Finite Group Databasebeta

← #10801709

Subgroups — #10801709

Order 768.

1010 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2323C211yesmaximal, characteristic, complemented
2C2×#5048C211yesmaximal, characteristic
3C2×C4×#215C211yesmaximal, characteristic, complemented
4#82236C211yesmaximal, characteristic
5C2×#4997C211yesmaximal, characteristic, complemented
6#71697C211yesmaximal, characteristic, complemented
7#71631C211yesmaximal, characteristic, complemented
8#69676C211yesmaximal, characteristic, complemented
9#69675C211yesmaximal, characteristic, complemented
10#69618C211yesmaximal, characteristic, complemented
11#69617C211yesmaximal, characteristic, complemented
12#69517C211yesmaximal, characteristic
13#69511C211yesmaximal, characteristic, complemented
14C4×#808C211yesmaximal, characteristic, complemented
15#69474C211yesmaximal, characteristic, complemented
16#21523—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×#930C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×#389—21no—
27C3×#389C2211yescharacteristic
28C3×#388—21no—
29C3×#388—21no—
30C3×#387—21no—
31C3×#380C2211yescharacteristic
32C3×C4×C4⋊C4C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36#5212C2211yescharacteristic
37#5048C418yescomplemented
38C4×#215—24no—
39#4997—24no—
40#4990—21no—
41#4989—21no—
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#808C2211yescharacteristic
48#4957—21no—
49#4956—21no—
50#4939C2211yescharacteristic
51C4×C4⋊Dic3C2211yescharacteristic
52#4928C2211yescharacteristic
53C2×#803C2211yescharacteristic
54C2×#803C2211yescharacteristic
55C2×C4×C3⋊C8C2211yescharacteristic
56C2×C4×C3⋊C8C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×#465—31no—
64C2×C4×SD16—31no—
65C2×#405—31no—
66#2324—31no—
67#2323D311yescharacteristic, complemented
68#1968—31no—
69#1941—31no—
70#1936—31no—
71#1935—31no—
72#1895—31no—
73#1894—31no—
74#1807—31no—
75C4×#106—31no—
76#1763—31no—
77#1762—31no—
78C22×C3×Q8C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×#67—21no—
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8C2×C414yescomplemented
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C3×#117—22no—
88C3×#117—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4D411yescharacteristic
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⋊C4D411yescharacteristic
100C2×C3×C22⋊C4D411yescharacteristic
101C2×C3×C42—21no—
102C2×C3×C42—21no—
103C2×C3×C42C2311yescharacteristic
104C2×C3×C42C2311yescharacteristic
105C2×#217—21no—
106C2×#215—22no—
107C2×#215—22no—
108C2×#215C2×C414yes—
109#930—22no—
110#930C2×C414yes—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4—21no—
114C2×(C2×D3)⋊C4—21no—
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×Dic3⋊C4—21no—
119C2×C4×Dic3—21no—
120C2×#208—22no—
121C2×#208—22no—
122C2×#208—22no—
123C22×C3⋊C8—22no—
124C22×C3⋊C8—22no—
125C22×C3⋊C8C2×C414yescomplemented
126C22×C3⋊C8C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128C22×C3⋊C8C2311yescharacteristic
129C22×C3⋊C8C2311yescharacteristic
130C4×D12—22no—
131C4×D12—22no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136C3×#95—21no—
137#835—22no—
138#835—22no—
139#835C2×C414yescomplemented
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143#809—22no—
144#809—22no—
145#809C2×C414yes—
146#808—22no—
147#808—22no—
148#808C2×C414yescomplemented
149#803—22no—
150#803—22no—
151C4×C3⋊C8—22no—
152C4×C3⋊C8D414yescomplemented
153C22×SD16—31no—
154C22×M16—31no—
155C23×C8—31no—
156C2×C22⋊Q8C2×D311yescharacteristic
157C2×#121—31no—
158C2×C4×Q8C2×D311yescharacteristic
159C2×C4×D4—31no—
160C2×#117C2×D311yescharacteristic
161#465—38no—
162C4×SD16—64no—
163C2×#106—31no—
164C2×#106—31no—
165C2×#106—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#102—31no—
170C2×#102—31no—
171C2×#102—31no—
172#405—64no—
173C2×#97—31no—
174C2×#97—31no—
175C2×C4×C8—31no—
176C2×C4×C8—31no—
177#391C2×D311yescharacteristic
178#390—31no—
179#390—61no—
180#389—21no—
181#389—31no—
182#389C2×D311yescharacteristic
183#388—21no—
184#388—21no—
185#388—61no—
186#387—21no—
187#386—61no—
188#385—61no—
189#380—31no—
190#380C2×D311yescharacteristic
191C4×C4⋊C4—31no—
192C4×C4⋊C4C2×D311yescharacteristic
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340—31no—
199C24×C3C2×D411yescharacteristic
200C23×D3—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8C22×C412yes—
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×C4C22×C412yes—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4#4112yes—
222C22×C3×C4C2411yescharacteristic
223C2×#67—24no—
224C2×#67—81no—
225C2×#67—81no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12C22×C412yes—
235C2×C4×D3—42no—
236C2×C4×D3—81no—
237C2×C4×D3—81no—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—42no—
247C3×C4⋊C4—81no—
248C3×C4⋊C4C22×C412yes—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—22no—
251C3×C22⋊C4—22no—
252C3×C22⋊C4—81no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—42no—
256C3×C42—81no—
257C3×C42C2×D412yes—
258C3×C42#4112yes—
259#217—81no—
260#215—28no—
261(C2×D3)⋊C4—24no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—81no—
264(C2×D3)⋊C4—81no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3C22×C412yes—
271Dic3⋊C4—81no—
272C4×Dic3—81no—
273#208—28no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—28no—
281C2×C3⋊C8C22×C412yes—
282C2×C3⋊C8C22×C412yes—
283C2×C3⋊C8C2×D412yes—
284C2×C3⋊C8C2×D412yes—
285C2×C3⋊C8#4112yes—
286C2×C3⋊C8#4112yes—
287C22×Q8C22×D311yescharacteristic
288C22×D4—31no—
289C22×D4—61no—
290C23×C4—61no—
291C23×C4C22×D311yescharacteristic
292C2×SD16—34no—
293C2×SD16—62no—
294C2×SD16—62no—
295C2×M16—62no—
296C2×M16—62no—
297C2×M16—62no—
298C22×C8—31no—
299C22×C8—34no—
300C22×C8—31no—
301C22×C8—31no—
302C22×C8—31no—
303C22×C8—62no—
304C22×C8—62no—
305C22⋊Q8—22no—
306C22⋊Q8C4×D314yescomplemented
307#121—34no—
308#121—62no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×D4—62no—
312C4×D4—62no—
313#117—22no—
314#117—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4#6711yescharacteristic
323C2×C4⋊C4#6711yescharacteristic
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4#6711yescharacteristic
335C2×C22⋊C4#6711yescharacteristic
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—61no—
339C2×C42C22×D311yescharacteristic
340C2×C42C22×D311yescharacteristic
341#106—34no—
342#106—62no—
343#106—62no—
344#103—34no—
345#103—62no—
346#103—62no—
347#102—34no—
348#102—62no—
349#102—62no—
350#97—62no—
351#97—62no—
352C4×C8—34no—
353C4×C8—62no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—61no—
360#95—61no—
361#95—61no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3C4×D412yes—
369C23×C3C22×D411yescharacteristic
370C22×D3—22no—
371C22×D3—41no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C3×Q8—22no—
378C3×Q8—22no—
379C3×Q8—41no—
380C3×Q8—41no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4C4×D412yes—
410C2×C3×C4#12112yes—
411C2×C3×C4#12112yes—
412C2×C3×C4C23×C411yescharacteristic
413C2×C3×C4C22×D411yescharacteristic
414C2×C3×C4C2×#4111yescharacteristic
415C2×C3×C4C2×#4111yescharacteristic
416C2×C3×C4C2×#4111yescharacteristic
417C2×C3×C4C2×#4111yescharacteristic
418#67—84no—
419C2×Dic3—22no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428D12—22no—
429D12—22no—
430D12—41no—
431D12—41no—
432C4×D3—84no—
433C3⋊C8—22no—
434C3⋊C8—22no—
435C3⋊C8—22no—
436C3⋊C8C4×D414yes—
437C24—61no—
438C24C2×#6711yescharacteristic
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C2×C4×D312yes—
445C2×D4—32no—
446C2×D4—64no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—241no—
453C2×D4—241no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—122no—
466C22×C4—241no—
467C22×C4—241no—
468C22×C4C2×C4×D312yes—
469C22×C4#23512yes—
470C22×C4C2×#6711yescharacteristic
471C22×C4C2×#6711yescharacteristic
472C22×C4C2×#6711yescharacteristic
473C22×C4C2×#6711yescharacteristic
474C22×C4C2×#6711yescharacteristic
475C22×C4C23×D311yescharacteristic
476SD16—68no—
477M16—68no—
478C2×C8—32no—
479C2×C8—32no—
480C2×C8—32no—
481C2×C8—32no—
482C2×C8—32no—
483C2×C8—32no—
484C2×C8—68no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—32no—
500C4⋊C4—42no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—81no—
507C4⋊C4—241no—
508C4⋊C4C2×C4×D312yes—
509C22⋊C4—24no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—64no—
513C22⋊C4—81no—
514C22⋊C4—122no—
515C22⋊C4—241no—
516C22⋊C4—241no—
517C22⋊C4—241no—
518C42—21no—
519C42—21no—
520C42—42no—
521C42—81no—
522C42—241no—
523C42D3×D412yes—
524C42#23712yes—
525C22×C3—22no—
526C22×C3—22no—
527C22×C3—22no—
528C22×C3—41no—
529C22×C3—41no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3C2×C4×D411yescharacteristic
533C22×C3C2×#12111yescharacteristic
534C22×C3#53211yescharacteristic
535C22×C3#57711yescharacteristic
536C22×C3#57711yescharacteristic
537C22×C3#57711yescharacteristic
538C22×C3#57711yescharacteristic
539C2×D3—42no—
540C2×D3—42no—
541C2×D3—42no—
542C2×D3—81no—
543C2×D3—81no—
544C2×D3—81no—
545C2×D3—81no—
546C3×C4—22no—
547C3×C4—22no—
548C3×C4—22no—
549C3×C4—42no—
550C3×C4—41no—
551C3×C4—41no—
552C3×C4—42no—
553C3×C4—81no—
554C3×C4—81no—
555C3×C4C2×C4×D411yescharacteristic
556C3×C4C4×#4111yescharacteristic
557C3×C4#52511yescharacteristic
558C3×C4#52911yescharacteristic
559Dic3—42no—
560Dic3—81no—
561Dic3—81no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—62no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23—121no—
575C23C4×#6712yes—
576C23C22×#6711yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—41no—
580Q8—41no—
581D4—62no—
582D4—62no—
583D4—121no—
584D4—121no—
585D4—244no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—62no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—244no—
624C2×C4C4×#6712yes—
625C2×C4#93912yes—
626C2×C4#93912yes—
627C2×C4C22×C4×D311yescharacteristic
628C2×C4C2×#23511yescharacteristic
629C2×C4C2×D3×D411yescharacteristic
630C2×C4C2×#23711yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4D3×#4111yescharacteristic
633C8—34no—
634C8—62no—
635C8—62no—
636C8—62no—
637C2×C3—42no—
638C2×C3C4×#12111yescharacteristic
639C2×C3#294911yescharacteristic
640C2×C3#295211yescharacteristic
641C2×C3#305711yescharacteristic
642C2×C3#306011yescharacteristic
643C2×C3#317311yescharacteristic
644C2×C3#317411yescharacteristic
645D3—82no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22—241no—
657C22—241no—
658C22—241no—
659C22—241no—
660C22C2×C4×#6711yescharacteristic
661C22C2×#93911yescharacteristic
662C22#583211yescharacteristic
663C22#583611yescharacteristic
664C22#583711yescharacteristic
665C22#583711yescharacteristic
666C22#583711yescharacteristic
667C4—22no—
668C4—22no—
669C4—22no—
670C4—42no—
671C4—42no—
672C4—41no—
673C4—41no—
674C4—81no—
675C4—81no—
676C4—122no—
677C4—241no—
678C4—241no—
679C4C4×#23711yescharacteristic
680C4#553811yescharacteristic
681C4C4×D3×D411yescharacteristic
682C4#554711yescharacteristic
683C3#2152311yescharacteristic, complemented, Sylow
684C2—42no—
685C2—242no—
686C2C4×#93911yescharacteristic
687C2#8254811yescharacteristic
688C2#8255811yescharacteristic
689C2#8339811yescharacteristic
690C2#8342011yescharacteristic
691C2#8344311yescharacteristic
692C2#8345611yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2323C2
socleC23×C3C22×D4
prime radicalC2×#930C22
cosocleC22×C3×C4C24
hypercenter#2323D3
hypocenterC3#21523
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
1. C3×#2323C2semidirect
3. C2×C4×#215C2semidirect
5. C2×#4997C2semidirect
6. #71697C2semidirect
7. #71631C2semidirect
8. #69676C2semidirect
9. #69675C2semidirect
10. #69618C2semidirect
11. #69617C2semidirect
13. #69511C2semidirect
14. C4×#808C2semidirect
15. #69474C2semidirect
37. #5048C4semidirect
67. #2323D3semidirect
84. C3×C22⋊Q8C2×C4semidirect
125. C22×C3⋊C8C2×C4semidirect
139. #835C2×C4semidirect
148. #808C2×C4semidirect
152. C4×C3⋊C8D4semidirect
306. C22⋊Q8C4×D3semidirect
683. C3#21523semidirect

Maximal subgroups

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

subgroupindex
C3×#23232
C2×#50482
C2×C4×#2152
#822362
C2×#49972
#716972
#716312
#696762
#696752
#696182
#696172
#695172
#695112
C4×#8082
#694742
#215233