2×C22⋊C4C22×D312yes—
361C2×C22⋊C4C22×D311yescharacteristic
362C2×C42—21no—
363C2×C42—32no—
364C2×C42—61no—
365#99—42no—
366#99—122no—
367#99—122no—
368#99—122no—
369#95—22no—
370#95—64no—
371#95—64no—
372#95—62no—
373C23×C3—21no—
374C23×C3—22no—
375C23×C3—42no—
376C23×C3—42no—
377C23×C3C23×C411yescharacteristic
378C23×C3C22×D411yescharacteristic
379C23×C3C22×D411yescharacteristic
380C23×C3H(5,𝔽2)12yes—
381C23×C3H(5,𝔽2)12yes—
382C22×D3—22no—
383C22×D3—24no—
384C22×D3—24no—
385C22×D3—22no—
386C22×D3—22no—
387C22×D3—22no—
388C22×D3—24no—
389C22×D3—24no—
390C22×D3—21no—
391C22×D3—22no—
392C22×D3—42no—
393C22×D3—42no—
394C22×D3—42no—
395C22×D3—48no—
396C22×D3—48no—
397C22×D3—42no—
398C22×D3C2×C22⋊C412yes—
399C22×D3C2×C22⋊C412yes—
400C3×D4—41no—
401C3×D4—41no—
402C3×D4—42no—
403C2×C3×C4—24no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—24no—
409C2×C3×C4—22no—
410C2×C3×C4—21no—
411C2×C3×C4—22no—
412C2×C3×C4—42no—
413C2×C3×C4—44no—
414C2×C3×C4—44no—
415C2×C3×C4—42no—
416C2×C3×C4—42no—
417C2×C3×C4C2×C22⋊C412yes—
418C2×C3×C4C2×C22⋊C412yes—
419#67—44no—
420#67—44no—
421#67—41no—
422#67—41no—
423#67—42no—
424#67—44no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—24no—
430C2×Dic3—22no—
431C2×Dic3—21no—
432C2×Dic3—22no—
433C2×Dic3—44no—
434C2×Dic3—42no—
435C2×Dic3—44no—
436C2×Dic3—44no—
437C2×Dic3—42no—
438C2×Dic3—42no—
439C2×Dic3C2×C22⋊C412yes—
440C2×Dic3C2×C22⋊C412yes—
441D12—44no—
442D12—44no—
443C4×D3—42no—
444C4×D3—42no—
445C4×D3—42no—
446C4×D3—42no—
447C4×D3—42no—
448C4×D3—42no—
449C4×D3—42no—
450C4×D3—42no—
451C4×D3—44no—
452C4×D3—88no—
453Dic6—44no—
454C24—31no—
455C24—32no—
456C24—32no—
457C24—31no—
458C24—31no—
459C24—62no—
460C24—62no—
461C24—124no—
462C24C2×C4×D311yescharacteristic
463C24C23×D311yescharacteristic
464#41—124no—
465#41—124no—
466#41—128no—
467C2×Q8—62no—
468C2×Q8—122no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—62no—
474C2×D4—62no—
475C2×D4—62no—
476C2×D4—62no—
477C2×D4—62no—
478C2×D4—61no—
479C2×D4—61no—
480C2×D4—62no—
481C2×D4—62no—
482C2×D4—124no—
483C2×D4—122no—
484C2×D4—124no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—31no—
490C22×C4—31no—
491C22×C4—32no—
492C22×C4—32no—
493C22×C4—32no—
494C22×C4—32no—
495C22×C4—32no—
496C22×C4—32no—
497C22×C4—32no—
498C22×C4—32no—
499C22×C4—32no—
500C22×C4—32no—
501C22×C4—31no—
502C22×C4—32no—
503C22×C4—62no—
504C22×C4—62no—
505C22×C4—62no—
506C22×C4—62no—
507C22×C4—62no—
508C22×C4—62no—
509C22×C4—122no—
510C22×C4—122no—
511C22×C4—124no—
512C22×C4—124no—
513C22×C4—122no—
514C22×C4—122no—
515C22×C4—124no—
516C22×C4—124no—
517C22×C4—122no—
518C22×C4—124no—
519C22×C4—128no—
520C22×C4—122no—
521C22×C4C2×C4×D311yescharacteristic
522C22×C4C2×C4×D312yes—
523C22×C4C2×C4×D312yes—
524C22×C4D3×D412yes—
525C22×C4D3×D411yescharacteristic
526C22×C4D3×D411yescharacteristic
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—42no—
530C4⋊C4—62no—
531C4⋊C4—62no—
532C4⋊C4—62no—
533C4⋊C4—62no—
534C4⋊C4—122no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—62no—
544C22⋊C4—62no—
545C22⋊C4—62no—
546C22⋊C4—62no—
547C22⋊C4—61no—
548C22⋊C4—61no—
549C22⋊C4—62no—
550C22⋊C4—62no—
551C22⋊C4—62no—
552C22⋊C4—62no—
553C22⋊C4—62no—
554C22⋊C4—62no—
555C22⋊C4—124no—
556C22⋊C4—128no—
557C42—42no—
558C42—62no—
559C42—62no—
560C42—122no—
561C22×C3—22no—
562C22×C3—22no—
563C22×C3—22no—
564C22×C3—42no—
565C22×C3—42no—
566C22×C3—41no—
567C22×C3—42no—
568C22×C3—82no—
569C22×C3C22×C22⋊C411yescharacteristic
570C22×C3#51812yes—
571C22×C3#53411yescharacteristic
572C22×C3#53411yescharacteristic
573C22×C3#53412yes—
574C2×D3—28no—
575C2×D3—22no—
576C2×D3—22no—
577C2×D3—22no—
578C2×D3—22no—
579C2×D3—22no—
580C2×D3—44no—
581C2×D3—44no—
582C2×D3—48no—
583C2×D3—44no—
584C2×D3—41no—
585C2×D3—42no—
586C2×D3—82no—
587C2×D3#41014yes—
588C3×C4—22no—
589C3×C4—22no—
590C3×C4—41no—
591C3×C4—42no—
592C3×C4—82no—
593C3×C4—82no—
594Dic3—22no—
595Dic3—22no—
596Dic3—41no—
597Dic3—42no—
598Dic3—82no—
599Dic3—82no—
600C23—21no—
601C23—22no—
602C23—32no—
603C23—32no—
604C23—42no—
605C23—42no—
606C23—64no—
607C23—64no—
608C23—64no—
609C23—62no—
610C23—62no—
611C23—64no—
612C23—62no—
613C23—62no—
614C23—61no—
615C23—62no—
616C23—122no—
617C23—122no—
618C23—122no—
619C23—122no—
620C23—128no—
621C23—128no—
622C23C22×C4×D311yescharacteristic
623C23C2×D3×D411yescharacteristic
624C23C2×D3×D411yescharacteristic
625C23#100412yes—
626C23#100912yes—
627Q8—124no—
628D4—41no—
629D4—41no—
630D4—42no—
631D4—124no—
632D4—124no—
633D4—124no—
634D4—124no—
635D4—124no—
636D4—121no—
637D4—121no—
638D4—122no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—24no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—24no—
645C2×C4—22no—
646C2×C4—21no—
647C2×C4—22no—
648C2×C4—32no—
649C2×C4—32no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—62no—
656C2×C4—62no—
657C2×C4—62no—
658C2×C4—62no—
659C2×C4—64no—
660C2×C4—62no—
661C2×C4—61no—
662C2×C4—62no—
663C2×C4—122no—
664C2×C4—122no—
665C2×C4—122no—
666C2×C4—122no—
667C2×C4—122no—
668C2×C4—122no—
669C2×C4—122no—
670C2×C4—122no—
671C2×C4—124no—
672C2×C4—124no—
673C2×C4—122no—
674C2×C4—124no—
675C2×C4—124no—
676C2×C4—122no—
677C2×C4—122no—
678C2×C4—248no—
679C2×C4D3×C22⋊C412yes—
680C2×C4D3×C22⋊C412yes—
681C2×C3—21no—
682C2×C3—22no—
683C2×C3—21no—
684C2×C3—22no—
685C2×C3—81no—
686C2×C3#230811yescharacteristic
687C2×C3C2×#41012yes—
688D3—24no—
689D3—42no—
690D3—81no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—34no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—41no—
699C22—68no—
700C22—62no—
701C22—62no—
702C22—62no—
703C22—62no—
704C22—62no—
705C22—82no—
706C22—124no—
707C22—124no—
708C22—128no—
709C22—124no—
710C22—122no—
711C22—121no—
712C22—242no—
713C22C2×D3×C22⋊C411yescharacteristic
714C22#554512yes—
715C22#558911yescharacteristic
716C22#560512yes—
717C22#565311yescharacteristic
718C4—22no—
719C4—22no—
720C4—42no—
721C4—41no—
722C4—62no—
723C4—62no—
724C4—82no—
725C4—82no—
726C4—122no—
727C4—121no—
728C4—242no—
729C4—242no—
730C3#2174811yescharacteristic, complemented, Sylow
731C2—21no—
732C2—22no—
733C2—21no—
734C2—22no—
735C2—64no—
736C2—81no—
737C2—122no—
738C2—241no—
739C2#7487411yescharacteristic
740C2D3×#41012yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1899C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5656C2
cosocleC24×C3C24
hypercenter#1899D3
hypocenterC3#21748
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1899

Complements

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

NH ≅ G/Ntype
2. #82267C2semidirect
3. D3×#381C2semidirect
4. C3×#1899C2semidirect
5. #71705C2semidirect
6. #71595C2semidirect
7. #70036C2semidirect
8. #70027C2semidirect
9. #69535C2semidirect
10. #68970C2semidirect
11. #68953C2semidirect
24. C2×#896C4semidirect
25. C2×#893C4semidirect
26. C2×#884C4semidirect
28. C2×#67⋊C4C4semidirect
54. #1899D3semidirect
63. C22×#67C2×C4semidirect
65. C23×Dic3C2×C4semidirect
77. C22×C4×D3C2×C4semidirect
82. C22×Dic6C2×C4semidirect
88. C2×C3×C4⋊C4C2×C4semidirect
89. C2×C3×C22⋊C4C2×C4semidirect
107. C2×(C2×D3)⋊C4C2×C4semidirect
110. C2×Dic3⋊C4C2×C4semidirect
111. C2×Dic3⋊C4C2×C4semidirect
113. C2×C4×Dic3C2×C4semidirect
347. C2×C4⋊C4C4×D3semidirect
358. C2×C22⋊C4C4×D3semidirect
730. C3#21748semidirect

Maximal subgroups

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

subgroupindex
C2×#56562
#822672
D3×#3812
C3×#18992
#717052
#715952
#700362
#700272
#695352
#689702
#689532
#217483