Finite Group Databasebeta

← #10855100

Subgroups — #10855100

Order 768.

846 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2415C211yesmaximal, characteristic
2C2×#5045C211yesmaximal, characteristic, complemented
3#82335C211yesmaximal, characteristic
4C2×#4995C211yesmaximal, characteristic, complemented
5#71697C211yesmaximal, characteristic, complemented
6#71694C211yesmaximal, characteristic, complemented
7#71692C211yesmaximal, characteristic, complemented
8#71611C211yesmaximal, characteristic
9#71576C211yesmaximal, characteristic, complemented
10#69686C211yesmaximal, characteristic, complemented
11#69667C211yesmaximal, characteristic, complemented
12#69609C211yesmaximal, characteristic, complemented
13#69602C211yesmaximal, characteristic, complemented
14#69519C211yesmaximal, characteristic, complemented
15#69506C211yesmaximal, characteristic, complemented
16#26154—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×#117C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C3×#397—21no—
25C3×#394—21no—
26C3×#391C2211yescharacteristic
27C3×#389—21no—
28C3×#389C2211yescharacteristic
29C3×#388—21no—
30C3×#380C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35#5220—21no—
36#5212C2211yescharacteristic
37#5212C2211yescharacteristic
38C2×#830C2211yescharacteristic
39C2×#830C2211yescharacteristic
40C2×#830C2211yescharacteristic
41#5045—24no—
42#4995—24no—
43#4980—21no—
44C2×#808C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#807C2211yescharacteristic
48C2×#807C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4951—21no—
51Dic3×C22⋊C4—21no—
52#4936C2211yescharacteristic
53C4×C4⋊Dic3C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#485—31no—
63C2×#428—31no—
64#2415D311yescharacteristic
65#2352—31no—
66#2097—31no—
67#2085—31no—
68#2076—31no—
69#2035—31no—
70#1928—31no—
71#1920—31no—
72#1894—31no—
73#1872—31no—
74#1870—31no—
75#1854—31no—
76#1854—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4D411yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C23×Dic3—21no—
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—41no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×#117—22no—
89C3×#117—41no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4D411yescharacteristic
102C2×C3×C22⋊C4D411yescharacteristic
103C2×C3×C42—21no—
104C2×C3×C42C2311yescharacteristic
105C2×#217—21no—
106C2×#217—21no—
107C2×#217—21no—
108#926—22no—
109#926—41no—
110C2×C4⋊Dic3—22no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3—41no—
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×#208—22no—
120C2×#208—22no—
121C2×#208—22no—
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130#835—22no—
131#835—22no—
132#835—22no—
133#835—41no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139#830—22no—
140#830—22no—
141#830—22no—
142#808—22no—
143#808—41no—
144#808D414yescomplemented
145#807—22no—
146#807—22no—
147#807—22no—
148C22×M16—31no—
149C2×C22⋊Q8C2×D311yescharacteristic
150C2×C22⋊Q8C2×D311yescharacteristic
151C2×C4×D4C2×D311yescharacteristic
152C2×#117—31no—
153C2×#117C2×D311yescharacteristic
154C22×C4⋊C4—31no—
155#485—64no—
156#428—64no—
157C2×#107—31no—
158C2×#107—31no—
159C2×#107—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×#106—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×C22⋊C8—31no—
170#397—21no—
171#394—21no—
172#391C2×D311yescharacteristic
173#389—21no—
174#389C2×D311yescharacteristic
175#388—21no—
176#388—61no—
177#387—61no—
178#386—61no—
179#382—31no—
180#380—31no—
181#380—31no—
182#380C2×D311yescharacteristic
183C4×C4⋊C4—31no—
184C4×C22⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C24×C3C2×D411yescharacteristic
194C24×C3C2×D411yescharacteristic
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×D4—22no—
202C2×C3×D4—41no—
203C2×C3×D4—41no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
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×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4#4112yes—
223C22×C3×C4#4112yes—
224C22×C3×C4C2411yescharacteristic
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—42no—
230C22×Dic3—42no—
231C22×Dic3—81no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4C2×D412yes—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—42no—
245C3×C22⋊C4—42no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—42no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—42no—
251#217—42no—
252#217—42no—
253#217—81no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—42no—
262C4⋊Dic3C2×D412yes—
263C4×Dic3—42no—
264C4×Dic3—81no—
265#208—28no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8C2×D412yes—
277C2×C3⋊C8#4112yes—
278C22×Q8C22×D311yescharacteristic
279C22×D4#6711yescharacteristic
280C23×C4—61no—
281C23×C4C22×D311yescharacteristic
282C23×C4C22×D311yescharacteristic
283C2×M16—62no—
284C2×M16—62no—
285C2×M16—62no—
286C22×C8—31no—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8—31no—
290C22⋊Q8—22no—
291C22⋊Q8—22no—
292C22⋊Q8—22no—
293C22⋊Q8—41no—
294C4×D4—22no—
295C4×D4—22no—
296#117—22no—
297#117—41no—
298#117—62no—
299#117—121no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—62no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—121no—
310C2×C4⋊C4#6711yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4#6711yescharacteristic
322C2×C22⋊C4#6711yescharacteristic
323C2×C42—21no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#107—62no—
328#107—62no—
329#107—62no—
330#106—34no—
331#106—62no—
332#106—121no—
333#103—62no—
334#103—62no—
335#103—62no—
336#103—121no—
337C22⋊C8—62no—
338C22⋊C8—62no—
339C22⋊C8—62no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3#12312yes—
363C23×C3C22×D411yescharacteristic
364C3×Q8—41no—
365C3×Q8—41no—
366C3×Q8—41no—
367C3×Q8—41no—
368C3×D4—44no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4#12312yes—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4H(5,𝔽2)11yescharacteristic
406C2×C3×C42−1+411yescharacteristic
407C2×Dic3—22no—
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—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—84no—
420C3⋊C8—22no—
421C3⋊C8—22no—
422C3⋊C8—41no—
423C3⋊C8—41no—
424C24C2×#6711yescharacteristic
425C24C2×#6711yescharacteristic
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×D4—22no—
433C2×D4—41no—
434C2×D4—41no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—122no—
454C22×C4—122no—
455C22×C4—241no—
456C22×C4#23712yes—
457C22×C4#23712yes—
458C22×C4C2×#6711yescharacteristic
459C22×C4C2×#6711yescharacteristic
460C22×C4C2×#6711yescharacteristic
461C22×C4C2×#6711yescharacteristic
462C22×C4C23×D311yescharacteristic
463M16—68no—
464C2×C8—32no—
465C2×C8—32no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—32no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—122no—
494C4⋊C4D3×D412yes—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—241no—
504C42—21no—
505C42—21no—
506C42—42no—
507C42—122no—
508C42—241no—
509C22×C3—22no—
510C22×C3—22no—
511C22×C3—22no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3C2×#12311yescharacteristic
524C22×C3D4211yescharacteristic
525C22×C3#54711yescharacteristic
526C22×C3#57511yescharacteristic
527C22×C3#57511yescharacteristic
528C22×C3#57711yescharacteristic
529C22×C3#57811yescharacteristic
530C3×C4—22no—
531C3×C4—41no—
532C3×C4—41no—
533C3×C4—41no—
534C3×C4—42no—
535C3×C4—81no—
536C3×C4—81no—
537C3×C4—81no—
538C3×C4#52511yescharacteristic
539C3×C4#53911yescharacteristic
540C3×C4#54811yescharacteristic
541C3×C4#55311yescharacteristic
542Dic3—42no—
543Dic3—81no—
544Dic3—81no—
545Dic3—81no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23#93512yes—
560C23C22×#6711yescharacteristic
561Q8—41no—
562Q8—41no—
563Q8—41no—
564Q8—41no—
565D4—44no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
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—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—62no—
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—121no—
609C2×C4—244no—
610C2×C4#93512yes—
611C2×C4C2×D3×D411yescharacteristic
612C2×C4C2×#23711yescharacteristic
613C2×C4C2×#23711yescharacteristic
614C2×C4#100411yescharacteristic
615C2×C4D3×#4111yescharacteristic
616C2×C4#101011yescharacteristic
617C8—62no—
618C8—62no—
619C8—121no—
620C8—121no—
621C2×C3—42no—
622C2×C3—81no—
623C2×C3#244311yescharacteristic
624C2×C3#309311yescharacteristic
625C2×C3#309711yescharacteristic
626C2×C3#326911yescharacteristic
627C2×C3#327511yescharacteristic
628C2×C3#335011yescharacteristic
629C2×C3#335211yescharacteristic
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22C2×#93511yescharacteristic
645C22D4×#6711yescharacteristic
646C22#582211yescharacteristic
647C22#582211yescharacteristic
648C22#582411yescharacteristic
649C22#583711yescharacteristic
650C22#583811yescharacteristic
651C4—22no—
652C4—42no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—81no—
657C4—81no—
658C4—81no—
659C4—122no—
660C4—241no—
661C4—241no—
662C4—241no—
663C4#559811yescharacteristic
664C4#560111yescharacteristic
665C4#560211yescharacteristic
666C4#560611yescharacteristic
667C3#2615411yescharacteristic, complemented, Sylow
668C2—42no—
669C2—81no—
670C2#8265511yescharacteristic
671C2#8291511yescharacteristic
672C2#8339711yescharacteristic
673C2#8352811yescharacteristic
674C2#8354411yescharacteristic
675C2#8359711yescharacteristic
676C2#8366111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2415C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#2415D3
hypocenterC3#26154
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2415

Complements

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

NH ≅ G/Ntype
2. C2×#5045C2semidirect
4. C2×#4995C2semidirect
5. #71697C2semidirect
6. #71694C2semidirect
7. #71692C2semidirect
9. #71576C2semidirect
10. #69686C2semidirect
11. #69667C2semidirect
12. #69609C2semidirect
13. #69602C2semidirect
14. #69519C2semidirect
15. #69506C2semidirect
144. #808D4semidirect
667. C3#26154semidirect

Maximal subgroups

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

subgroupindex
C3×#24152
C2×#50452
#823352
C2×#49952
#716972
#716942
#716922
#716112
#715762
#696862
#696672
#696092
#696022
#695192
#695062
#261543