Finite Group Databasebeta

← #10824800

Subgroups — #10824800

Order 768.

932 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4869C211yesmaximal, characteristic, complemented
2C2×#4720C211yesmaximal, characteristic, complemented
3#73872C211yesmaximal, characteristic, complemented
4#73872C211yesmaximal, characteristic, complemented
5C3×#2077C211yesmaximal, characteristic, complemented
6#70568C211yesmaximal, characteristic, complemented
7#69961C211yesmaximal, characteristic
8#69524C211yesmaximal, characteristic, complemented
9#69523C211yesmaximal, characteristic, complemented
10#68901C211yesmaximal, characteristic, complemented
11#68900C211yesmaximal, characteristic, complemented
12#68862C211yesmaximal, characteristic, complemented
13#68786C211yesmaximal, characteristic, complemented
14#68780C211yesmaximal, characteristic, complemented
15#68780C211yesmaximal, characteristic, complemented
16#23712—31nomaximal, Sylow
17C2×#930C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C22×D24C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#893C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25C3×#380C2211yescharacteristic
26C3×#380C2211yescharacteristic
27C2×#830C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#815C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34#4988C2211yescharacteristic
35#4988C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38#4972C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4957—21no—
47#4957—21no—
48#4956—21no—
49#4956—21no—
50#4869—24no—
51#4720—24no—
52#4670—21no—
53#4670—21no—
54#4668—21no—
55#4668—21no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#442—31no—
63C2×#433—31no—
64#2324—31no—
65#2324—31no—
66#2088—31no—
67#2077D311yescharacteristic, complemented
68#2067—31no—
69#2065—31no—
70#2054—31no—
71#2034—31no—
72#2033—31no—
73#1999—31no—
74#1776—31no—
75C4×#102—31no—
76C4×#102—31no—
77C23×C3×C4C2311yescharacteristic
78C22×#67—21no—
79C22×#67—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×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×#217D411yescharacteristic
93C2×#217D411yescharacteristic
94#930—22no—
95#930—22no—
96#930—22no—
97#930—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—
106#926—22no—
107#926—22no—
108C2×C4⋊Dic3D411yescharacteristic
109C2×C4⋊Dic3D411yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3C2311yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C2×D24—22no—
121C2×D24—22no—
122C2×D24—22no—
123#893—22no—
124#893—22no—
125#893—22no—
126#893—22no—
127C3×#106—22no—
128C3×C22⋊C8—22no—
129C3×#95—21no—
130C3×#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135#830—22no—
136#821—22no—
137#821—22no—
138#815D414yescomplemented
139Dic3⋊C8—22no—
140Dic3⋊C8—22no—
141C8×Dic3—22no—
142#809—22no—
143#809—22no—
144#809—22no—
145#809—22no—
146#808—22no—
147#807—22no—
148C22×D8—31no—
149C2×#121—31no—
150C2×#121—31no—
151C2×C4×D4—31no—
152C2×C4×D4—31no—
153C2×#117—31no—
154#442—64no—
155#433—64no—
156C2×#107—31no—
157C2×#106—31no—
158C2×#106C2×D311yescharacteristic
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×#102—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×C22⋊C8—31no—
168C2×C22⋊C8C2×D311yescharacteristic
169C2×#97—31no—
170C2×C4×C8—31no—
171#390—31no—
172#390—31no—
173#390—61no—
174#390—61no—
175#389—31no—
176#389—31no—
177#388—61no—
178#388—61no—
179#386—61no—
180#386—61no—
181#385—61no—
182#385—61no—
183#380C2×D311yescharacteristic
184#380C2×D311yescharacteristic
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340C2×D311yescharacteristic
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
193C24×C3C2×D411yescharacteristic
194C23×D3—21no—
195C23×D3—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C2×#67—24no—
209C2×#67—24no—
210C2×#67—81no—
211C2×#67—81no—
212C2×#67—81no—
213C2×#67—81no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
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×C8C2×D412yes—
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×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C22⋊C4—81no—
253C3×C22⋊C4—81no—
254#217—22no—
255#217—22no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—24no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—81no—
261(C2×D3)⋊C4—81no—
262(C2×D3)⋊C4—81no—
263(C2×D3)⋊C4—81no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—22no—
267C4⋊Dic3—22no—
268C4⋊Dic3#4112yes—
269C4⋊Dic3#4112yes—
270Dic3⋊C4—81no—
271Dic3⋊C4—81no—
272C4×Dic3—21no—
273C4×Dic3—21no—
274C4×Dic3—81no—
275C4×Dic3—81no—
276C4×Dic3C2×D412yes—
277C4×Dic3#4112yes—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8C2×D412yes—
284D24—28no—
285C22×D4—31no—
286C22×D4—31no—
287C22×D4—61no—
288C22×D4—61no—
289C23×C4—61no—
290C23×C4—61no—
291C23×C4C22×D311yescharacteristic
292C2×D8—62no—
293C2×D8—62no—
294C2×D8—62no—
295C22×C8—31no—
296C22×C8—31no—
297C22×C8C22×D311yescharacteristic
298C22×C8C22×D311yescharacteristic
299#121—62no—
300#121—62no—
301#121—62no—
302#121—62no—
303C4×D4—62no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—62no—
307#117—62no—
308#117—62no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—31no—
322C2×C22⋊C4—31no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C42—31no—
332C2×C42—31no—
333C2×C42—61no—
334C2×C42—61no—
335#107—62no—
336#106—22no—
337#106—62no—
338C4⋊C8—62no—
339C4⋊C8—62no—
340#102—62no—
341#102—62no—
342#102—62no—
343#102—62no—
344#102—62no—
345#102—62no—
346C22⋊C8—22no—
347C22⋊C8—62no—
348#97—34no—
349C4×C8—62no—
350#95—21no—
351#95—21no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3C22×D411yescharacteristic
364C22×D3—22no—
365C22×D3—22no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C22×D3—41no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
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×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×C4C22×D411yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403#67—84no—
404#67—84no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3#12112yes—
422C2×Dic3#12112yes—
423D12—22no—
424D12—22no—
425D12—22no—
426D12—22no—
427D12—41no—
428D12—41no—
429D12—41no—
430D12—41no—
431C4×D3—84no—
432C4×D3—84no—
433C3×C8—22no—
434C3×C8—41no—
435C3⋊C8—22no—
436C3⋊C8—41no—
437C24—61no—
438C24—61no—
439C24D3×D411yescharacteristic
440C2×D4—32no—
441C2×D4—32no—
442C2×D4—64no—
443C2×D4—64no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—241no—
455C2×D4—241no—
456C2×D4—241no—
457C2×D4—241no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—31no—
469C22×C4—31no—
470C22×C4—31no—
471C22×C4—31no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—122no—
475C22×C4—122no—
476C22×C4—241no—
477C22×C4—241no—
478C22×C4—241no—
479C22×C4—241no—
480C22×C4D3×D411yescharacteristic
481C22×C4C23×D311yescharacteristic
482D8—68no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—32no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8D3×D412yes—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—32no—
504C4⋊C4—32no—
505C4⋊C4—62no—
506C4⋊C4—62no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—241no—
510C4⋊C4—241no—
511C22⋊C4—64no—
512C22⋊C4—64no—
513C22⋊C4—62no—
514C22⋊C4—62no—
515C22⋊C4—81no—
516C22⋊C4—81no—
517C22⋊C4—122no—
518C22⋊C4—122no—
519C22⋊C4—241no—
520C22⋊C4—241no—
521C22⋊C4—241no—
522C22⋊C4—241no—
523C42—32no—
524C42—32no—
525C42—61no—
526C42—61no—
527C42—241no—
528C42—241no—
529C22×C3—41no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3—41no—
533C22×C3—41no—
534C22×C3—41no—
535C22×C3—41no—
536C22×C3C2×#12111yescharacteristic
537C22×C3#52511yescharacteristic
538C22×C3#52511yescharacteristic
539C22×C3#57611yescharacteristic
540C22×C3#57611yescharacteristic
541C22×C3#57711yescharacteristic
542C22×C3#57711yescharacteristic
543C2×D3—42no—
544C2×D3—42no—
545C2×D3—42no—
546C2×D3—42no—
547C2×D3—42no—
548C2×D3—42no—
549C2×D3—81no—
550C2×D3—81no—
551C2×D3—81no—
552C2×D3—81no—
553C2×D3—81no—
554C2×D3—81no—
555C2×D3—81no—
556C2×D3—81no—
557C3×C4—41no—
558C3×C4—81no—
559C3×C4—81no—
560C3×C4—81no—
561C3×C4—81no—
562C3×C4#52511yescharacteristic
563C3×C4#52911yescharacteristic
564C3×C4#52911yescharacteristic
565C3×C4#53211yescharacteristic
566Dic3—22no—
567Dic3—22no—
568Dic3—42no—
569Dic3—42no—
570Dic3—41no—
571Dic3—41no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—62no—
580C23—62no—
581C23—121no—
582C23—121no—
583C23—121no—
584C23—121no—
585C23—121no—
586C23—121no—
587C23—121no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23C2×D3×D411yescharacteristic
594D4—62no—
595D4—62no—
596D4—62no—
597D4—62no—
598D4—121no—
599D4—121no—
600D4—121no—
601D4—121no—
602D4—244no—
603D4—244no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—32no—
608C2×C4—32no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—61no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—244no—
642C2×C4—244no—
643C2×C4C2×D3×D411yescharacteristic
644C2×C4D3×#4111yescharacteristic
645C2×C4D3×#4111yescharacteristic
646C2×C4D3×#4111yescharacteristic
647C2×C4D3×#4111yescharacteristic
648C2×C4D3×#4111yescharacteristic
649C8—22no—
650C8—41no—
651C8—62no—
652C8—121no—
653C2×C3—81no—
654C2×C3#239211yescharacteristic
655C2×C3#306211yescharacteristic
656C2×C3#306211yescharacteristic
657C2×C3#311811yescharacteristic
658C2×C3#311911yescharacteristic
659C2×C3#312211yescharacteristic
660C2×C3#312511yescharacteristic
661D3—82no—
662D3—82no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—122no—
671C22—122no—
672C22—122no—
673C22—122no—
674C22—122no—
675C22—122no—
676C22—241no—
677C22—241no—
678C22—241no—
679C22—241no—
680C22—241no—
681C22—241no—
682C22—241no—
683C22—241no—
684C22D3×#12111yescharacteristic
685C22#565611yescharacteristic
686C22#565611yescharacteristic
687C22#576211yescharacteristic
688C22#576911yescharacteristic
689C22#577411yescharacteristic
690C22#577811yescharacteristic
691C4—41no—
692C4—62no—
693C4—62no—
694C4—81no—
695C4—81no—
696C4—81no—
697C4—81no—
698C4—121no—
699C4—122no—
700C4—121no—
701C4—122no—
702C4#565611yescharacteristic
703C4#566811yescharacteristic
704C4#570611yescharacteristic
705C4#570611yescharacteristic
706C3#2371211yescharacteristic, complemented, Sylow
707C2—81no—
708C2—242no—
709C2—242no—
710C2#7745711yescharacteristic
711C2#8066111yescharacteristic
712C2#8080911yescharacteristic
713C2#8131211yescharacteristic
714C2#8135411yescharacteristic
715C2#8141711yescharacteristic
716C2#8149611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2077C2
socleC23×C3C22×D4
prime radical#69961C2
hypercenter#2077D3
hypocenterC3#23712
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2077

Complements

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

NH ≅ G/Ntype
1. C2×#4869C2semidirect
2. C2×#4720C2semidirect
3. #73872C2semidirect
4. #73872C2semidirect
5. C3×#2077C2semidirect
6. #70568C2semidirect
8. #69524C2semidirect
9. #69523C2semidirect
10. #68901C2semidirect
11. #68900C2semidirect
12. #68862C2semidirect
13. #68786C2semidirect
14. #68780C2semidirect
15. #68780C2semidirect
67. #2077D3semidirect
138. #815D4semidirect
706. C3#23712semidirect

Maximal subgroups

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

subgroupindex
C2×#48692
C2×#47202
#738722
C3×#20772
#705682
#699612
#695242
#695232
#689012
#689002
#688622
#687862
#687802
#237123