Finite Group Databasebeta

← #10801750

Subgroups — #10801750

Order 768.

1014 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#815C211yesmaximal, characteristic
2Dic3×C22⋊Q8C211yesmaximal, characteristic
3C2×#5051C211yesmaximal, characteristic
4C2×#4789C211yesmaximal, characteristic, complemented
5#73866C211yesmaximal, characteristic
6C3×#1892C211yesmaximal, characteristic
7#71972C211yesmaximal, characteristic
8#70259C211yesmaximal, characteristic, complemented
9#70256C211yesmaximal, characteristic, complemented
10#70220C211yesmaximal, characteristic, complemented
11#69511C211yesmaximal, characteristic
12#68860C211yesmaximal, characteristic, complemented
13#68855C211yesmaximal, characteristic, complemented
14#68786C211yesmaximal, characteristic, complemented
15#68783C211yesmaximal, characteristic, complemented
16#21527—31nomaximal, Sylow
17C23×C3×C8C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C2×#924C2211yescharacteristic
22C22×C4×Dic3C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24C2×#887C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C3×#380C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30#5212C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#815—24no—
34C2×#815C2211yescharacteristic
35C2×#815C2211yescharacteristic
36C2×#815C2211yescharacteristic
37C2×#815C2211yescharacteristic
38#5051C418yescomplemented
39#4984C2211yescharacteristic
40#4980—21no—
41Dic3×C4⋊C4—21no—
42Dic3×C4⋊C4C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44#4973C2211yescharacteristic
45#4972C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4950—21no—
51#4946—21no—
52Dic3×C22⋊C4—21no—
53Dic3×C22⋊C4—21no—
54#4789—24no—
55#4652—21no—
56#4649—21no—
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×#467—31no—
64C2×#441—31no—
65C22×#97—31no—
66C4×C22⋊Q8—31no—
67C4×C22⋊Q8—31no—
68#1971—31no—
69#1943—31no—
70#1935—31no—
71#1935—31no—
72#1892—31no—
73#1892D311yescharacteristic
74#1807—31no—
75#1776—31no—
76#1763—31no—
77#1763—31no—
78C22×C3×Q8C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C23×Dic3D411yescharacteristic
81C23×Dic3D411yescharacteristic
82C22×Dic6C2311yescharacteristic
83C22×C3×C8—22no—
84C22×C3×C8—22no—
85C22×C3×C8—22no—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8C2×C414yes—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×#217—21no—
97C2×#217—21no—
98Q8×Dic3—22no—
99Q8×Dic3—22no—
100C2×#216—22no—
101C2×#216—22no—
102C2×#216C2×C414yescomplemented
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108#924—22no—
109#924C2×C414yes—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×C4×Dic3—22no—
114C2×C4×Dic3—22no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3D411yescharacteristic
120C2×C4×Dic3D411yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8—22no—
124C22×C3⋊C8—22no—
125C22×C3⋊C8C2×C414yes—
126C22×C3⋊C8C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128#887—22no—
129#887—22no—
130C3×#103—22no—
131C3×#103—22no—
132C3×#95—21no—
133#835C2×C414yescomplemented
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143#816—22no—
144#816—22no—
145#815—22no—
146#815—22no—
147#815—22no—
148#815D414yes—
149#810C2×C414yescomplemented
150#808—22no—
151#808—22no—
152#807C2×C414yescomplemented
153C22×Q16—31no—
154C23×C8—31no—
155C23×C8C2×D311yescharacteristic
156C2×C22⋊Q8—31no—
157C2×C22⋊Q8C2×D311yescharacteristic
158C2×C4×Q8—31no—
159C2×C4×Q8—31no—
160C22×C42—31no—
161#467—38no—
162#441—64no—
163C2×#107—31no—
164C2×#106—31no—
165C2×#106—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103C2×D311yescharacteristic
171C2×#103C2×D311yescharacteristic
172C2×#97—31no—
173C2×#97—31no—
174C2×#97—31no—
175C2×#97—31no—
176C2×#97—64no—
177#391—31no—
178#391—31no—
179#389—31no—
180#389—31no—
181#387—61no—
182#387—61no—
183#385—61no—
184#385—61no—
185#380—31no—
186#380C2×D311yescharacteristic
187C4×C4⋊C4—31no—
188C4×C4⋊C4—31no—
189C4×C4⋊C4—61no—
190C4×C4⋊C4—61no—
191C4×C22⋊C4—61no—
192C4×C22⋊C4—61no—
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340C2×D311yescharacteristic
198#340C2×D311yescharacteristic
199C24×C3C2×D411yescharacteristic
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8C22×C412yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4C22×C412yes—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4#4112yes—
217C22×C3×C4C2411yescharacteristic
218C22×Dic3—22no—
219C22×Dic3—22no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—22no—
223C22×Dic3—22no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—41no—
227C22×Dic3—41no—
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6C22×C412yes—
238C2×C3×C8—28no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8C2×D412yes—
244C2×C3×C8#4112yes—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4C22×C412yes—
252C3×C22⋊C4—24no—
253C3×C22⋊C4—42no—
254#217—24no—
255#217—42no—
256#216—28no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—42no—
263C4⋊Dic3—42no—
264C4⋊Dic3C22×C412yes—
265Dic3⋊C4—24no—
266Dic3⋊C4—42no—
267Dic3⋊C4—42no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—22no—
271C4×Dic3—22no—
272C4×Dic3—42no—
273C4×Dic3—42no—
274C4×Dic3—42no—
275C4×Dic3—42no—
276C4×Dic3C2×D412yes—
277C4×Dic3#4112yes—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—28no—
281C2×C3⋊C8C22×C412yes—
282C2×C3⋊C8C22×C412yes—
283C2×C3⋊C8C2×D412yes—
284C2×C3⋊C8#4112yes—
285C22×Q8—31no—
286C22×Q8C22×D311yescharacteristic
287C23×C4—31no—
288C23×C4—31no—
289C23×C4C22×D311yescharacteristic
290C2×Q16—34no—
291C2×Q16—62no—
292C2×Q16—62no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8—31no—
297C22×C8—34no—
298C22×C8—31no—
299C22×C8—62no—
300C22×C8—62no—
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303C22⋊Q8—22no—
304C22⋊Q8—34no—
305C22⋊Q8—62no—
306C22⋊Q8C4×D314yes—
307C4×Q8—62no—
308C4×Q8—62no—
309C4×Q8—62no—
310C4×Q8—62no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4C22×D311yescharacteristic
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C42—31no—
328C2×C42—31no—
329C2×C42—31no—
330C2×C42—31no—
331C2×C42—62no—
332C2×C42—62no—
333C2×C42—61no—
334C2×C42—61no—
335C2×C42—61no—
336C2×C42—61no—
337#107—34no—
338#106—62no—
339#106—62no—
340#103—22no—
341#103—22no—
342#103—34no—
343#103—34no—
344#103—62no—
345#103—62no—
346#97—34no—
347#97—62no—
348#97—62no—
349#97—62no—
350#95—21no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3C4×D412yes—
365C23×C3#13712yes—
366C23×C3#13712yes—
367C23×C3C22×D411yescharacteristic
368C3×Q8—22no—
369C3×Q8—22no—
370C3×Q8—41no—
371C3×Q8—41no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4C4×D412yes—
391C2×C3×C4C23×C411yescharacteristic
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—22no—
402C2×Dic3—22no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—44no—
410C2×Dic3—44no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3#12112yes—
422C2×Dic3#12112yes—
423Dic6—22no—
424Dic6—22no—
425Dic6—41no—
426Dic6—41no—
427C3×C8—22no—
428C3×C8—22no—
429C3⋊C8—22no—
430C3⋊C8C4×D414yes—
431C24D3×D411yescharacteristic
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—32no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×Q8—61no—
442C2×Q8—61no—
443C2×Q8C2×C4×D312yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—31no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—121no—
465C22×C4—121no—
466C22×C4C2×C4×D312yes—
467C22×C4D3×D411yescharacteristic
468C22×C4#23712yes—
469C22×C4C23×D311yescharacteristic
470Q16—68no—
471C2×C8—28no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—32no—
477C2×C8—32no—
478C2×C8—32no—
479C2×C8—32no—
480C2×C8—68no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8D3×D412yes—
484C2×C8#23712yes—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—32no—
492C4⋊C4—64no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—122no—
499C4⋊C4—122no—
500C4⋊C4—122no—
501C4⋊C4—122no—
502C4⋊C4C2×C4×D312yes—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—64no—
506C22⋊C4—122no—
507C42—32no—
508C42—32no—
509C42—62no—
510C42—62no—
511C42—61no—
512C42—61no—
513C42—122no—
514C42—122no—
515C42—122no—
516C42—122no—
517C22×C3—22no—
518C22×C3—22no—
519C22×C3—22no—
520C22×C3—22no—
521C22×C3—22no—
522C22×C3—22no—
523C22×C3#44114yescomplemented
524C22×C3C2×C4×D411yescharacteristic
525C22×C3C2×#12111yescharacteristic
526C22×C3#53211yescharacteristic
527C22×C3C2×#13711yescharacteristic
528C22×C3C2×#13711yescharacteristic
529C22×C3#57511yescharacteristic
530C22×C3#57511yescharacteristic
531C3×C4—22no—
532C3×C4—42no—
533C3×C4—42no—
534C3×C4—81no—
535C3×C4—81no—
536C3×C4C2×C4×D411yescharacteristic
537C3×C4C4×#4111yescharacteristic
538C3×C4#52511yescharacteristic
539C3×C4#52911yescharacteristic
540Dic3—22no—
541Dic3—22no—
542Dic3—42no—
543Dic3—41no—
544Dic3—41no—
545Dic3—42no—
546Dic3—81no—
547Dic3—81no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23#67⋊C412yes—
553C23#91512yes—
554C23#91812yes—
555C23C2×D3×D411yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—41no—
559Q8—41no—
560Q8—62no—
561Q8—62no—
562Q8—121no—
563Q8—121no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—32no—
573C2×C4—32no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—124no—
597C2×C4—124no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4#67⋊C412yes—
609C2×C4C22×C4×D311yescharacteristic
610C2×C4C2×D3×D411yescharacteristic
611C2×C4C2×#23711yescharacteristic
612C2×C4C2×#23711yescharacteristic
613C2×C4D3×#4111yescharacteristic
614C2×C4D3×#4111yescharacteristic
615C8—22no—
616C8—22no—
617C8—34no—
618C8—62no—
619C2×C3—24no—
620C2×C3C4×#12111yescharacteristic
621C2×C3C2×#44111yescharacteristic
622C2×C3#294311yescharacteristic
623C2×C3#305611yescharacteristic
624C2×C3#305611yescharacteristic
625C2×C3#318111yescharacteristic
626C2×C3#318111yescharacteristic
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22#478914yescomplemented
634C22C2×#67⋊C411yescharacteristic
635C22#559611yescharacteristic
636C22D3×#12111yescharacteristic
637C22C2×#91511yescharacteristic
638C22C2×#91811yescharacteristic
639C22#576811yescharacteristic
640C22#576811yescharacteristic
641C4—22no—
642C4—42no—
643C4—42no—
644C4—62no—
645C4—62no—
646C4—81no—
647C4—81no—
648C4—121no—
649C4—121no—
650C4—122no—
651C4—122no—
652C4—241no—
653C4—241no—
654C4C4×#23711yescharacteristic
655C4C4×D3×D411yescharacteristic
656C4#558611yescharacteristic
657C4#564911yescharacteristic
658C3#2152711yescharacteristic, complemented, Sylow
659C2—24no—
660C2#7466011yescharacteristic
661C2C2×#478911yescharacteristic
662C2#7881911yescharacteristic
663C2#8063911yescharacteristic
664C2#8065811yescharacteristic
665C2#8074511yescharacteristic
666C2#8075911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1892C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
cosocleC22×C3×C4C24
hypercenter#1892D3
hypocenterC3#21527
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
4. C2×#4789C2semidirect
8. #70259C2semidirect
9. #70256C2semidirect
10. #70220C2semidirect
12. #68860C2semidirect
13. #68855C2semidirect
14. #68786C2semidirect
15. #68783C2semidirect
38. #5051C4semidirect
102. C2×#216C2×C4semidirect
133. #835C2×C4semidirect
149. #810C2×C4semidirect
152. #807C2×C4semidirect
523. C22×C3#441semidirect
633. C22#4789semidirect
658. C3#21527semidirect

Maximal subgroups

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

subgroupindex
C22×#8152
Dic3×C22⋊Q82
C2×#50512
C2×#47892
#738662
C3×#18922
#719722
#702592
#702562
#702202
#695112
#688602
#688552
#687862
#687832
#215273