Finite Group Databasebeta

← #10824532

Subgroups — #10824532

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×#4757C211yesmaximal, characteristic, complemented
2C2×#4756C211yesmaximal, characteristic
3#76295C211yesmaximal, characteristic, complemented
4#74321C211yesmaximal, characteristic, complemented
5C3×#2071C211yesmaximal, characteristic, complemented
6#70217C211yesmaximal, characteristic, complemented
7#70212C211yesmaximal, characteristic
8#69659C211yesmaximal, characteristic
9#69569C211yesmaximal, characteristic, complemented
10#68913C211yesmaximal, characteristic, complemented
11#68901C211yesmaximal, characteristic, complemented
12#68782C211yesmaximal, characteristic, complemented
13#68781C211yesmaximal, characteristic, complemented
14#68630C211yesmaximal, characteristic, complemented
15#68604C211yesmaximal, characteristic, complemented
16#23692—31nomaximal, Sylow
17C2×D4×Dic3C2211yescharacteristic
18C22×#214C2211yescharacteristic
19C22×#213C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×#875C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C3×#390C2211yescharacteristic
26C3×#384C2211yescharacteristic
27#5221C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#815C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#4990—21no—
37#4990—21no—
38#4989—21no—
39#4989—21no—
40#4988C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42#4973C2211yescharacteristic
43#4972C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C2×#807C2211yescharacteristic
48#4945—21no—
49#4944—21no—
50#4943—21no—
51#4942—21no—
52#4932C2211yescharacteristic
53#4928C2211yescharacteristic
54#4757—24no—
55#4756—24no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#440—31no—
63C2×C4×D8—31no—
64#2326—31no—
65#2319—31no—
66#2083—31no—
67#2071D311yescharacteristic, complemented
68#2068—31no—
69#2067—31no—
70#2055—31no—
71#2054—31no—
72#2042—31no—
73#2012—31no—
74#1768—31no—
75#1762—31no—
76C4×#103—31no—
77C22×C3×D4C2311yescharacteristic
78C23×Dic3—21no—
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C42C2311yescharacteristic
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95D4×Dic3—22no—
96D4×Dic3D414yes—
97C2×#214—22no—
98C2×#214—22no—
99C2×#214—22no—
100C2×#213—22no—
101C2×#213—22no—
102C2×#213—22no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#893—22no—
120#893—22no—
121#887—22no—
122#887—22no—
123#875—22no—
124#875—22no—
125C3×#102—22no—
126C3×#102—22no—
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#816—22no—
139#816—22no—
140#815—22no—
141Dic3⋊C8—22no—
142Dic3⋊C8—22no—
143C8×Dic3—22no—
144#810—22no—
145#809—22no—
146#807—22no—
147#807—22no—
148C22×SD16—31no—
149C22×D8—31no—
150C2×#124—31no—
151C2×C4×Q8—31no—
152C2×C4×D4—31no—
153C2×C4×D4—31no—
154#440—64no—
155C4×D8—64no—
156C2×#107—31no—
157C2×#107—31no—
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×#97—31no—
169C2×C4×C8—31no—
170#391—31no—
171#390—31no—
172#390—31no—
173#390C2×D311yescharacteristic
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#384C2×D311yescharacteristic
183#381—61no—
184#380—61no—
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
193C24×C3—21no—
194C23×D3—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4C2×D412yes—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—24no—
210C22×Dic3—21no—
211C22×Dic3—81no—
212C22×Dic3—81no—
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12#4112yes—
223C2×C4×D3—42no—
224C2×C4×D3—81no—
225C2×C4×D3—81no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6#4112yes—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—81no—
245C3×C4⋊C4—81no—
246C3×C22⋊C4—81no—
247C3×C22⋊C4—81no—
248C3×C42—21no—
249C3×C42—21no—
250#217—24no—
251#217—81no—
252#217—81no—
253#214—28no—
254#213—28no—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—81no—
259(C2×D3)⋊C4—81no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3C2×D412yes—
264Dic3⋊C4—42no—
265Dic3⋊C4—81no—
266Dic3⋊C4—81no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—42no—
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×Q8—31no—
280C22×D4—31no—
281C22×D4C22×D311yescharacteristic
282C23×C4—61no—
283C23×C4—61no—
284C2×SD16—62no—
285C2×SD16—62no—
286C2×SD16—62no—
287C2×D8—62no—
288C2×D8—62no—
289C2×D8—62no—
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—34no—
299C4×D4—62no—
300C4×D4—62no—
301C4×D4—62no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
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—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#107—62no—
328#107—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—62no—
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—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3C22×D411yescharacteristic
358C22×D3—22no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C3×D4—22no—
366C3×D4—22no—
367C3×D4—41no—
368C3×D4—41no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—41no—
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×C4C22×D411yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—84no—
410C2×Dic3#12112yes—
411C2×Dic3#12112yes—
412D12—22no—
413D12—22no—
414D12—41no—
415D12—41no—
416C4×D3—84no—
417Dic6—22no—
418Dic6—22no—
419Dic6—41no—
420Dic6—41no—
421C3×C8—41no—
422C3×C8—41no—
423C3⋊C8—22no—
424C3⋊C8—22no—
425C24—21no—
426C24—61no—
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—32no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4D3×D412yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—31no—
452C22×C4—64no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—122no—
457C22×C4—241no—
458C22×C4—241no—
459C22×C4—241no—
460C22×C4—241no—
461C22×C4D3×D411yescharacteristic
462C22×C4D3×D411yescharacteristic
463C22×C4C23×D311yescharacteristic
464SD16—68no—
465D8—68no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—32no—
473C2×C8—32no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—32no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4—122no—
491C4⋊C4—241no—
492C4⋊C4—241no—
493C22⋊C4—64no—
494C22⋊C4—64no—
495C22⋊C4—64no—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C22⋊C4—122no—
499C22⋊C4—241no—
500C22⋊C4—241no—
501C22⋊C4—241no—
502C22⋊C4—241no—
503C42—21no—
504C42—21no—
505C42—32no—
506C42—32no—
507C42—61no—
508C42—61no—
509C42—122no—
510C42—241no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—81no—
515C22×C3—81no—
516C22×C3—81no—
517C22×C3—81no—
518C22×C3C2×#12111yescharacteristic
519C22×C3#52511yescharacteristic
520C22×C3#53211yescharacteristic
521C22×C3#57611yescharacteristic
522C22×C3#57611yescharacteristic
523C22×C3#57711yescharacteristic
524C22×C3#57811yescharacteristic
525C2×D3—42no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—81no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—81no—
532C3×C4—41no—
533C3×C4—41no—
534C3×C4—81no—
535C3×C4—81no—
536C3×C4—81no—
537C3×C4#52511yescharacteristic
538C3×C4#52511yescharacteristic
539C3×C4#52911yescharacteristic
540C3×C4#52911yescharacteristic
541Dic3—22no—
542Dic3—22no—
543Dic3—42no—
544Dic3—41no—
545Dic3—41no—
546Dic3—42no—
547Dic3—81no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—62no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23C2×D3×D411yescharacteristic
563Q8—62no—
564Q8—62no—
565Q8—121no—
566Q8—121no—
567D4—22no—
568D4—22no—
569D4—41no—
570D4—41no—
571D4—62no—
572D4—62no—
573D4—121no—
574D4—121no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—32no—
582C2×C4—32no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—62no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—62no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—244no—
612C2×C4—244no—
613C2×C4C2×D3×D411yescharacteristic
614C2×C4D3×#4111yescharacteristic
615C2×C4D3×#4111yescharacteristic
616C2×C4D3×#4111yescharacteristic
617C2×C4D3×#4111yescharacteristic
618C2×C4D3×#4111yescharacteristic
619C8—41no—
620C8—41no—
621C8—62no—
622C8—62no—
623C2×C3—82no—
624C2×C3#239211yescharacteristic
625C2×C3#304111yescharacteristic
626C2×C3#304611yescharacteristic
627C2×C3#311911yescharacteristic
628C2×C3#312411yescharacteristic
629C2×C3#317511yescharacteristic
630C2×C3#317911yescharacteristic
631D3—82no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22—241no—
643C22—241no—
644C22—241no—
645C22—241no—
646C22D3×#12111yescharacteristic
647C22#565611yescharacteristic
648C22#566811yescharacteristic
649C22#575711yescharacteristic
650C22#576211yescharacteristic
651C22#577411yescharacteristic
652C22#577911yescharacteristic
653C4—41no—
654C4—41no—
655C4—62no—
656C4—62no—
657C4—81no—
658C4—81no—
659C4—81no—
660C4—121no—
661C4—122no—
662C4—121no—
663C4—122no—
664C4—241no—
665C4#565611yescharacteristic
666C4#565611yescharacteristic
667C4#570611yescharacteristic
668C4#570611yescharacteristic
669C3#2369211yescharacteristic, complemented, Sylow
670C2—82no—
671C2—242no—
672C2#7745711yescharacteristic
673C2#8033411yescharacteristic
674C2#8043111yescharacteristic
675C2#8127311yescharacteristic
676C2#8131211yescharacteristic
677C2#8155811yescharacteristic
678C2#8166711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2071C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#2071D3
hypocenterC3#23692
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
1. C2×#4757C2semidirect
3. #76295C2semidirect
4. #74321C2semidirect
5. C3×#2071C2semidirect
6. #70217C2semidirect
9. #69569C2semidirect
10. #68913C2semidirect
11. #68901C2semidirect
12. #68782C2semidirect
13. #68781C2semidirect
14. #68630C2semidirect
15. #68604C2semidirect
67. #2071D3semidirect
669. C3#23692semidirect

Maximal subgroups

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

subgroupindex
C2×#47572
C2×#47562
#762952
#743212
C3×#20712
#702172
#702122
#696592
#695692
#689132
#689012
#687822
#687812
#686302
#686042
#236923