Finite Group Databasebeta

← #10828534

Subgroups — #10828534

Order 768.

1104 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×#5479C211yesmaximal, characteristic
2#82069C211yesmaximal, characteristic, complemented
3C2×#4723C211yesmaximal, characteristic, complemented
4C2×#4737C211yesmaximal, characteristic
5C3×#1817C211yesmaximal, characteristic, complemented
6#69933C211yesmaximal, characteristic
7#69931C211yesmaximal, characteristic, complemented
8#68684C211yesmaximal, characteristic, complemented
9#68683C211yesmaximal, characteristic, complemented
10#68671C211yesmaximal, characteristic, complemented
11#68670C211yesmaximal, characteristic, complemented
12#68634C211yesmaximal, characteristic, complemented
13#68634C211yesmaximal, characteristic, complemented
14#68630C211yesmaximal, characteristic, complemented
15#68630C211yesmaximal, characteristic, complemented
16#24068—31nomaximal, Sylow
17C22×C3×C42C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C2×#929—21no—
20C2×C4×#67—21no—
21C2×#926C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C2×#924C2211yescharacteristic
24#5479—24no—
25#5479—24no—
26C2×#876—21no—
27C2×#875C2211yescharacteristic
28C2×C4×D12C2211yescharacteristic
29C2×#872—21no—
30C2×#870C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×C22⋊C8C2211yescharacteristic
33C2×C3×C22⋊C8C2211yescharacteristic
34#5210—21no—
35#5210—21no—
36C2×#821C2211yescharacteristic
37C2×#821C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#821C2211yescharacteristic
40C2×#818C2211yescharacteristic
41C2×#818C2211yescharacteristic
42C2×#817C2211yescharacteristic
43C2×#817C2211yescharacteristic
44C2×#816C2211yescharacteristic
45C2×#816C2211yescharacteristic
46C2×#816C2211yescharacteristic
47C2×#816C2211yescharacteristic
48C4×Dic3⋊C4—21no—
49C4×Dic3⋊C4—21no—
50#4928C2211yescharacteristic
51#4928C2211yescharacteristic
52#4928C2211yescharacteristic
53#4928C2211yescharacteristic
54#4737—24no—
55#4723—24no—
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62#4551C2211yescharacteristic
63#4551C2211yescharacteristic
64C2×#529—31no—
65C2×#485—31no—
66C2×#482—31no—
67#2296—31no—
68#2059—31no—
69#2059—31no—
70#2042—31no—
71#2042—31no—
72#1875—31no—
73#1873—31no—
74#1865—31no—
75#1864—31no—
76#1863—31no—
77#1862—31no—
78#1817D311yescharacteristic, complemented
79C23×C3×C4D411yescharacteristic
80C23×C3×C4D411yescharacteristic
81C23×C3×C4C2311yescharacteristic
82C22×#67—21no—
83C22×D12C2311yescharacteristic
84C22×C4×D3—21no—
85C22×Dic6C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C2×C3×C42—22no—
91C2×C3×C42—22no—
92C2×C3×C42D411yescharacteristic
93C2×C3×C42D411yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99#930—22no—
100#930D414yes—
101#929—28no—
102C4×#67—28no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107#926—22no—
108#926—22no—
109#926—22no—
110#926—22no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117#924—22no—
118#924D414yes—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×Dic3⋊C4—21no—
123C2×Dic3⋊C4—21no—
124C2×Dic3⋊C4—21no—
125C2×Dic3⋊C4—21no—
126C2×Dic3⋊C4—21no—
127C2×C4×Dic3—21no—
128C2×C4×Dic3—21no—
129C2×C4×Dic3—21no—
130#876—28no—
131#875—22no—
132#875—22no—
133C4×D12—22no—
134C4×D12—22no—
135#872—28no—
136#870—22no—
137#870—22no—
138C4×Dic6—22no—
139C4×Dic6—22no—
140C3×C22⋊C8—22no—
141C3×C22⋊C8—22no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146#821—22no—
147#821—22no—
148#821—22no—
149#821—22no—
150#818—22no—
151#818—22no—
152#817—22no—
153#817—22no—
154#816—22no—
155#816—22no—
156#816—22no—
157#816—22no—
158#529—64no—
159#529—64no—
160C2×#126—61no—
161C2×#125—31no—
162C2×#124—31no—
163C2×#123—61no—
164C2×C22⋊Q8—31no—
165C2×#121—31no—
166C2×C4×Q8—31no—
167C2×C4×D4—31no—
168C2×C4×D4—61no—
169C2×#117—31no—
170C2×#117—31no—
171C2×#117—61no—
172C22×C42C2×D311yescharacteristic
173#485—64no—
174#482—64no—
175C2×#107—31no—
176C2×#107—31no—
177C2×#106—31no—
178C2×#106—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#103—31no—
182C2×#103—31no—
183C2×#102—31no—
184C2×#102—31no—
185C2×#102—31no—
186C2×#102—31no—
187C2×C22⋊C8C2×D311yescharacteristic
188C2×C22⋊C8C2×D311yescharacteristic
189#389—31no—
190#389—31no—
191#389—31no—
192#389—31no—
193#385—61no—
194#385—61no—
195C4×C4⋊C4—61no—
196C4×C4⋊C4—61no—
197#340—31no—
198#340—31no—
199#340—31no—
200#340—31no—
201#336C2×D311yescharacteristic
202#336C2×D311yescharacteristic
203#336C2×D311yescharacteristic
204#336C2×D311yescharacteristic
205C24×C3C2×D411yescharacteristic
206C23×D3—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—22no—
214C22×C3×C4—41no—
215C22×C3×C4—41no—
216C22×C3×C4—41no—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D412yes—
223C22×C3×C4C2×D412yes—
224C22×C3×C4C2411yescharacteristic
225C2×#67—24no—
226C2×#67—42no—
227C2×#67—42no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3—21no—
232C22×Dic3—21no—
233C22×Dic3—21no—
234C22×Dic3—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12—21no—
240C2×D12C2×D412yes—
241C2×C4×D3—24no—
242C2×C4×D3—42no—
243C2×C4×D3—42no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6C2×D412yes—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C3×C42—22no—
263C3×C42—22no—
264C3×C42—21no—
265C3×C42—21no—
266C3×C42—21no—
267C3×C42—21no—
268#217—24no—
269#217—24no—
270#217—42no—
271(C2×D3)⋊C4—24no—
272(C2×D3)⋊C4—24no—
273(C2×D3)⋊C4—24no—
274(C2×D3)⋊C4—24no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282C4⋊Dic3—21no—
283C4⋊Dic3C2×D412yes—
284C4⋊Dic3C2×D412yes—
285C4⋊Dic3#4112yes—
286C4⋊Dic3#4112yes—
287Dic3⋊C4—24no—
288Dic3⋊C4—24no—
289Dic3⋊C4—24no—
290Dic3⋊C4—24no—
291Dic3⋊C4—42no—
292Dic3⋊C4—42no—
293Dic3⋊C4—42no—
294Dic3⋊C4—42no—
295C4×Dic3—24no—
296C4×Dic3—24no—
297C4×Dic3—42no—
298C22×Q8—31no—
299C22×D4—31no—
300C22×D4—61no—
301C23×C4—61no—
302C23×C4D1211yescharacteristic
303C23×C4D1211yescharacteristic
304C23×C4C22×D311yescharacteristic
305C22×C8C22×D311yescharacteristic
306C22×C8C22×D311yescharacteristic
307C22×C8C22×D311yescharacteristic
308C22×C8C22×D311yescharacteristic
309#126—68no—
310#125—62no—
311#125—62no—
312#124—62no—
313#124—62no—
314#123—68no—
315C22⋊Q8—34no—
316C22⋊Q8—62no—
317#121—34no—
318#121—62no—
319C4×Q8—62no—
320C4×Q8—62no—
321C4×D4—62no—
322C4×D4—62no—
323C4×D4—68no—
324#117—68no—
325#117—62no—
326#117—62no—
327#117—62no—
328#117—62no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C22⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C22⋊C4—61no—
346C2×C22⋊C4—61no—
347C2×C22⋊C4—61no—
348C2×C22⋊C4—61no—
349C2×C22⋊C4—61no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—61no—
353C2×C42—61no—
354C2×C42—61no—
355C2×C42D1211yescharacteristic
356C2×C42D1211yescharacteristic
357C2×C42C22×D311yescharacteristic
358C2×C42C22×D311yescharacteristic
359#107—62no—
360#107—62no—
361#106—62no—
362#106—62no—
363#103—62no—
364#103—62no—
365#103—62no—
366#103—62no—
367#102—62no—
368#102—62no—
369#102—62no—
370#102—62no—
371C22⋊C8—22no—
372C22⋊C8—22no—
373#95—61no—
374#95—61no—
375#95—61no—
376#95—61no—
377C23×C3—21no—
378C23×C3—21no—
379C23×C3—21no—
380C23×C3—21no—
381C23×C3—21no—
382C23×C3—21no—
383C23×C3#12012yes—
384C23×C3C22×D411yescharacteristic
385C22×D3—22no—
386C22×D3—41no—
387C22×D3—41no—
388C22×D3—41no—
389C22×D3—41no—
390C22×D3—41no—
391C22×D3—41no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—22no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—44no—
410C2×C3×C4—44no—
411C2×C3×C4#12012yes—
412C2×C3×C4#13512yes—
413C2×C3×C4C22×D411yescharacteristic
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4C2×#4111yescharacteristic
416C2×C3×C4C2×#4111yescharacteristic
417C2×C3×C42−1+411yescharacteristic
418C2×C3×C42−1+411yescharacteristic
419#67—48no—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439C2×Dic3—41no—
440C2×Dic3—41no—
441D12—22no—
442D12—22no—
443D12—41no—
444D12—41no—
445C4×D3—48no—
446Dic6—22no—
447Dic6—22no—
448Dic6—41no—
449Dic6—41no—
450C3×C8—41no—
451C3×C8—41no—
452C3×C8—41no—
453C3×C8—41no—
454C24—61no—
455C24C2×D1211yescharacteristic
456C2×Q8—32no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×Q8—61no—
460C2×Q8—61no—
461C2×Q8—61no—
462C2×D4—32no—
463C2×D4—64no—
464C2×D4—61no—
465C2×D4—61no—
466C2×D4—61no—
467C2×D4—61no—
468C2×D4—61no—
469C2×D4—122no—
470C2×D4—122no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—64no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—61no—
486C22×C4—61no—
487C22×C4—61no—
488C22×C4—61no—
489C22×C4—122no—
490C22×C4—122no—
491C22×C4C2×D1211yescharacteristic
492C22×C4C2×D1211yescharacteristic
493C22×C4C2×D1211yescharacteristic
494C22×C4C2×D1211yescharacteristic
495C22×C4C2×D1211yescharacteristic
496C22×C4D3×D412yes—
497C22×C4D3×D412yes—
498C22×C4C23×D311yescharacteristic
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C4⋊C4—32no—
512C4⋊C4—32no—
513C4⋊C4—32no—
514C4⋊C4—32no—
515C4⋊C4—64no—
516C4⋊C4—64no—
517C4⋊C4—64no—
518C4⋊C4—64no—
519C4⋊C4—61no—
520C4⋊C4—61no—
521C4⋊C4—61no—
522C4⋊C4—61no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—122no—
528C4⋊C4—122no—
529C4⋊C4—122no—
530C4⋊C4—122no—
531C22⋊C4—64no—
532C22⋊C4—64no—
533C22⋊C4—64no—
534C22⋊C4—64no—
535C22⋊C4—64no—
536C22⋊C4—64no—
537C22⋊C4—122no—
538C42—22no—
539C42—22no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—64no—
545C42—64no—
546C42—122no—
547C22×C3—22no—
548C22×C3—22no—
549C22×C3—22no—
550C22×C3—41no—
551C22×C3—41no—
552C22×C3—41no—
553C22×C3—41no—
554C22×C3C2×#12011yescharacteristic
555C22×C3#52511yescharacteristic
556C22×C3#53611yescharacteristic
557C22×C3C2×#13511yescharacteristic
558C22×C3#57511yescharacteristic
559C22×C3#57511yescharacteristic
560C22×C3#57511yescharacteristic
561C2×D3—42no—
562C2×D3—42no—
563C2×D3—42no—
564C2×D3—81no—
565C2×D3—81no—
566C2×D3—81no—
567C2×D3—81no—
568C3×C4—22no—
569C3×C4—22no—
570C3×C4—41no—
571C3×C4—41no—
572C3×C4—41no—
573C3×C4#54711yescharacteristic
574C3×C4#54711yescharacteristic
575C3×C4#54711yescharacteristic
576C3×C4#54711yescharacteristic
577Dic3—42no—
578Dic3—42no—
579Dic3—42no—
580Dic3—42no—
581Dic3—42no—
582Dic3—81no—
583Dic3—81no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—62no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23—121no—
595C23—121no—
596C23—121no—
597C23#88212yes—
598C23C22×D1211yescharacteristic
599Q8—62no—
600Q8—62no—
601Q8—121no—
602Q8—121no—
603D4—62no—
604D4—62no—
605D4—128no—
606D4—121no—
607D4—121no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—62no—
628C2×C4—62no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—62no—
632C2×C4—128no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4#88212yes—
650C2×C4#90412yes—
651C2×C4C2×D3×D411yescharacteristic
652C2×C4C2×D3×D411yescharacteristic
653C2×C4D3×#4111yescharacteristic
654C2×C4D3×#4111yescharacteristic
655C2×C4#101011yescharacteristic
656C2×C4#101011yescharacteristic
657C8—41no—
658C8—41no—
659C8—41no—
660C8—41no—
661C2×C3—42no—
662C2×C3#240611yescharacteristic
663C2×C3#300211yescharacteristic
664C2×C3#300511yescharacteristic
665C2×C3#310211yescharacteristic
666C2×C3#311111yescharacteristic
667C2×C3#323311yescharacteristic
668C2×C3#324011yescharacteristic
669D3—82no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—122no—
678C22—122no—
679C22—122no—
680C22—241no—
681C22—241no—
682C22—241no—
683C22—241no—
684C22C2×#88211yescharacteristic
685C22#552411yescharacteristic
686C22#552611yescharacteristic
687C22C2×#90411yescharacteristic
688C22#574811yescharacteristic
689C22#574811yescharacteristic
690C22#574811yescharacteristic
691C4—22no—
692C4—22no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—122no—
697C4—122no—
698C4—122no—
699C4—122no—
700C4—122no—
701C4—241no—
702C4—241no—
703C4#565511yescharacteristic
704C4#565511yescharacteristic
705C4#565511yescharacteristic
706C4#565511yescharacteristic
707C3#2406811yescharacteristic, complemented, Sylow
708C2—42no—
709C2—242no—
710C2#7406311yescharacteristic
711C2#7861011yescharacteristic
712C2#7863811yescharacteristic
713C2#7927511yescharacteristic
714C2#7929911yescharacteristic
715C2#7932411yescharacteristic
716C2#7934511yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1817C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1817D3
hypocenterC3#24068
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. #82069C2semidirect
3. C2×#4723C2semidirect
5. C3×#1817C2semidirect
7. #69931C2semidirect
8. #68684C2semidirect
9. #68683C2semidirect
10. #68671C2semidirect
11. #68670C2semidirect
12. #68634C2semidirect
13. #68634C2semidirect
14. #68630C2semidirect
15. #68630C2semidirect
78. #1817D3semidirect
707. C3#24068semidirect

Maximal subgroups

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

subgroupindex
C2×#54792
#820692
C2×#47232
C2×#47372
C3×#18172
#699332
#699312
#686842
#686832
#686712
#686702
#686342
#686302
#240683