Finite Group Databasebeta

← #10824556

Subgroups — #10824556

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
1#82455C211yesmaximal, characteristic, complemented
2C2×C4×#215C211yesmaximal, characteristic, complemented
3C2×#4711C211yesmaximal, characteristic
4#73761C211yesmaximal, characteristic, complemented
5C3×#1763C211yesmaximal, characteristic, complemented
6#70280C211yesmaximal, characteristic, complemented
7#70279C211yesmaximal, characteristic, complemented
8#70277C211yesmaximal, characteristic
9#70247C211yesmaximal, characteristic, complemented
10#69832C211yesmaximal, characteristic, complemented
11C4×#809C211yesmaximal, characteristic, complemented
12#68902C211yesmaximal, characteristic, complemented
13#68900C211yesmaximal, characteristic, complemented
14#68862C211yesmaximal, characteristic
15#68835C211yesmaximal, characteristic, complemented
16#23693—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C22×D24C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#817C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C4×#215—24no—
35#4989—21no—
36#4989—21no—
37#4988C2211yescharacteristic
38#4988C2211yescharacteristic
39#4988C2211yescharacteristic
40#4975C2211yescharacteristic
41#4972C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#808C2211yescharacteristic
47#4940—21no—
48#4940—21no—
49#4940—21no—
50#4940—21no—
51#4932C2211yescharacteristic
52C2×C4×C3⋊C8C2211yescharacteristic
53#4711—24no—
54#4667—21no—
55#4664—21no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#442—31no—
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#1763D311yescharacteristic, complemented
76C4×#102—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C42—21no—
91C2×C3×C42—21no—
92C2×C3×C42C2311yescharacteristic
93#947—22no—
94#947—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×(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×Dic3⋊C4—21no—
114C2×C4×Dic3C2311yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C2×D24—22no—
119C2×D24—22no—
120C2×D24—22no—
121C4×D12—22no—
122C4×D12—22no—
123C4×D12—22no—
124C4×D12D414yes—
125C3×#103—22no—
126C3×#103—22no—
127C3×#97—22no—
128C3×#95—21no—
129#835—22no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95D411yescharacteristic
135C3⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137C3⋊#95D411yescharacteristic
138#821—22no—
139#821—22no—
140#817—22no—
141Dic3⋊C8—22no—
142Dic3⋊C8—22no—
143#809—22no—
144#809—22no—
145#809—22no—
146#808—22no—
147C4×C3⋊C8—22no—
148C22×SD16—31no—
149C22×D8—31no—
150C2×#124—31no—
151C2×C4×Q8C2×D311yescharacteristic
152C2×C4×D4—31no—
153C2×C4×D4—31no—
154#442—64no—
155C4×SD16—64no—
156C2×#106—31no—
157C2×#106—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103C2×D311yescharacteristic
162C2×#103C2×D311yescharacteristic
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
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⋊C4C2×D311yescharacteristic
186#376—31no—
187#340—31no—
188#340—31no—
189#340C2×D311yescharacteristic
190#340C2×D311yescharacteristic
191#336—31no—
192#336—31no—
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×Q8#4112yes—
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×D12C2×D412yes—
225C2×D12#4112yes—
226C2×C4×D3—24no—
227C2×C4×D3—42no—
228C2×C4×D3—81no—
229C2×C4×D3—81no—
230C2×C4×D3—81no—
231C2×C4×D3—81no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8#4112yes—
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—42no—
245C3×C42—42no—
246C3×C42—81no—
247C3×C42C2×D412yes—
248C3×C42#4112yes—
249#215—28no—
250(C2×D3)⋊C4—24no—
251(C2×D3)⋊C4—24no—
252(C2×D3)⋊C4—24no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—81no—
255(C2×D3)⋊C4—81no—
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—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3C2×D412yes—
264Dic3⋊C4—81no—
265Dic3⋊C4—81no—
266Dic3⋊C4—81no—
267Dic3⋊C4—81no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8#4112yes—
278D24—28no—
279C22×Q8C22×D311yescharacteristic
280C22×D4—31no—
281C22×D4—31no—
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—22no—
297C4×Q8—22no—
298C4×D4—34no—
299C4×D4—62no—
300C4×D4—62no—
301C4×D4—62no—
302C2×C4⋊C4—21no—
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⋊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×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—31no—
325C2×C42—31no—
326C2×C42C22×D311yescharacteristic
327#106—62no—
328#106—62no—
329C4⋊C8—62no—
330C4⋊C8—62no—
331#103—22no—
332#103—22no—
333#103—62no—
334#102—62no—
335#102—62no—
336#102—62no—
337#102—62no—
338#102—62no—
339#97—22no—
340C4×C8—62no—
341#95—21no—
342#95—31no—
343#95—31no—
344#95—31no—
345#95—31no—
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—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—22no—
374C2×C3×C4—41no—
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×C4—41no—
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—22no—
394C2×Dic3—21no—
395C2×Dic3—21no—
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—
411D12—22no—
412D12—22no—
413D12—22no—
414D12—22no—
415D12—41no—
416D12—41no—
417D12—41no—
418D12—41no—
419C4×D3—84no—
420C4×D3—84no—
421C3×C8—22no—
422C3×C8—41no—
423C3⋊C8—22no—
424C3⋊C8—41no—
425C24—61no—
426C24—61no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8#23512yes—
433C2×D4—32no—
434C2×D4—32no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—31no—
452C22×C4—31no—
453C22×C4—61no—
454C22×C4—64no—
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—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—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—32no—
485C4⋊C4—42no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—241no—
490C4⋊C4—241no—
491C4⋊C4—241no—
492C4⋊C4—241no—
493C22⋊C4—64no—
494C22⋊C4—64no—
495C22⋊C4—64no—
496C22⋊C4—122no—
497C22⋊C4—241no—
498C22⋊C4—241no—
499C22⋊C4—241no—
500C22⋊C4—241no—
501C22⋊C4—241no—
502C22⋊C4—241no—
503C42—42no—
504C42—61no—
505C42—61no—
506C42—61no—
507C42—61no—
508C42—81no—
509C42#23512yes—
510C42D3×D412yes—
511C22×C3C2×#12111yescharacteristic
512C22×C3#52511yescharacteristic
513C22×C3#53211yescharacteristic
514C22×C3#57611yescharacteristic
515C22×C3#57711yescharacteristic
516C22×C3#57711yescharacteristic
517C22×C3#57711yescharacteristic
518C2×D3—42no—
519C2×D3—42no—
520C2×D3—42no—
521C2×D3—42no—
522C2×D3—42no—
523C2×D3—42no—
524C2×D3—81no—
525C2×D3—81no—
526C2×D3—81no—
527C2×D3—81no—
528C2×D3—81no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—81no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—42no—
535C3×C4—81no—
536C3×C4—81no—
537C3×C4—81no—
538C3×C4#52511yescharacteristic
539C3×C4#52511yescharacteristic
540C3×C4#52911yescharacteristic
541C3×C4#52911yescharacteristic
542Dic3—41no—
543Dic3—41no—
544Dic3—41no—
545Dic3—41no—
546Dic3—42no—
547Dic3—81no—
548C23—62no—
549C23—62no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23C2×D3×D411yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567D4—62no—
568D4—62no—
569D4—62no—
570D4—62no—
571D4—121no—
572D4—121no—
573D4—121no—
574D4—121no—
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—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—62no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—244no—
610C2×C4—244no—
611C2×C4#88412yes—
612C2×C4#88412yes—
613C2×C4C2×#23511yescharacteristic
614C2×C4C2×#23511yescharacteristic
615C2×C4C2×#23511yescharacteristic
616C2×C4C2×D3×D411yescharacteristic
617C2×C4D3×#4111yescharacteristic
618C2×C4D3×#4111yescharacteristic
619C8—22no—
620C8—41no—
621C8—62no—
622C8—121no—
623C2×C3#239211yescharacteristic
624C2×C3#304211yescharacteristic
625C2×C3#304511yescharacteristic
626C2×C3#311911yescharacteristic
627C2×C3#312511yescharacteristic
628C2×C3#317311yescharacteristic
629C2×C3#317911yescharacteristic
630D3—82no—
631D3—82no—
632C22—122no—
633C22—122no—
634C22—122no—
635C22—122no—
636C22—122no—
637C22—122no—
638C22—241no—
639C22—241no—
640C22—241no—
641C22—241no—
642C22—241no—
643C22—241no—
644C22—241no—
645C22—241no—
646C22C2×#88411yescharacteristic
647C22#554711yescharacteristic
648C22#554811yescharacteristic
649C22#577411yescharacteristic
650C22#577811yescharacteristic
651C22#577811yescharacteristic
652C22#577811yescharacteristic
653C4—22no—
654C4—22no—
655C4—42no—
656C4—81no—
657C4—81no—
658C4—81no—
659C4—122no—
660C4—121no—
661C4—121no—
662C4—121no—
663C4—121no—
664C4—241no—
665C4#547911yescharacteristic
666C4#552811yescharacteristic
667C4#554711yescharacteristic
668C4#554711yescharacteristic
669C3#2369311yescharacteristic, complemented, Sylow
670C2—242no—
671C2—242no—
672C2#7358111yescharacteristic
673C2#7891111yescharacteristic
674C2#7897611yescharacteristic
675C2#7977011yescharacteristic
676C2#7981211yescharacteristic
677C2#7982311yescharacteristic
678C2#7985011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1763C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1763D3
hypocenterC3#23693
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
1. #82455C2semidirect
2. C2×C4×#215C2semidirect
4. #73761C2semidirect
5. C3×#1763C2semidirect
6. #70280C2semidirect
7. #70279C2semidirect
9. #70247C2semidirect
10. #69832C2semidirect
11. C4×#809C2semidirect
12. #68902C2semidirect
13. #68900C2semidirect
15. #68835C2semidirect
75. #1763D3semidirect
669. C3#23693semidirect

Maximal subgroups

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

subgroupindex
#824552
C2×C4×#2152
C2×#47112
#737612
C3×#17632
#702802
#702792
#702772
#702472
#698322
C4×#8092
#689022
#689002
#688622
#688352
#236933