Finite Group Databasebeta

← #10824554

Subgroups — #10824554

Order 768.

892 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#442C211yesmaximal, characteristic
2#82328C211yesmaximal, characteristic, complemented
3#82325C211yesmaximal, characteristic, complemented
4C2×C4×#214C211yesmaximal, characteristic, complemented
5#72045C211yesmaximal, characteristic
6#72040C211yesmaximal, characteristic
7C4×#832C211yesmaximal, characteristic, complemented
8#70589C211yesmaximal, characteristic, complemented
9#70567C211yesmaximal, characteristic, complemented
10#70217C211yesmaximal, characteristic, complemented
11#70193C211yesmaximal, characteristic, complemented
12#70178C211yesmaximal, characteristic, complemented
13#70165C211yesmaximal, characteristic, complemented
14#69832C211yesmaximal, characteristic, complemented
15#69813C211yesmaximal, characteristic, complemented
16#23693—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#936C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#442—24no—
24C2×C3×#106C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28#5221C2211yescharacteristic
29#5221C2211yescharacteristic
30#5221C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34#5212—21no—
35#5211—21no—
36#5210—21no—
37#5210—21no—
38C2×#816C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42C4×#214—24no—
43#4984C2211yescharacteristic
44#4984C2211yescharacteristic
45#4975C2211yescharacteristic
46#4973C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#808C2211yescharacteristic
49#4951—21no—
50#4951—21no—
51#4944—21no—
52#4944—21no—
53C4×C4⋊Dic3C2211yescharacteristic
54#4932C2211yescharacteristic
55C2×C4×C3⋊C8C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#442D311yescharacteristic
63C2×C4×SD16—31no—
64#2326—31no—
65#2319—31no—
66#2084—31no—
67#2071—31no—
68#2066—31no—
69#2065—31no—
70#2055—31no—
71#2054—31no—
72#2042—31no—
73#2012—31no—
74#1768—31no—
75#1763—31no—
76C4×#102—31no—
77C22×C3×D4C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C23×C3×C4—21no—
81C22×Dic6C2311yescharacteristic
82C2×C3×D8—22no—
83C2×C3×D8—22no—
84C2×C3×D8—22no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C3×C4×D4D414yes—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C42C2311yescharacteristic
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104#936—22no—
105#936—22no—
106C2×#214—22no—
107C2×#214—22no—
108C2×#214—22no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C4×Dic6—22no—
124C4×Dic6—22no—
125C3×#106—22no—
126C3×#102—22no—
127C3×#102—22no—
128C3×#97—22no—
129#832—22no—
130#832—22no—
131#832—22no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95D411yescharacteristic
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140C3⋊#95D411yescharacteristic
141#816—22no—
142#816—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—22no—
145#810—22no—
146#808—22no—
147C4×C3⋊C8—22no—
148C22×SD16—31no—
149C22×D8C2×D311yescharacteristic
150C2×#124—31no—
151C2×C4×Q8—31no—
152C2×C4×D4C2×D311yescharacteristic
153C2×C4×D4C2×D311yescharacteristic
154#442—24no—
155C4×SD16—64no—
156C2×#106—31no—
157C2×#106C2×D311yescharacteristic
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×#97C2×D311yescharacteristic
169C2×C4×C8—31no—
170#391—31no—
171#390—31no—
172#390—31no—
173#390—31no—
174#389—31no—
175#389—31no—
176#388—61no—
177#388—61no—
178#388—61no—
179#388—61no—
180#385—61no—
181#385—61no—
182#384—31no—
183#381—61no—
184#380—61no—
185C4×C4⋊C4—31no—
186#376—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C24×C3—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4C2×D412yes—
206C2×C3×D4#4112yes—
207C22×C3×C4—21no—
208C22×C3×C4—24no—
209C22×C3×C4—21no—
210C22×C3×C4—42no—
211C22×C3×C4—81no—
212C22×C3×C4—81no—
213C22×C3×C4—81no—
214C22×C3×C4—81no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2411yescharacteristic
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6#4112yes—
232C3×D8—28no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8#4112yes—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4C2×D412yes—
243C3×C22⋊C4—24no—
244C3×C22⋊C4—42no—
245C3×C42C2×D412yes—
246C3×C42#4112yes—
247#217—24no—
248#217—24no—
249#217—81no—
250#217—81no—
251#217—81no—
252#217—81no—
253#217—81no—
254#217—81no—
255#214—28no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—81no—
263C4⋊Dic3—81no—
264Dic3⋊C4—42no—
265Dic3⋊C4—81no—
266Dic3⋊C4—81no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—42no—
272C4×Dic3—81no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8#4112yes—
279C22×Q8—31no—
280C22×D4C22×D311yescharacteristic
281C22×D4C22×D311yescharacteristic
282C23×C4—21no—
283C23×C4—21no—
284C2×SD16—62no—
285C2×SD16—62no—
286C2×SD16—62no—
287C2×D8—22no—
288C2×D8—22no—
289C2×D8—22no—
290C22×C8—31no—
291C22×C8—31no—
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294#124—62no—
295#124—62no—
296C4×Q8—62no—
297C4×Q8—62no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4#6714yes—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C42—31no—
323C2×C42—31no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#106—22no—
328#106—62no—
329C4⋊C8—62no—
330C4⋊C8—62no—
331#103—62no—
332#103—62no—
333#103—62no—
334#102—22no—
335#102—22no—
336#102—62no—
337#102—62no—
338#102—62no—
339#97—22no—
340C4×C8—62no—
341#95—31no—
342#95—31no—
343#95—31no—
344#95—31no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3—22no—
351C23×C3—22no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3C22×D411yescharacteristic
365C3×D4—22no—
366C3×D4—22no—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—41no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4—41no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—84no—
384C2×C3×C4—84no—
385C2×C3×C4#12112yes—
386C2×C3×C4#12112yes—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—22no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417Dic6—22no—
418Dic6—22no—
419Dic6—41no—
420Dic6—41no—
421C3×C8—22no—
422C3×C8—41no—
423C3⋊C8—22no—
424C3⋊C8—41no—
425C24—21no—
426C24—21no—
427C2×Q8—32no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4#23512yes—
444C2×D4C2×#6712yes—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—31no—
452C22×C4—42no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—81no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4D3×D411yescharacteristic
462C22×C4D3×D411yescharacteristic
463C22×C4C23×D311yescharacteristic
464SD16—68no—
465D8—28no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—32no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8#23512yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—122no—
488C4⋊C4—241no—
489C4⋊C4—241no—
490C4⋊C4—241no—
491C4⋊C4—241no—
492C4⋊C4C2×#6712yes—
493C22⋊C4—24no—
494C22⋊C4—42no—
495C22⋊C4—64no—
496C22⋊C4—64no—
497C22⋊C4—241no—
498C22⋊C4—241no—
499C22⋊C4—241no—
500C22⋊C4—241no—
501C22⋊C4—241no—
502C22⋊C4—241no—
503C42—61no—
504C42—61no—
505C42—61no—
506C42—61no—
507C42—122no—
508C42—241no—
509C42#23712yes—
510C42C2×#6712yes—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—42no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—81no—
518C22×C3—81no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3—81no—
525C22×C3C2×#12111yescharacteristic
526C22×C3#52511yescharacteristic
527C22×C3#53211yescharacteristic
528C22×C3#57611yescharacteristic
529C22×C3#57711yescharacteristic
530C22×C3#57711yescharacteristic
531C22×C3#57711yescharacteristic
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—42no—
535C3×C4—81no—
536C3×C4#52511yescharacteristic
537C3×C4#52511yescharacteristic
538C3×C4#52911yescharacteristic
539C3×C4#52911yescharacteristic
540Dic3—41no—
541Dic3—41no—
542Dic3—42no—
543Dic3—41no—
544Dic3—41no—
545Dic3—81no—
546Dic3—81no—
547Dic3—81no—
548C23—22no—
549C23—22no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23C2×D3×D411yescharacteristic
563Q8—62no—
564Q8—62no—
565Q8—121no—
566Q8—121no—
567D4—22no—
568D4—22no—
569D4—22no—
570D4—22no—
571D4—41no—
572D4—41no—
573D4—41no—
574D4—41no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—62no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—84no—
599C2×C4—84no—
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—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4#93912yes—
612C2×C4#93912yes—
613C2×C4C2×#23511yescharacteristic
614C2×C4C2×#23511yescharacteristic
615C2×C4C2×#23711yescharacteristic
616C2×C4D3×#4111yescharacteristic
617C2×C4D3×#4111yescharacteristic
618C2×C4C22×#6711yescharacteristic
619C8—22no—
620C8—41no—
621C8—62no—
622C8—121no—
623C2×C3—82no—
624C2×C3—82no—
625C2×C3#239211yescharacteristic
626C2×C3#304211yescharacteristic
627C2×C3#304511yescharacteristic
628C2×C3#311911yescharacteristic
629C2×C3#312511yescharacteristic
630C2×C3#317311yescharacteristic
631C2×C3#317911yescharacteristic
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22#554711yescharacteristic
647C22#554811yescharacteristic
648C22#577411yescharacteristic
649C22#577511yescharacteristic
650C22#577511yescharacteristic
651C22#577511yescharacteristic
652C22C2×#93911yescharacteristic
653C4—22no—
654C4—22no—
655C4—42no—
656C4—81no—
657C4—121no—
658C4—121no—
659C4—122no—
660C4—121no—
661C4—121no—
662C4—241no—
663C4—241no—
664C4—241no—
665C4#553811yescharacteristic
666C4#555711yescharacteristic
667C4#579111yescharacteristic
668C4#582811yescharacteristic
669C3#2369311yescharacteristic, complemented, Sylow
670C2—82no—
671C2—82no—
672C2#7985911yescharacteristic
673C2#7986111yescharacteristic
674C2#7987311yescharacteristic
675C2#7987511yescharacteristic
676C2#8234511yescharacteristic
677C2#8393511yescharacteristic
678C2#8393611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#442C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
hypercenterC2×#442D3
hypocenterC3#23693
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#442

Complements

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

NH ≅ G/Ntype
2. #82328C2semidirect
3. #82325C2semidirect
4. C2×C4×#214C2semidirect
7. C4×#832C2semidirect
8. #70589C2semidirect
9. #70567C2semidirect
10. #70217C2semidirect
11. #70193C2semidirect
12. #70178C2semidirect
13. #70165C2semidirect
14. #69832C2semidirect
15. #69813C2semidirect
669. C3#23693semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4422
#823282
#823252
C2×C4×#2142
#720452
#720402
C4×#8322
#705892
#705672
#702172
#701932
#701782
#701652
#698322
#698132
#236933