Finite Group Databasebeta

← #10824682

Subgroups — #10824682

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
1C3×#2324C211yesmaximal, characteristic, complemented
2C2×C4×#213C211yesmaximal, characteristic, complemented
3#82236C211yesmaximal, characteristic, complemented
4C2×#5000C211yesmaximal, characteristic, complemented
5#71684C211yesmaximal, characteristic
6#71679C211yesmaximal, characteristic, complemented
7#71609C211yesmaximal, characteristic, complemented
8#69653C211yesmaximal, characteristic, complemented
9#69650C211yesmaximal, characteristic, complemented
10#69630C211yesmaximal, characteristic, complemented
11#69623C211yesmaximal, characteristic, complemented
12#69523C211yesmaximal, characteristic, complemented
13#69519C211yesmaximal, characteristic, complemented
14C4×#807C211yesmaximal, characteristic, complemented
15#69474C211yesmaximal, characteristic, complemented
16#23703—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C22×#213C2211yescharacteristic
21C2×#930C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C3×#390—21no—
24C3×#390C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×#388—21no—
27C3×#386—21no—
28C3×#385—21no—
29C3×#380C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34#5212C2211yescharacteristic
35C2×#830C2211yescharacteristic
36C2×#830C2211yescharacteristic
37C4×#213—24no—
38#5000—24no—
39#4990—21no—
40#4989—21no—
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#807C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4957—21no—
48#4956—21no—
49#4939C2211yescharacteristic
50C4×C4⋊Dic3C2211yescharacteristic
51#4928C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×#803C2211yescharacteristic
55C2×C4×C3⋊C8C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×C4×D8—31no—
63C2×#433—31no—
64#2324—31no—
65#2324D311yescharacteristic, complemented
66#2087—31no—
67#2076—31no—
68#2067—31no—
69#2067—31no—
70#2054—31no—
71#2033—31no—
72#2033—31no—
73#1999—31no—
74C4×#107—31no—
75#1762—31no—
76#1762—31no—
77C22×C3×D4—21no—
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C23×C3×C4C2311yescharacteristic
81C22×#67—21no—
82C22×D12C2311yescharacteristic
83C22×C4×D3—21no—
84C3×#121—22no—
85C3×#121—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×#117—22no—
89C3×#117—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
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×C3×C42C2311yescharacteristic
106C2×#217—21no—
107C2×#213—22no—
108C2×#213—22no—
109C2×#213—22no—
110#930—22no—
111#930—22no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4—21no—
114C2×(C2×D3)⋊C4—21no—
115C2×(C2×D3)⋊C4—21no—
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×Dic3⋊C4—21no—
120C2×C4×Dic3—21no—
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C4×D12—22no—
126C4×D12—22no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130#832—22no—
131#832—22no—
132#832—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#830—22no—
137#830—22no—
138#809—22no—
139#809—22no—
140#809—22no—
141#807—22no—
142#807—22no—
143#807—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—22no—
146#803—22no—
147C4×C3⋊C8D414yescomplemented
148C22×D8—31no—
149C2×#121—31no—
150C2×#121C2×D311yescharacteristic
151C2×C4×D4—31no—
152C2×C4×D4C2×D311yescharacteristic
153C2×#117C2×D311yescharacteristic
154C4×D8—64no—
155#433—64no—
156C2×#107—31no—
157C2×#107—31no—
158C2×#107—31no—
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⋊C8—31no—
169C2×#97—31no—
170C2×C4×C8—31no—
171#390—21no—
172#390—31no—
173#390—61no—
174#390C2×D311yescharacteristic
175#389—31no—
176#389C2×D311yescharacteristic
177#388—21no—
178#388—61no—
179#386—21no—
180#386—61no—
181#385—21no—
182#385—61no—
183#380—31no—
184#380C2×D311yescharacteristic
185C4×C4⋊C4—31no—
186C4×C4⋊C4C2×D311yescharacteristic
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193C24×C3—21no—
194C24×C3C2×D411yescharacteristic
195C23×D3—21no—
196C2×C3×D4—24no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—81no—
203C2×C3×D4—81no—
204C2×C3×D4#4112yes—
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—42no—
215C22×C3×C4—81no—
216C22×C3×C4—81no—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
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×D12#4112yes—
235C2×C4×D3—42no—
236C2×C4×D3—81no—
237C2×C4×D3—81no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—81no—
246C3×C4⋊C4#4112yes—
247C3×C22⋊C4—24no—
248C3×C22⋊C4—22no—
249C3×C22⋊C4—22no—
250C3×C22⋊C4—42no—
251C3×C22⋊C4—81no—
252C3×C22⋊C4—81no—
253C3×C22⋊C4—81no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42—81no—
257C3×C42C2×D412yes—
258C3×C42#4112yes—
259#217—81no—
260#213—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⋊Dic3#4112yes—
271Dic3⋊C4—81no—
272C4×Dic3—81no—
273C2×C3⋊C8—21no—
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—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8C2×D412yes—
284C2×C3⋊C8C2×D412yes—
285C22×D4—21no—
286C22×D4—31no—
287C22×D4—61no—
288C22×D4C22×D311yescharacteristic
289C23×C4—21no—
290C23×C4—61no—
291C23×C4C22×D311yescharacteristic
292C2×D8—62no—
293C2×D8—62no—
294C2×D8—62no—
295C22×C8—31no—
296C22×C8—31no—
297C22×C8—31no—
298C22×C8—31no—
299#121—22no—
300#121—22no—
301#121—62no—
302#121—62no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—62no—
306C4×D4—62no—
307#117—22no—
308#117—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4#6711yescharacteristic
315C2×C4⋊C4#6711yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4#6711yescharacteristic
330C2×C22⋊C4#6711yescharacteristic
331C2×C42—21no—
332C2×C42—61no—
333C2×C42C22×D311yescharacteristic
334C2×C42C22×D311yescharacteristic
335#107—62no—
336#107—62no—
337#107—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—62no—
347C22⋊C8—62no—
348#97—62no—
349C4×C8—34no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3—21no—
357C23×C3—22no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3C22×D411yescharacteristic
371C22×D3—22no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C3×D4—22no—
379C3×D4—22no—
380C3×D4—41no—
381C3×D4—41no—
382C3×D4—84no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
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×C4—84no—
410C2×C3×C4#12112yes—
411C2×C3×C4#12112yes—
412C2×C3×C4C22×D411yescharacteristic
413C2×C3×C4C2×#4111yescharacteristic
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—41no—
436C3⋊C8—41no—
437C24—21no—
438C24—61no—
439C24C2×#6711yescharacteristic
440C2×D4—24no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—32no—
447C2×D4—64no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—81no—
454C2×D4—81no—
455C2×D4—241no—
456C2×D4—241no—
457C2×D4#23512yes—
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—42no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—122no—
474C22×C4—241no—
475C22×C4—241no—
476C22×C4C2×#6711yescharacteristic
477C22×C4C2×#6711yescharacteristic
478C22×C4C2×#6711yescharacteristic
479C22×C4C2×#6711yescharacteristic
480C22×C4C2×#6711yescharacteristic
481C22×C4C23×D311yescharacteristic
482D8—68no—
483C2×C8—32no—
484C2×C8—32no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—32no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—81no—
509C4⋊C4—241no—
510C4⋊C4#23512yes—
511C22⋊C4—24no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—42no—
515C22⋊C4—64no—
516C22⋊C4—81no—
517C22⋊C4—81no—
518C22⋊C4—81no—
519C22⋊C4—122no—
520C22⋊C4—241no—
521C22⋊C4—241no—
522C22⋊C4—241no—
523C42—21no—
524C42—21no—
525C42—81no—
526C42—241no—
527C42D3×D412yes—
528C42#23712yes—
529C22×C3—41no—
530C22×C3—41no—
531C22×C3—42no—
532C22×C3—41no—
533C22×C3—41no—
534C22×C3—42no—
535C22×C3—41no—
536C22×C3—42no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3—81no—
540C22×C3—81no—
541C22×C3—81no—
542C22×C3—81no—
543C22×C3C2×#12111yescharacteristic
544C22×C3#52511yescharacteristic
545C22×C3#52511yescharacteristic
546C22×C3#57611yescharacteristic
547C22×C3#57611yescharacteristic
548C22×C3#57611yescharacteristic
549C22×C3#57811yescharacteristic
550C2×D3—42no—
551C2×D3—42no—
552C2×D3—42no—
553C2×D3—81no—
554C2×D3—81no—
555C2×D3—81no—
556C2×D3—81no—
557C3×C4—22no—
558C3×C4—22no—
559C3×C4—42no—
560C3×C4—41no—
561C3×C4—41no—
562C3×C4—41no—
563C3×C4—81no—
564C3×C4—81no—
565C3×C4#52511yescharacteristic
566C3×C4#52911yescharacteristic
567C3×C4#52911yescharacteristic
568C3×C4#53211yescharacteristic
569Dic3—42no—
570Dic3—81no—
571Dic3—81no—
572C23—22no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—62no—
587C23—121no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23C22×#6711yescharacteristic
594D4—22no—
595D4—22no—
596D4—41no—
597D4—41no—
598D4—62no—
599D4—62no—
600D4—84no—
601D4—121no—
602D4—121no—
603D4—244no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—62no—
631C2×C4—84no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—244no—
641C2×C4#93912yes—
642C2×C4#93912yes—
643C2×C4C2×#23511yescharacteristic
644C2×C4C2×#23511yescharacteristic
645C2×C4C2×D3×D411yescharacteristic
646C2×C4C2×#23711yescharacteristic
647C2×C4D3×#4111yescharacteristic
648C2×C4D3×#4111yescharacteristic
649C8—62no—
650C8—62no—
651C8—121no—
652C8—121no—
653C2×C3—81no—
654C2×C3—82no—
655C2×C3#239211yescharacteristic
656C2×C3#305811yescharacteristic
657C2×C3#306111yescharacteristic
658C2×C3#312211yescharacteristic
659C2×C3#312211yescharacteristic
660C2×C3#312411yescharacteristic
661C2×C3#312411yescharacteristic
662D3—82no—
663C22—41no—
664C22—41no—
665C22—42no—
666C22—41no—
667C22—42no—
668C22—41no—
669C22—42no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—122no—
678C22—122no—
679C22—122no—
680C22—241no—
681C22—241no—
682C22—241no—
683C22—241no—
684C22#579111yescharacteristic
685C22C2×#93911yescharacteristic
686C22#582811yescharacteristic
687C22#583511yescharacteristic
688C22#583511yescharacteristic
689C22#583511yescharacteristic
690C22#583811yescharacteristic
691C4—22no—
692C4—22no—
693C4—42no—
694C4—41no—
695C4—41no—
696C4—41no—
697C4—81no—
698C4—81no—
699C4—122no—
700C4—241no—
701C4—241no—
702C4#553811yescharacteristic
703C4#554711yescharacteristic
704C4#554811yescharacteristic
705C4#555711yescharacteristic
706C3#2370311yescharacteristic, complemented, Sylow
707C2—81no—
708C2—82no—
709C2—242no—
710C2#8234511yescharacteristic
711C2#8254211yescharacteristic
712C2#8256911yescharacteristic
713C2#8337811yescharacteristic
714C2#8342111yescharacteristic
715C2#8343511yescharacteristic
716C2#8346111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2324C2
socleC23×C3C22×D4
prime radical#71684C2
hypercenter#2324D3
hypocenterC3#23703
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2324

Complements

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

NH ≅ G/Ntype
1. C3×#2324C2semidirect
2. C2×C4×#213C2semidirect
3. #82236C2semidirect
4. C2×#5000C2semidirect
6. #71679C2semidirect
7. #71609C2semidirect
8. #69653C2semidirect
9. #69650C2semidirect
10. #69630C2semidirect
11. #69623C2semidirect
12. #69523C2semidirect
13. #69519C2semidirect
14. C4×#807C2semidirect
15. #69474C2semidirect
65. #2324D3semidirect
147. C4×C3⋊C8D4semidirect
706. C3#23703semidirect

Maximal subgroups

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

subgroupindex
C3×#23242
C2×C4×#2132
#822362
C2×#50002
#716842
#716792
#716092
#696532
#696502
#696302
#696232
#695232
#695192
C4×#8072
#694742
#237033