Finite Group Databasebeta

← #10828537

Subgroups — #10828537

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×#5580C211yesmaximal, characteristic
2C2×#4866C211yesmaximal, characteristic
3C2×#4815C211yesmaximal, characteristic, complemented
4#75963C211yesmaximal, characteristic, complemented
5C3×#1865C211yesmaximal, characteristic, complemented
6#71628C211yesmaximal, characteristic, complemented
7#70566C211yesmaximal, characteristic, complemented
8#70279C211yesmaximal, characteristic, complemented
9#70269C211yesmaximal, characteristic, complemented
10#70245C211yesmaximal, characteristic, complemented
11#69625C211yesmaximal, characteristic, complemented
12#68881C211yesmaximal, characteristic
13#68828C211yesmaximal, characteristic, complemented
14#68671C211yesmaximal, characteristic, complemented
15#68664C211yesmaximal, characteristic, complemented
16#24068—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C2×#946C2211yescharacteristic
20#5580—24no—
21#5580—24no—
22C2×#898—21no—
23C2×#895C2211yescharacteristic
24C2×#894—21no—
25C2×#892C2211yescharacteristic
26C2×#892C2211yescharacteristic
27C2×#890C2211yescharacteristic
28C2×C4×D12—21no—
29C2×C4×D12C2211yescharacteristic
30C2×#872—21no—
31C2×C42×D3C2211yescharacteristic
32C2×C3×#106C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×#103C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#821C2211yescharacteristic
40C2×(C2×D3)⋊C8C2211yescharacteristic
41C2×(C2×D3)⋊C8C2211yescharacteristic
42C2×#817C2211yescharacteristic
43#4989—21no—
44#4984C2211yescharacteristic
45Dic3×C4⋊C4—21no—
46#4972C2211yescharacteristic
47C2×#809C2211yescharacteristic
48C2×#809C2211yescharacteristic
49C2×#807C2211yescharacteristic
50C2×#807C2211yescharacteristic
51#4928C2211yescharacteristic
52#4866—24no—
53#4815—24no—
54#4668—21no—
55#4649—21no—
56C3×#340C2211yescharacteristic
57C3×#336C2211yescharacteristic
58#4551C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
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#1865D311yescharacteristic, complemented
75#1864—31no—
76#1863—31no—
77#1862—31no—
78#1817—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×D12—21no—
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C22×C4×D3D411yescharacteristic
84C22×C4×D3D411yescharacteristic
85C22×C4×D3C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97#947—22no—
98#947—22no—
99#946—22no—
100#946D414yes—
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—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3D411yescharacteristic
120C2×C4×Dic3D411yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124#898—28no—
125#895—22no—
126#895D414yes—
127#894—28no—
128#892—22no—
129#892—22no—
130#892—22no—
131#892—22no—
132#890—22no—
133#890—22no—
134C4×D12—28no—
135C4×D12—22no—
136C4×D12—22no—
137#872—28no—
138C42×D3—22no—
139C42×D3—22no—
140C3×#106—22no—
141C3×#103—22no—
142C3×#103—22no—
143C3×#95—21no—
144#835—22no—
145#835—22no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149#821—22no—
150#821—22no—
151(C2×D3)⋊C8—22no—
152(C2×D3)⋊C8—22no—
153#817—22no—
154#809—22no—
155#809—22no—
156#807—22no—
157#807—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×Q8C2×D311yescharacteristic
167C2×C4×D4—31no—
168C2×C4×D4—61no—
169C2×#117—31no—
170C2×#117—31no—
171C2×#117—61no—
172C22×C42—31no—
173#485—64no—
174#482—64no—
175C2×#107—31no—
176C2×#107—31no—
177C2×#106—31no—
178C2×#106C2×D311yescharacteristic
179C2×#103—31no—
180C2×#103—31no—
181C2×#103C2×D311yescharacteristic
182C2×#103C2×D311yescharacteristic
183C2×#102—31no—
184C2×#102—31no—
185C2×#102—31no—
186C2×#102—31no—
187C2×C22⋊C8—31no—
188C2×C22⋊C8—31no—
189#389—31no—
190#389—31no—
191#389—31no—
192#389C2×D311yescharacteristic
193#385—61no—
194#385—61no—
195C4×C4⋊C4—61no—
196C4×C4⋊C4—61no—
197#340—31no—
198#340—31no—
199#340—31no—
200#340C2×D311yescharacteristic
201#336—31no—
202#336—31no—
203#336—31no—
204#336C2×D311yescharacteristic
205C23×D3—21no—
206C23×D3C2×D411yescharacteristic
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8C2×D412yes—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2411yescharacteristic
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—24no—
231C2×D12—21no—
232C2×D12—42no—
233C2×D12—42no—
234C2×D12C2×D412yes—
235C2×C4×D3—21no—
236C2×C4×D3—21no—
237C2×C4×D3—21no—
238C2×C4×D3—21no—
239C2×C4×D3—24no—
240C2×C4×D3—22no—
241C2×C4×D3—22no—
242C2×C4×D3—22no—
243C2×C4×D3—41no—
244C2×C4×D3—41no—
245C2×C4×D3—41no—
246C2×C4×D3—42no—
247C2×C4×D3—42no—
248C2×C4×D3C2×D412yes—
249C2×C4×D3C2×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—
256C3×C4⋊C4—24no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—42no—
262C3×C4⋊C4C2×D412yes—
263C3×C4⋊C4#4112yes—
264C3×C42—24no—
265C3×C42—21no—
266C3×C42—21no—
267(C2×D3)⋊C4—24no—
268(C2×D3)⋊C4—24no—
269(C2×D3)⋊C4—24no—
270(C2×D3)⋊C4—24no—
271(C2×D3)⋊C4—24no—
272(C2×D3)⋊C4—24no—
273(C2×D3)⋊C4—42no—
274C4⋊Dic3—24no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—42no—
280C4⋊Dic3C2×D412yes—
281C4⋊Dic3#4112yes—
282Dic3⋊C4—24no—
283Dic3⋊C4—24no—
284Dic3⋊C4—42no—
285Dic3⋊C4—42no—
286C4×Dic3—24no—
287C4×Dic3—21no—
288C4×Dic3—21no—
289C4×Dic3—22no—
290C4×Dic3—22no—
291C4×Dic3—42no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C22×Q8C22×D311yescharacteristic
299C22×D4—31no—
300C22×D4—61no—
301C23×C4—31no—
302C23×C4—31no—
303C23×C4—31no—
304C23×C4—61no—
305C22×C8—31no—
306C22×C8—31no—
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—22no—
320C4×Q8—22no—
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—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C4⋊C4C22×D311yescharacteristic
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—21no—
351C2×C42—31no—
352C2×C42—31no—
353C2×C42—31no—
354C2×C42—62no—
355C2×C42—62no—
356C2×C42—61no—
357C2×C42—61no—
358C2×C42C22×D311yescharacteristic
359#107—62no—
360#107—62no—
361#106—22no—
362#106—62no—
363#103—22no—
364#103—22no—
365#103—62no—
366#103—62no—
367#102—62no—
368#102—62no—
369#102—62no—
370#102—62no—
371C22⋊C8—62no—
372C22⋊C8—62no—
373#95—21no—
374#95—61no—
375#95—61no—
376#95—61no—
377C23×C3C22×D411yescharacteristic
378C22×D3—21no—
379C22×D3—21no—
380C22×D3—21no—
381C22×D3—21no—
382C22×D3—21no—
383C22×D3—21no—
384C22×D3—22no—
385C22×D3—41no—
386C22×D3—41no—
387C22×D3—41no—
388C22×D3—41no—
389C22×D3—41no—
390C22×D3—41no—
391C22×D3#12012yes—
392C3×Q8—22no—
393C3×Q8—22no—
394C3×Q8—41no—
395C3×Q8—41no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4#13512yes—
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4C22×D411yescharacteristic
416C2×C3×C4C2×#4111yescharacteristic
417C2×C3×C4C2×#4111yescharacteristic
418C2×C3×C42−1+411yescharacteristic
419C2×C3×C42−1+411yescharacteristic
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437C2×Dic3#12012yes—
438D12—22no—
439D12—22no—
440D12—48no—
441D12—41no—
442D12—41no—
443C4×D3—22no—
444C4×D3—22no—
445C4×D3—22no—
446C4×D3—22no—
447C4×D3—44no—
448C4×D3—48no—
449C4×D3—44no—
450C3×C8—41no—
451C3×C8—41no—
452C3⋊C8—41no—
453C3⋊C8—41no—
454C24—31no—
455C24—61no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8D3×D412yes—
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—31no—
476C22×C4—32no—
477C22×C4—32no—
478C22×C4—31no—
479C22×C4—31no—
480C22×C4—62no—
481C22×C4—62no—
482C22×C4—64no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—61no—
486C22×C4—61no—
487C22×C4—61no—
488C22×C4—61no—
489C22×C4—62no—
490C22×C4—61no—
491C22×C4—121no—
492C22×C4—121no—
493C22×C4—121no—
494C22×C4—122no—
495C22×C4—122no—
496C22×C4D3×D411yescharacteristic
497C22×C4D3×D411yescharacteristic
498C22×C4C23×D311yescharacteristic
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—61no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C2×C8—61no—
510C2×C8—61no—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—32no—
517C4⋊C4—32no—
518C4⋊C4—42no—
519C4⋊C4—64no—
520C4⋊C4—64no—
521C4⋊C4—64no—
522C4⋊C4—61no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—122no—
527C4⋊C4—122no—
528C4⋊C4—122no—
529C4⋊C4#23512yes—
530C4⋊C4D3×D412yes—
531C22⋊C4—64no—
532C22⋊C4—64no—
533C22⋊C4—64no—
534C22⋊C4—64no—
535C22⋊C4—64no—
536C22⋊C4—64no—
537C22⋊C4—122no—
538C42—24no—
539C42—21no—
540C42—21no—
541C42—62no—
542C42—62no—
543C42—64no—
544C42—61no—
545C42—61no—
546C42—122no—
547C22×C3C2×#12011yescharacteristic
548C22×C3#52511yescharacteristic
549C22×C3#53611yescharacteristic
550C22×C3C2×#13511yescharacteristic
551C22×C3#57511yescharacteristic
552C22×C3#57511yescharacteristic
553C22×C3#57511yescharacteristic
554C2×D3—22no—
555C2×D3—22no—
556C2×D3—22no—
557C2×D3—41no—
558C2×D3—41no—
559C2×D3—41no—
560C2×D3—41no—
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—42no—
570C3×C4—42no—
571C3×C4—41no—
572C3×C4—42no—
573C3×C4—81no—
574C3×C4#54711yescharacteristic
575C3×C4#54711yescharacteristic
576C3×C4#54711yescharacteristic
577C3×C4#54711yescharacteristic
578Dic3—22no—
579Dic3—42no—
580Dic3—41no—
581Dic3—42no—
582Dic3—41no—
583Dic3—81no—
584C23—32no—
585C23—61no—
586C23—61no—
587C23—61no—
588C23—61no—
589C23—61no—
590C23—61no—
591C23—62no—
592C23—121no—
593C23—121no—
594C23—121no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23C2×D3×D411yescharacteristic
599Q8—22no—
600Q8—22no—
601Q8—41no—
602Q8—41no—
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—21no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—32no—
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—41no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—62no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—62no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—62no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—62no—
640C2×C4—124no—
641C2×C4—128no—
642C2×C4—124no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4#91612yes—
651C2×C4C2×#23511yescharacteristic
652C2×C4C2×D3×D411yescharacteristic
653C2×C4C2×D3×D411yescharacteristic
654C2×C4#100711yescharacteristic
655C2×C4D3×#4111yescharacteristic
656C2×C4#101011yescharacteristic
657C8—41no—
658C8—41no—
659C8—121no—
660C8—121no—
661C2×C3#240611yescharacteristic
662C2×C3#300211yescharacteristic
663C2×C3#300511yescharacteristic
664C2×C3#310211yescharacteristic
665C2×C3#311111yescharacteristic
666C2×C3#323311yescharacteristic
667C2×C3#324011yescharacteristic
668D3—42no—
669D3—82no—
670C22—62no—
671C22—62no—
672C22—62no—
673C22—121no—
674C22—121no—
675C22—121no—
676C22—121no—
677C22—122no—
678C22—122no—
679C22—122no—
680C22—241no—
681C22—241no—
682C22—241no—
683C22—241no—
684C22#554711yescharacteristic
685C22D3×#12011yescharacteristic
686C22#562811yescharacteristic
687C22C2×#91611yescharacteristic
688C22#576811yescharacteristic
689C22#577711yescharacteristic
690C22#577711yescharacteristic
691C4—22no—
692C4—42no—
693C4—41no—
694C4—42no—
695C4—42no—
696C4—62no—
697C4—81no—
698C4—121no—
699C4—121no—
700C4—122no—
701C4—122no—
702C4—241no—
703C4#550711yescharacteristic
704C4#555111yescharacteristic
705C4#563111yescharacteristic
706C4#565511yescharacteristic
707C3#2406811yescharacteristic, complemented, Sylow
708C2—122no—
709C2—242no—
710C2#7460011yescharacteristic
711C2#7965211yescharacteristic
712C2#7984711yescharacteristic
713C2#8003811yescharacteristic
714C2#8022711yescharacteristic
715C2#8101611yescharacteristic
716C2#8123911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1865C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#1865D3
hypocenterC3#24068
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
3. C2×#4815C2semidirect
4. #75963C2semidirect
5. C3×#1865C2semidirect
6. #71628C2semidirect
7. #70566C2semidirect
8. #70279C2semidirect
9. #70269C2semidirect
10. #70245C2semidirect
11. #69625C2semidirect
13. #68828C2semidirect
14. #68671C2semidirect
15. #68664C2semidirect
74. #1865D3semidirect
707. C3#24068semidirect

Maximal subgroups

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

subgroupindex
C2×#55802
C2×#48662
C2×#48152
#759632
C3×#18652
#716282
#705662
#702792
#702692
#702452
#696252
#688812
#688282
#686712
#686642
#240683