Finite Group Databasebeta

← #10804181

Subgroups — #10804181

Order 768.

1318 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5608C211yesmaximal, characteristic
2#83762C211yesmaximal, characteristic, complemented
3#78630C211yesmaximal, characteristic, complemented
4C3×#1900C211yesmaximal, characteristic, complemented
5#72003C211yesmaximal, characteristic, complemented
6#71683C211yesmaximal, characteristic, complemented
7#70081C212yesmaximal, complemented
8#70080C212yesmaximal, complemented
9#69986C211yesmaximal, characteristic, complemented
10#68993C212yesmaximal, complemented
11#68990C212yesmaximal, complemented
12#21757—31nomaximal, Sylow
13C22×#237C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×C3×#121C2211yescharacteristic
17C2×#939C412yes—
18C2×#938C412yes—
19C2×#938C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C22×#208C2211yescharacteristic
22C2×D3×M16C2212yes—
23#5608—22no—
24#5608—22no—
25#5608—22no—
26#5608—22no—
27C2×#897C2211yescharacteristic
28C2×D3×C4⋊C4C2211yescharacteristic
29C2×#884C412yes—
30C2×#878C412yes—
31C2×C3×#100C2212yes—
32C3×#407C2211yescharacteristic
33C2×#833C2212yes—
34#5207C2211yescharacteristic
35C2×#824C2212yes—
36C2×#824C2212yes—
37#5129—22no—
38#5126C2212yes—
39#5126C2212yes—
40#5126C2212yes—
41#5110—22no—
42#4727—22no—
43#4727—22no—
44C3×#343C2212yes—
45#4554C2212yes—
46#4554C2212yes—
47#4530C2212yes—
48C2×#537—31no—
49#2879—31no—
50#2879—31no—
51#1901—31no—
52#1901—32no—
53#1901—31no—
54#1900—32no—
55#1900—31no—
56#1900D311yescharacteristic, complemented
57#1821—32no—
58#1785—32no—
59C22×C3×D4C2×C412yes—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67C2×C412yes—
63C22×#67C2×C412yes—
64C22×#67C2×C411yescharacteristic
65C22×#67C2×C411yescharacteristic
66C23×Dic3C2×C412yes—
67C2×#237—22no—
68C2×#237—22no—
69C2×#237—42no—
70C2×#237—42no—
71C2×#235—42no—
72C2×#235—42no—
73C2×#235D412yes—
74C2×#235D412yes—
75C2×#235D412yes—
76C2×#235D412yes—
77C22×D12C2×C411yescharacteristic
78C22×C4×D3C2×C412yes—
79C22×C4×D3C2311yescharacteristic
80C22×C4×D3C2311yescharacteristic
81C22×Dic6C2×C411yescharacteristic
82C22×Dic6C2311yescharacteristic
83C2×C3×M16—21no—
84C2×C3×M16—42no—
85C2×C3×M16C2312yes—
86C2×C3×M16C2312yes—
87C22×C3×C8—21no—
88C3×#121—21no—
89C3×#121—21no—
90C3×#121—21no—
91C3×#121—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4C2×C412yes—
94C2×#217C2×C412yes—
95C2×#217C2×C412yes—
96#939—24no—
97#939—24no—
98#938—24no—
99#938—24no—
100#938—22no—
101#938—22no—
102C2×(C2×D3)⋊C4C2×C412yes—
103C2×C4⋊Dic3C2×C412yes—
104C2×C4⋊Dic3C2311yescharacteristic
105#924—21no—
106#924—21no—
107#924—21no—
108#924—21no—
109C2×Dic3⋊C4C2×C412yes—
110C2×Dic3⋊C4C2×C412yes—
111C2×#208—21no—
112C2×#208—42no—
113C2×#208C2312yes—
114C2×#208C2312yes—
115C22×C3⋊C8—21no—
116D3×M16—42no—
117D3×M16—42no—
118C2×ΓL(1,𝔽25)—22no—
119C2×D3×C8—22no—
120#897—22no—
121#897—22no—
122D3×C4⋊C4—22no—
123D3×C4⋊C4—22no—
124#884—24no—
125#884—24no—
126#878—24no—
127#878—24no—
128C3×#100—42no—
129C3×C22⋊C8—21no—
130C3×C22⋊C8—21no—
131#833—42no—
132#830—21no—
133#830—21no—
134#824—42no—
135#824—42no—
136(C2×D3)⋊C8—24no—
137(C2×D3)⋊C8—24no—
138(C2×D3)⋊C8—24no—
139C22×#41—31no—
140C22×#41—31no—
141C22×M16—31no—
142C22×M16—32no—
143C22×M16C2×D311yescharacteristic
144#537—62no—
145#537—62no—
146#537—62no—
147#537—62no—
148C2×C22⋊Q8—31no—
149C2×C22⋊Q8—31no—
150C2×C22⋊Q8—32no—
151C2×#121—32no—
152C2×#121—32no—
153C2×#121—31no—
154C2×#121—32no—
155C2×#121C2×D311yescharacteristic
156C22×C4⋊C4—31no—
157C2×#101—32no—
158C2×#101—32no—
159C2×#100—32no—
160C2×#100C2×D312yes—
161#408—62no—
162#408—62no—
163#408—62no—
164#408—62no—
165#407—31no—
166#407—32no—
167#407—32no—
168#407—32no—
169#407C2×D311yescharacteristic
170#343—32no—
171#343—32no—
172#343—32no—
173#343C2×D312yes—
174C24×C3C22×C412yes—
175C24×C3C22×C411yescharacteristic
176C23×D3C22×C411yescharacteristic
177C23×D3C22×C411yescharacteristic
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—82no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4C22×C411yescharacteristic
188C22×C3×C4C22×C412yes—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C2×#67—24no—
195C2×#67—24no—
196C2×#67—42no—
197C2×#67—42no—
198C2×#67—44no—
199C2×#67—82no—
200C2×#67C22⋊C412yes—
201C2×#67C22⋊C412yes—
202C2×#67C22⋊C412yes—
203C2×#67C22⋊C412yes—
204C22×Dic3—24no—
205C22×Dic3—44no—
206C22×Dic3C22×C412yes—
207C22×Dic3C22×C411yescharacteristic
208C22×Dic3C22×C411yescharacteristic
209C22×Dic3C22×C412yes—
210#237—44no—
211#237—44no—
212#237—44no—
213#237—44no—
214#235—42no—
215#235—42no—
216#235—42no—
217#235—42no—
218#235—42no—
219#235—42no—
220#235—42no—
221#235—42no—
222C2×D12—42no—
223C2×D12C22⋊C412yes—
224C2×D12C22⋊C412yes—
225C2×C4×D3—22no—
226C2×C4×D3—22no—
227C2×C4×D3—22no—
228C2×C4×D3—22no—
229C2×C4×D3—24no—
230C2×C4×D3—82no—
231C2×C4×D3C2×D412yes—
232C2×C4×D3C2×D412yes—
233C2×C4×D3C2×D412yes—
234C2×C4×D3C2×D412yes—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—22no—
238C2×Dic6—22no—
239C2×Dic6—42no—
240C2×Dic6C22⋊C412yes—
241C2×Dic6C22⋊C412yes—
242C3×M16—22no—
243C3×M16—22no—
244C3×M16—22no—
245C3×M16—22no—
246C3×M16—42no—
247C2×C3×C8—22no—
248C2×C3×C8—22no—
249C2×C3×C8—42no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C22⋊C4—22no—
253C3×C22⋊C4—22no—
254#217—22no—
255#217—22no—
256#217—24no—
257(C2×D3)⋊C4—24no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—24no—
261Dic3⋊C4—22no—
262Dic3⋊C4—22no—
263Dic3⋊C4—24no—
264#208—22no—
265#208—22no—
266#208—22no—
267#208—22no—
268#208—42no—
269C2×C3⋊C8—22no—
270C2×C3⋊C8—22no—
271C2×C3⋊C8—42no—
272ΓL(1,𝔽25)—44no—
273D3×C8—44no—
274C2×#41—32no—
275C2×#41—32no—
276C2×#41—32no—
277C2×#41—32no—
278C2×#41—62no—
279C2×#41—62no—
280C2×#41—122no—
281C2×#41—122no—
282C2×#41—122no—
283C2×#41—122no—
284C22×Q8—31no—
285C22×Q8—31no—
286C22×D4—31no—
287C22×D4—31no—
288C22×D4—32no—
289C22×D4—32no—
290C22×D4—31no—
291C22×D4C4×D312yes—
292C22×D4C22×D311yescharacteristic
293C23×C4—31no—
294C23×C4—31no—
295C23×C4—32no—
296C23×C4—32no—
297C23×C4C22×D311yescharacteristic
298C2×M16—21no—
299C2×M16—32no—
300C2×M16—32no—
301C2×M16—42no—
302C2×M16—62no—
303C2×M16—61no—
304C2×M16—122no—
305C2×M16—122no—
306C2×M16—122no—
307C2×M16C22×D312yes—
308C2×M16C22×D312yes—
309C22×C8—21no—
310C22×C8—62no—
311C22×C8—61no—
312C22⋊Q8—64no—
313C22⋊Q8—64no—
314C22⋊Q8—62no—
315C22⋊Q8—62no—
316C22⋊Q8—61no—
317C22⋊Q8—61no—
318C22⋊Q8—61no—
319C22⋊Q8—61no—
320#121—21no—
321#121—21no—
322#121—21no—
323#121—21no—
324#121—64no—
325#121—64no—
326#121—62no—
327#121—62no—
328#121—64no—
329#121—64no—
330#121—64no—
331#121—64no—
332C2×C4⋊C4—32no—
333C2×C4⋊C4—32no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—31no—
336C2×C4⋊C4—62no—
337C2×C4⋊C4—62no—
338C2×C4⋊C4C22×D311yescharacteristic
339C2×C22⋊C4—32no—
340C2×C22⋊C4—32no—
341C2×C22⋊C4—32no—
342C2×C22⋊C4C4×D312yes—
343#101—122no—
344#101—122no—
345#100—42no—
346#100—122no—
347C22⋊C8—21no—
348C22⋊C8—21no—
349C22⋊C8—64no—
350C22⋊C8—64no—
351C22⋊C8—64no—
352C22⋊C8—61no—
353C22⋊C8—61no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3—24no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—42no—
360C23×C3—42no—
361C23×C3C2×C22⋊C411yescharacteristic
362C23×C3C2×C22⋊C411yescharacteristic
363C23×C3C23×C411yescharacteristic
364C22×D3—22no—
365C22×D3—22no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3C2×C22⋊C412yes—
371C22×D3C2×C22⋊C412yes—
372C3×D4—41no—
373C3×D4—41no—
374C3×D4—42no—
375C3×D4—82no—
376C3×D4—82no—
377C2×C3×C4—22no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—42no—
386C2×C3×C4C2×C22⋊C411yescharacteristic
387C2×C3×C4C2×C22⋊C411yescharacteristic
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4H(5,𝔽2)12yes—
391C2×C3×C4H(5,𝔽2)12yes—
392#67—42no—
393#67—42no—
394#67—42no—
395#67—42no—
396#67—48no—
397#67—84no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—24no—
403C2×Dic3—48no—
404C2×Dic3—42no—
405C2×Dic3C2×C22⋊C412yes—
406C2×Dic3C2×C22⋊C412yes—
407D12—42no—
408D12—42no—
409C4×D3—22no—
410C4×D3—22no—
411C4×D3—22no—
412C4×D3—22no—
413C4×D3—22no—
414C4×D3—22no—
415C4×D3—22no—
416C4×D3—22no—
417C4×D3—84no—
418Dic6—42no—
419Dic6—42no—
420Dic6—41no—
421Dic6—41no—
422Dic6—42no—
423C3×C8—42no—
424C3×C8—42no—
425C3⋊C8—42no—
426C3⋊C8—42no—
427C24—31no—
428C24—31no—
429C24C2×C4×D312yes—
430C24C2×C4×D311yescharacteristic
431#41—122no—
432#41—122no—
433#41—122no—
434#41—122no—
435#41—122no—
436#41—122no—
437#41—122no—
438#41—122no—
439#41—124no—
440#41—124no—
441#41—124no—
442#41—124no—
443C2×Q8—32no—
444C2×Q8—32no—
445C2×Q8—61no—
446C2×Q8—61no—
447C2×Q8—62no—
448C2×Q8—62no—
449C2×Q8—122no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—32no—
457C2×D4—32no—
458C2×D4—32no—
459C2×D4—32no—
460C2×D4—32no—
461C2×D4—32no—
462C2×D4—64no—
463C2×D4—64no—
464C2×D4—82no—
465C2×D4—122no—
466C2×D4—122no—
467C2×D4—122no—
468C2×D4—124no—
469C2×D4—242no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—31no—
473C22×C4—31no—
474C22×C4—32no—
475C22×C4—32no—
476C22×C4—32no—
477C22×C4—32no—
478C22×C4—32no—
479C22×C4—32no—
480C22×C4—64no—
481C22×C4—64no—
482C22×C4—62no—
483C22×C4—62no—
484C22×C4—62no—
485C22×C4—62no—
486C22×C4—124no—
487C22×C4—242no—
488C22×C4C2×C4×D311yescharacteristic
489C22×C4C2×C4×D312yes—
490C22×C4D3×D411yescharacteristic
491C22×C4D3×D411yescharacteristic
492C22×C4D3×D411yescharacteristic
493C22×C4D3×D411yescharacteristic
494C22×C4C23×D311yescharacteristic
495M16—22no—
496M16—22no—
497M16—22no—
498M16—22no—
499M16—42no—
500M16—62no—
501M16—62no—
502M16—62no—
503M16—62no—
504M16—124no—
505M16—122no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—42no—
509C2×C8—62no—
510C2×C8—62no—
511C2×C8—124no—
512C2×C8—122no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—64no—
516C4⋊C4—64no—
517C4⋊C4—62no—
518C4⋊C4—62no—
519C4⋊C4—61no—
520C4⋊C4—61no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—64no—
524C22⋊C4—64no—
525C22⋊C4—62no—
526C22⋊C4—62no—
527C22×C3—21no—
528C22×C3—21no—
529C22×C3—21no—
530C22×C3—21no—
531C22×C3—21no—
532C22×C3—21no—
533C22×C3—21no—
534C22×C3—21no—
535C22×C3—42no—
536C22×C3—42no—
537C22×C3—42no—
538C22×C3—82no—
539C22×C3—84no—
540C22×C3C22×C22⋊C411yescharacteristic
541C22×C3#51812yes—
542C2×D3—22no—
543C2×D3—22no—
544C2×D3—22no—
545C2×D3—22no—
546C2×D3—22no—
547C2×D3—22no—
548C2×D3—44no—
549C2×D3—82no—
550C3×C4—21no—
551C3×C4—21no—
552C3×C4—42no—
553C3×C4#53411yescharacteristic
554C3×C4#53411yescharacteristic
555C3×C4#53412yes—
556Dic3—22no—
557Dic3—22no—
558Dic3—42no—
559Dic3—42no—
560C23—22no—
561C23—22no—
562C23—24no—
563C23—32no—
564C23—32no—
565C23—41no—
566C23—41no—
567C23—42no—
568C23—42no—
569C23—62no—
570C23—62no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23—121no—
575C23D3×C22⋊C411yescharacteristic
576C23D3×C22⋊C411yescharacteristic
577C23C22×C4×D311yescharacteristic
578Q8—122no—
579Q8—122no—
580Q8—121no—
581Q8—121no—
582Q8—122no—
583D4—41no—
584D4—41no—
585D4—42no—
586D4—82no—
587D4—82no—
588D4—122no—
589D4—122no—
590D4—122no—
591D4—122no—
592D4—122no—
593D4—122no—
594D4—128no—
595D4—244no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—32no—
605C2×C4—32no—
606C2×C4—42no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—62no—
614C2×C4—62no—
615C2×C4—62no—
616C2×C4—62no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—64no—
620C2×C4—128no—
621C2×C4—122no—
622C2×C4—244no—
623C2×C4D3×C22⋊C411yescharacteristic
624C2×C4D3×C22⋊C411yescharacteristic
625C2×C4C2×D3×D411yescharacteristic
626C2×C4C2×D3×D411yescharacteristic
627C2×C4#100412yes—
628C2×C4#100912yes—
629C8—42no—
630C8—42no—
631C8—122no—
632C8—122no—
633C2×C3—22no—
634C2×C3—41no—
635C2×C3—41no—
636C2×C3—82no—
637C2×C3#230811yescharacteristic
638C2×C3#288811yescharacteristic
639C2×C3#288911yescharacteristic
640D3—42no—
641D3—42no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—62no—
654C22—62no—
655C22—62no—
656C22—62no—
657C22—62no—
658C22—62no—
659C22—82no—
660C22—84no—
661C22—124no—
662C22—242no—
663C22C2×D3×C22⋊C411yescharacteristic
664C22#554512yes—
665C4—21no—
666C4—21no—
667C4—42no—
668C4—62no—
669C4—62no—
670C4—122no—
671C4—122no—
672C4#558911yescharacteristic
673C4#560512yes—
674C4#565311yescharacteristic
675C3#2175711yescharacteristic, complemented, Sylow
676C2—22no—
677C2—41no—
678C2—41no—
679C2—82no—
680C2—122no—
681C2—122no—
682C2#7487411yescharacteristic
683C2#7869711yescharacteristic
684C2#7871511yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1900C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5608C2
cosocleC22×C3×C4C24
hypercenter#1900D3
hypocenterC3#21757
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1900

Complements

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

NH ≅ G/Ntype
2. #83762C2semidirect
3. #78630C2semidirect
4. C3×#1900C2semidirect
5. #72003C2semidirect
6. #71683C2semidirect
7. #70081C2semidirect
8. #70080C2semidirect
9. #69986C2semidirect
10. #68993C2semidirect
11. #68990C2semidirect
56. #1900D3semidirect
675. C3#21757semidirect

Maximal subgroups

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

subgroupindex
C2×#56082
#837622
#786302
C3×#19002
#720032
#716832
#700812
#700802
#699862
#689932
#689902
#217573