Finite Group Databasebeta

← #10824502

Subgroups — #10824502

Order 768.

908 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4817C211yesmaximal, characteristic, complemented
2C2×#4788C211yesmaximal, characteristic, complemented
3#76483C211yesmaximal, characteristic, complemented
4#74153C211yesmaximal, characteristic
5C3×#2074C211yesmaximal, characteristic, complemented
6#70280C211yesmaximal, characteristic, complemented
7#70280C211yesmaximal, characteristic, complemented
8#69688C211yesmaximal, characteristic
9#69541C211yesmaximal, characteristic, complemented
10#68901C211yesmaximal, characteristic, complemented
11#68900C211yesmaximal, characteristic, complemented
12#68782C211yesmaximal, characteristic, complemented
13#68780C211yesmaximal, characteristic, complemented
14#68629C211yesmaximal, characteristic
15#68604C211yesmaximal, characteristic, complemented
16#23690—31nomaximal, Sylow
17C2×Q8×Dic3C2211yescharacteristic
18C22×#215C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#874C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×#382C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#815C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#4988C2211yescharacteristic
37#4988C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4932C2211yescharacteristic
45#4928C2211yescharacteristic
46#4817—24no—
47#4788—24no—
48#4665—21no—
49#4665—21no—
50#4665—21no—
51#4665—21no—
52#4654—21no—
53#4653—21no—
54#4651—21no—
55#4650—21no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#440—31no—
63C2×C4×SD16—31no—
64#2328—31no—
65#2320—31no—
66#2085—31no—
67#2074D311yescharacteristic, complemented
68#2067—31no—
69#2065—31no—
70#2055—31no—
71#2055—31no—
72#2041—31no—
73#2014—31no—
74#1768—31no—
75#1762—31no—
76C4×#102—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12—21no—
79C22×D12—21no—
80C22×D12C2311yescharacteristic
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C22×C4×D3—21no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90Q8×Dic3—22no—
91Q8×Dic3D414yescomplemented
92C2×#215—22no—
93C2×#215—22no—
94C2×#215—22no—
95C2×#215—22no—
96C2×#215—22no—
97C2×#215—22no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115#893—22no—
116#893—22no—
117#893—22no—
118#893—22no—
119#874—22no—
120#874—22no—
121C3×#103—22no—
122C3×#103—22no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134C3⋊#95D411yescharacteristic
135C3⋊#95D411yescharacteristic
136#821—22no—
137#821—22no—
138#821—22no—
139#821—22no—
140#815—22no—
141Dic3⋊C8—22no—
142Dic3⋊C8—22no—
143C8×Dic3—22no—
144#809—22no—
145#809—22no—
146#808—22no—
147#807—22no—
148C22×SD16—31no—
149C22×SD16—31no—
150C2×#127—31no—
151C2×C4×Q8—31no—
152C2×C4×D4—31no—
153C2×C4×D4—31no—
154#440—64no—
155C4×SD16—64no—
156C2×#107—31no—
157C2×#106—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103C2×D311yescharacteristic
161C2×#103C2×D311yescharacteristic
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97—31no—
169C2×C4×C8—31no—
170#391—31no—
171#391C2×D311yescharacteristic
172#390—31no—
173#390—31no—
174#389—31no—
175#389—31no—
176#387—61no—
177#387—61no—
178#386—61no—
179#386—61no—
180#386—61no—
181#386—61no—
182#383—61no—
183#382—61no—
184#382C2×D311yescharacteristic
185C4×C4⋊C4—31no—
186#376—31no—
187#340—31no—
188#340—31no—
189#340C2×D311yescharacteristic
190#340C2×D311yescharacteristic
191#336—31no—
192#336C2×D311yescharacteristic
193C23×D3—21no—
194C23×D3—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8C2×D412yes—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—24no—
225C2×D12—24no—
226C2×D12—81no—
227C2×D12—81no—
228C2×D12—81no—
229C2×D12—81no—
230C2×D12#4112yes—
231C2×D12#4112yes—
232C2×C4×D3—42no—
233C2×C4×D3—42no—
234C2×C4×D3—81no—
235C2×C4×D3—81no—
236C2×C4×D3—81no—
237C2×C4×D3—81no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C42—21no—
251C3×C42—21no—
252#215—28no—
253#215—28no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—81no—
257(C2×D3)⋊C4—81no—
258(C2×D3)⋊C4—81no—
259(C2×D3)⋊C4—81no—
260C4⋊Dic3—21no—
261C4⋊Dic3—24no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3C2×D412yes—
265Dic3⋊C4—81no—
266Dic3⋊C4—81no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—24no—
270C4×Dic3—81no—
271C4×Dic3C2×D412yes—
272C4×Dic3#4112yes—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8#4112yes—
278C2×C3⋊C8#4112yes—
279C22×Q8C22×D311yescharacteristic
280C22×D4—31no—
281C22×D4—31no—
282C22×D4—61no—
283C22×D4—61no—
284C23×C4—61no—
285C23×C4—61no—
286C2×SD16—62no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—62no—
290C2×SD16—62no—
291C2×SD16—62no—
292C22×C8—31no—
293C22×C8—31no—
294C22×C8C22×D311yescharacteristic
295C22×C8C22×D311yescharacteristic
296#127—62no—
297#127—62no—
298C4×Q8—34no—
299C4×Q8—62no—
300C4×D4—62no—
301C4×D4—62no—
302C4×D4—62no—
303C4×D4—62no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C42—31no—
319C2×C42—31no—
320C2×C42—61no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323#107—62no—
324#106—62no—
325C4⋊C8—62no—
326C4⋊C8—62no—
327#103—22no—
328#103—22no—
329#102—62no—
330#102—62no—
331#102—62no—
332#102—62no—
333#102—62no—
334#102—62no—
335#97—62no—
336C4×C8—62no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
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×C3C22×D411yescharacteristic
351C22×D3—22no—
352C22×D3—22no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C3×Q8—22no—
366C3×Q8—22no—
367C3×Q8—41no—
368C3×Q8—41no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
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×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—21no—
396C2×Dic3—21no—
397C2×Dic3—22no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3#12112yes—
410C2×Dic3#12112yes—
411D12—22no—
412D12—22no—
413D12—22no—
414D12—22no—
415D12—41no—
416D12—41no—
417D12—41no—
418D12—41no—
419D12—84no—
420D12—84no—
421C4×D3—84no—
422C4×D3—84no—
423C3×C8—41no—
424C3×C8—41no—
425C3⋊C8—22no—
426C3⋊C8—22no—
427C24—61no—
428C24—61no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8D3×D412yes—
435C2×D4—32no—
436C2×D4—32no—
437C2×D4—64no—
438C2×D4—64no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—241no—
450C2×D4—241no—
451C2×D4—241no—
452C2×D4—241no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—31no—
460C22×C4—31no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—122no—
464C22×C4—122no—
465C22×C4—241no—
466C22×C4—241no—
467C22×C4—241no—
468C22×C4—241no—
469C22×C4D3×D411yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4C23×D311yescharacteristic
472SD16—68no—
473SD16—68no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—32no—
481C2×C8—32no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—32no—
493C4⋊C4—64no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—241no—
498C4⋊C4—241no—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—241no—
502C22⋊C4—241no—
503C22⋊C4—241no—
504C22⋊C4—241no—
505C42—21no—
506C42—21no—
507C42—32no—
508C42—32no—
509C42—64no—
510C42—61no—
511C42—61no—
512C42—241no—
513C22×C3C2×#12111yescharacteristic
514C22×C3#52511yescharacteristic
515C22×C3#53211yescharacteristic
516C22×C3#57611yescharacteristic
517C22×C3#57711yescharacteristic
518C22×C3#57711yescharacteristic
519C22×C3#57711yescharacteristic
520C2×D3—42no—
521C2×D3—42no—
522C2×D3—42no—
523C2×D3—42no—
524C2×D3—42no—
525C2×D3—42no—
526C2×D3—81no—
527C2×D3—81no—
528C2×D3—81no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—81no—
532C2×D3—81no—
533C2×D3—81no—
534C3×C4—41no—
535C3×C4—41no—
536C3×C4—42no—
537C3×C4—81no—
538C3×C4—81no—
539C3×C4—81no—
540C3×C4#52511yescharacteristic
541C3×C4#52511yescharacteristic
542C3×C4#52911yescharacteristic
543C3×C4#52911yescharacteristic
544Dic3—22no—
545Dic3—22no—
546Dic3—41no—
547Dic3—41no—
548Dic3—42no—
549Dic3—81no—
550C23—62no—
551C23—62no—
552C23—121no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23—121no—
563C23—121no—
564C23C2×D3×D411yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—41no—
569D4—62no—
570D4—62no—
571D4—62no—
572D4—62no—
573D4—121no—
574D4—121no—
575D4—121no—
576D4—121no—
577D4—244no—
578D4—244no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—32no—
587C2×C4—32no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—62no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—244no—
616C2×C4—244no—
617C2×C4C2×D3×D411yescharacteristic
618C2×C4D3×#4111yescharacteristic
619C2×C4D3×#4111yescharacteristic
620C2×C4D3×#4111yescharacteristic
621C2×C4D3×#4111yescharacteristic
622C2×C4D3×#4111yescharacteristic
623C8—41no—
624C8—41no—
625C8—62no—
626C8—62no—
627C2×C3#239211yescharacteristic
628C2×C3#304311yescharacteristic
629C2×C3#304511yescharacteristic
630C2×C3#311811yescharacteristic
631C2×C3#312311yescharacteristic
632C2×C3#317411yescharacteristic
633C2×C3#317911yescharacteristic
634D3—82no—
635D3—82no—
636C22—122no—
637C22—122no—
638C22—122no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22—241no—
643C22—241no—
644C22—241no—
645C22—241no—
646C22—241no—
647C22—241no—
648C22—241no—
649C22—241no—
650C22D3×#12111yescharacteristic
651C22#565611yescharacteristic
652C22#566811yescharacteristic
653C22#576911yescharacteristic
654C22#577011yescharacteristic
655C22#577811yescharacteristic
656C22#577811yescharacteristic
657C4—41no—
658C4—41no—
659C4—42no—
660C4—62no—
661C4—62no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—121no—
666C4—122no—
667C4—121no—
668C4—241no—
669C4#565611yescharacteristic
670C4#565611yescharacteristic
671C4#570611yescharacteristic
672C4#570611yescharacteristic
673C3#2369011yescharacteristic, complemented, Sylow
674C2—242no—
675C2—242no—
676C2#7745711yescharacteristic
677C2#8060011yescharacteristic
678C2#8060211yescharacteristic
679C2#8149611yescharacteristic
680C2#8149811yescharacteristic
681C2#8163011yescharacteristic
682C2#8172911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2074C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#2074D3
hypocenterC3#23690
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2074

Complements

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

NH ≅ G/Ntype
1. C2×#4817C2semidirect
2. C2×#4788C2semidirect
3. #76483C2semidirect
5. C3×#2074C2semidirect
6. #70280C2semidirect
7. #70280C2semidirect
9. #69541C2semidirect
10. #68901C2semidirect
11. #68900C2semidirect
12. #68782C2semidirect
13. #68780C2semidirect
15. #68604C2semidirect
67. #2074D3semidirect
91. Q8×Dic3D4semidirect
673. C3#23690semidirect

Maximal subgroups

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

subgroupindex
C2×#48172
C2×#47882
#764832
#741532
C3×#20742
#702802
#696882
#695412
#689012
#689002
#687822
#687802
#686292
#686042
#236903