Finite Group Databasebeta

← #11025034

Subgroups — #11025034

Order 768.

2364 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#541C211yesmaximal, characteristic, complemented
2C2×#5725C211yesmaximal, characteristic, complemented
3#83227C211yesmaximal, characteristic, complemented
4#83117C214yesmaximal, complemented
5#83106C214yesmaximal, complemented
6#83072C211yesmaximal, characteristic, complemented
7#83070C214yesmaximal, complemented
8#83067C211yesmaximal, characteristic, complemented
9#83059C214yesmaximal, complemented
10#82812C212yesmaximal, complemented
11#82755C212yesmaximal, complemented
12#82469C212yesmaximal, complemented
13#82457C212yesmaximal, complemented
14#82286C211yesmaximal, characteristic, complemented
15#82268C211yesmaximal, characteristic, complemented
16#37701—31nomaximal, Sylow
17C22×#239C2212yes—
18C3×#541—28no—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×#126C2214yes—
21C2×C3×#125C2214yes—
22C2×C3×#125C2211yescharacteristic
23C2×C3×C22⋊Q8C2212yes—
24C2×C3×C4×Q8C2212yes—
25C2×C3×#117C2211yescharacteristic
26C2×#947C2212yes—
27C2×Q8×Dic3C2212yes—
28C2×#930C2212yes—
29C2×C4×#67C2211yescharacteristic
30#5725—28no—
31C2×#895C2214yes—
32C2×#893C2214yes—
33C2×#874C2211yescharacteristic
34C2×C42×D3C2211yescharacteristic
35#5229C2212yes—
36C4×#217C2211yescharacteristic
37#4994C2214yes—
38#4993C2214yes—
39#4992C2214yes—
40#4991C2214yes—
41#4991C2214yes—
42#4991C2214yes—
43#4991C2214yes—
44#4990C2214yes—
45#4989C2212yes—
46#4989C2214yes—
47#4989C2214yes—
48#4988C2212yes—
49#4988C2214yes—
50#4988C2212yes—
51#4987C2214yes—
52#4987C2212yes—
53#4980C2214yes—
54#4980C2214yes—
55#4978C2212yes—
56#4978C2214yes—
57#4972C2212yes—
58#4961C2214yes—
59#4960C2214yes—
60#4960C2214yes—
61#4957C2214yes—
62#4956C2212yes—
63#4956C2214yes—
64#4944C2212yes—
65#4940C2214yes—
66#4940C2214yes—
67#4939C2212yes—
68#4939C2214yes—
69C4×(C2×D3)⋊C4C2211yescharacteristic
70C4×(C2×D3)⋊C4C2214yes—
71#4936C2211yescharacteristic
72#4932C2211yescharacteristic
73C4×Dic3⋊C4C2211yescharacteristic
74C2×#541D311yescharacteristic, complemented
75C2×#532—31no—
76#2851—31no—
77#2714—34no—
78#2711—34no—
79#2641—31no—
80#2619—31no—
81#2467—34no—
82#2460—34no—
83#2448—32no—
84#2426—32no—
85#2407—32no—
86#2341—32no—
87#2327—31no—
88#2290—31no—
89C22×C3×Q8C2312yes—
90C23×C3×C4C2311yescharacteristic
91C22×#67C2311yescharacteristic
92C22×#67C2314yes—
93C2×#239—28no—
94C2×#239—28no—
95C2×#239—44no—
96C22×D12C2312yes—
97C22×D12C2314yes—
98C22×D12C2314yes—
99C22×C4×D3C2314yes—
100C22×C4×D3C2311yescharacteristic
101C22×C4×D3C2312yes—
102C3×C4⋊Q8D418yescomplemented
103C3×#126—216no—
104C3×#125—216no—
105C3×#125—24no—
106C3×C22⋊Q8—28no—
107C3×C4×Q8—28no—
108C3×#117—24no—
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C4⋊C4C2314yes—
112C2×C3×C4⋊C4C2314yes—
113C2×C3×C4⋊C4C2314yes—
114C2×C3×C4⋊C4C2314yes—
115C2×C3×C22⋊C4C2312yes—
116C2×C3×C22⋊C4C2314yes—
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2314yes—
119C2×C3×C42C2311yescharacteristic
120C2×#217C2311yescharacteristic
121C2×#217C2312yes—
122#947—28no—
123Q8×Dic3—44no—
124#930—44no—
125C4×#67—24no—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2311yescharacteristic
129C2×(C2×D3)⋊C4C2312yes—
130C2×(C2×D3)⋊C4C2314yes—
131C2×(C2×D3)⋊C4C2314yes—
132C2×(C2×D3)⋊C4C2314yes—
133C2×(C2×D3)⋊C4C2314yes—
134C2×(C2×D3)⋊C4C2314yes—
135C2×(C2×D3)⋊C4C2314yes—
136C2×(C2×D3)⋊C4C2314yes—
137C2×(C2×D3)⋊C4C2314yes—
138C2×C4⋊Dic3C2312yes—
139C2×C4⋊Dic3C2314yes—
140C2×Dic3⋊C4C2311yescharacteristic
141C2×Dic3⋊C4C2312yes—
142C2×Dic3⋊C4C2314yes—
143C2×C4×Dic3C2311yescharacteristic
144C2×C4×Dic3C2311yescharacteristic
145C2×C4×Dic3C2311yescharacteristic
146C2×C4×Dic3C2312yes—
147C2×C4×Dic3C2314yes—
148#895—216no—
149#893—216no—
150#874—24no—
151C42×D3—24no—
152C3⋊#95C2312yes—
153C3⋊#95C2312yes—
154C3⋊#95C2312yes—
155C3⋊#95C2314yes—
156C3⋊#95C2312yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2312yes—
162C22×#41—32no—
163#541—28no—
164#532—68no—
165C2×C4⋊Q8C2×D311yescharacteristic
166C2×#127—31no—
167C2×#126C2×D314yes—
168C2×#125C2×D311yescharacteristic
169C2×#125C2×D314yes—
170C2×#124—32no—
171C2×C22⋊Q8C2×D312yes—
172C2×#121—32no—
173C2×#121—34no—
174C2×C4×Q8—32no—
175C2×C4×Q8C2×D312yes—
176C2×C4×D4—31no—
177C2×C4×D4—34no—
178C2×#117C2×D311yescharacteristic
179C22×C42—31no—
180#399—34no—
181#397—34no—
182#396—34no—
183#394—34no—
184#394—34no—
185#394—34no—
186#394—34no—
187#394—34no—
188#392—34no—
189#392—34no—
190#391—32no—
191#390—32no—
192#390—34no—
193#390—32no—
194#390—34no—
195#390—32no—
196#390—34no—
197#389—32no—
198#388—34no—
199#388—32no—
200#388—34no—
201#388—34no—
202#387—34no—
203#387—34no—
204#387—34no—
205#387—32no—
206#386—34no—
207#386—32no—
208#385—34no—
209#385—34no—
210#385—32no—
211#382—31no—
212#380—32no—
213#380—34no—
214C4×C4⋊C4—31no—
215C4×C22⋊C4—31no—
216C4×C22⋊C4—31no—
217C4×C22⋊C4—34no—
218#376—31no—
219C24×C3C2411yescharacteristic
220C23×D3C2414yes—
221C23×D3C2411yescharacteristic
222C2×C3×Q8—24no—
223C2×C3×Q8—24no—
224C2×C3×Q8C2×D418yes—
225C22×C3×C4—22no—
226C22×C3×C4—22no—
227C22×C3×C4—22no—
228C22×C3×C4C2412yes—
229C22×C3×C4C2414yes—
230C22×C3×C4C2411yescharacteristic
231C22×C3×C4C2412yes—
232C22×C3×C4C2414yes—
233C22×C3×C4C2411yescharacteristic
234C2×#67—22no—
235C2×#67—42no—
236C2×#67—44no—
237C2×#67—44no—
238C2×#67—44no—
239C22×Dic3C2411yescharacteristic
240C22×Dic3C2411yescharacteristic
241C22×Dic3C2414yes—
242C22×Dic3C2412yes—
243C22×Dic3C2412yes—
244C22×Dic3C2411yescharacteristic
245#239—432no—
246C2×D12—28no—
247C2×D12—28no—
248C2×D12—24no—
249C2×D12—28no—
250C2×D12—24no—
251C2×D12—24no—
252C2×D12—44no—
253C2×D12—44no—
254C2×D12—44no—
255C2×C4×D3—22no—
256C2×C4×D3—22no—
257C2×C4×D3—24no—
258C2×C4×D3—28no—
259C2×C4×D3—28no—
260C2×C4×D3—42no—
261C2×C4×D3—42no—
262C2×C4×D3—44no—
263C2×C4×D3#4114yes—
264C3×C4⋊C4—24no—
265C3×C4⋊C4—24no—
266C3×C4⋊C4—28no—
267C3×C4⋊C4—28no—
268C3×C4⋊C4—28no—
269C3×C4⋊C4C2×D4116yes—
270C3×C22⋊C4—24no—
271C3×C22⋊C4—28no—
272C3×C42—22no—
273C3×C42—28no—
274C3×C42C2×D414yes—
275#217—22no—
276#217—42no—
277(C2×D3)⋊C4—22no—
278(C2×D3)⋊C4—28no—
279(C2×D3)⋊C4—28no—
280(C2×D3)⋊C4—44no—
281(C2×D3)⋊C4—44no—
282(C2×D3)⋊C4—44no—
283(C2×D3)⋊C4—42no—
284(C2×D3)⋊C4—44no—
285(C2×D3)⋊C4—44no—
286(C2×D3)⋊C4—44no—
287(C2×D3)⋊C4—44no—
288(C2×D3)⋊C4—44no—
289C4⋊Dic3—24no—
290C4⋊Dic3—44no—
291Dic3⋊C4—22no—
292Dic3⋊C4—44no—
293Dic3⋊C4—42no—
294C4×Dic3—24no—
295C4×Dic3—22no—
296C4×Dic3—22no—
297C4×Dic3—22no—
298C4×Dic3—44no—
299C2×#41—68no—
300C2×#41—68no—
301C2×#41—124no—
302C22×Q8C22×D312yes—
303C22×D4—31no—
304C22×D4—34no—
305C22×D4—32no—
306C22×D4—34no—
307C22×D4—34no—
308C23×C4—31no—
309C23×C4—34no—
310C23×C4—32no—
311C23×C4C22×D311yescharacteristic
312C4⋊Q8#6718yescomplemented
313#127—64no—
314#126—216no—
315#125—216no—
316#125—24no—
317#124—68no—
318C22⋊Q8—28no—
319#121—616no—
320#121—124no—
321C4×Q8—28no—
322C4×Q8—124no—
323C4×D4—64no—
324C4×D4—616no—
325#117—24no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—32no—
328C2×C4⋊C4—34no—
329C2×C4⋊C4—32no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4C22×D312yes—
332C2×C4⋊C4C22×D312yes—
333C2×C4⋊C4C22×D314yes—
334C2×C4⋊C4C22×D314yes—
335C2×C4⋊C4C22×D314yes—
336C2×C4⋊C4C22×D314yes—
337C2×C22⋊C4—31no—
338C2×C22⋊C4—34no—
339C2×C22⋊C4—34no—
340C2×C22⋊C4—31no—
341C2×C22⋊C4—32no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—32no—
344C2×C22⋊C4—34no—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—34no—
350C2×C22⋊C4—34no—
351C2×C22⋊C4C22×D312yes—
352C2×C22⋊C4C22×D314yes—
353C2×C42—32no—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—31no—
357C2×C42—34no—
358C2×C42—64no—
359C2×C42C22×D311yescharacteristic
360C2×C42C22×D314yes—
361C2×C42C22×D311yescharacteristic
362#95—32no—
363#95—32no—
364#95—32no—
365#95—34no—
366#95—32no—
367#95—32no—
368#95—34no—
369#95—34no—
370#95—34no—
371#95—34no—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—21no—
375C23×C3—22no—
376C23×C3—22no—
377C23×C3C2511yescharacteristic
378C22×D3—21no—
379C22×D3—21no—
380C22×D3—22no—
381C22×D3—22no—
382C22×D3—24no—
383C22×D3—24no—
384C22×D3—24no—
385C22×D3—24no—
386C22×D3—24no—
387C22×D3—24no—
388C22×D3—24no—
389C22×D3C2×#4112yes—
390C3×Q8—216no—
391C2×C3×C4—24no—
392C2×C3×C4—24no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—24no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—24no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—24no—
404C2×C3×C4—44no—
405C2×C3×C4C22×D414yes—
406C2×C3×C4C22×D418yes—
407C2×C3×C4C22×D412yes—
408C2×C3×C4C2×#4112yes—
409C2×C3×C4H(5,𝔽2)12yes—
410#67—44no—
411#67—88no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—22no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—22no—
418C2×Dic3—24no—
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—24no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—24no—
427C2×Dic3C2×#4112yes—
428D12—416no—
429D12—416no—
430D12—48no—
431C4×D3—28no—
432C4×D3—416no—
433C4×D3—48no—
434C24—31no—
435C24—34no—
436C24C23×D311yescharacteristic
437#41—1232no—
438C2×Q8—24no—
439C2×Q8—24no—
440C2×Q8C2×#6718yes—
441C2×D4—68no—
442C2×D4—62no—
443C2×D4—64no—
444C2×D4—64no—
445C2×D4—68no—
446C2×D4—68no—
447C2×D4—64no—
448C2×D4—124no—
449C2×D4—124no—
450C2×D4—124no—
451C2×D4—122no—
452C2×D4—124no—
453C2×D4—124no—
454C2×D4—124no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—31no—
459C22×C4—34no—
460C22×C4—31no—
461C22×C4—31no—
462C22×C4—34no—
463C22×C4—32no—
464C22×C4—32no—
465C22×C4—68no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—64no—
469C22×C4—68no—
470C22×C4—124no—
471C22×C4—122no—
472C22×C4—122no—
473C22×C4C23×D312yes—
474C22×C4C23×D314yes—
475C22×C4C23×D311yescharacteristic
476C22×C4C23×D312yes—
477C22×C4C23×D314yes—
478C22×C4C23×D311yescharacteristic
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—62no—
485C4⋊C4—64no—
486C4⋊C4—124no—
487C4⋊C4—122no—
488C4⋊C4—124no—
489C4⋊C4C2×#67116yes—
490C22⋊C4—24no—
491C22⋊C4—28no—
492C22⋊C4—62no—
493C22⋊C4—62no—
494C22⋊C4—68no—
495C22⋊C4—68no—
496C22⋊C4—124no—
497C22⋊C4—122no—
498C22⋊C4—124no—
499C22⋊C4—124no—
500C22⋊C4—124no—
501C22⋊C4—124no—
502C22⋊C4—124no—
503C22⋊C4—124no—
504C22⋊C4—124no—
505C22⋊C4—122no—
506C42—22no—
507C42—28no—
508C42—62no—
509C42—62no—
510C42—64no—
511C42—62no—
512C42—124no—
513C42C2×#6714yes—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—42no—
517C22×C3—41no—
518C22×C3—42no—
519C22×C3C23×D411yescharacteristic
520C22×C3C22×#4111yescharacteristic
521C22×C3C2×H(5,𝔽2)11yescharacteristic
522C22×C3#58512yes—
523C22×C3#58512yes—
524C2×D3—22no—
525C2×D3—22no—
526C2×D3—22no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3—44no—
530C2×D3—44no—
531C2×D3—44no—
532C2×D3—44no—
533C2×D3—44no—
534C2×D3—44no—
535C2×D3—44no—
536C3×C4—28no—
537C3×C4—24no—
538C3×C4—41no—
539C3×C4—42no—
540C3×C4—44no—
541C3×C4#54614yes—
542Dic3—22no—
543Dic3—41no—
544Dic3—42no—
545Dic3—41no—
546Dic3—42no—
547Dic3—44no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—22no—
552C23—21no—
553C23—32no—
554C23—61no—
555C23—62no—
556C23—62no—
557C23—61no—
558C23—64no—
559C23—64no—
560C23—64no—
561C23—64no—
562C23—64no—
563C23—64no—
564C23—64no—
565C23C24×D311yescharacteristic
566Q8—216no—
567D4—1216no—
568D4—1216no—
569D4—124no—
570D4—128no—
571D4—248no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—21no—
585C2×C4—32no—
586C2×C4—44no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—68no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—62no—
593C2×C4—64no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—64no—
599C2×C4—62no—
600C2×C4—64no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—1216no—
604C2×C4—128no—
605C2×C4D3×#4112yes—
606C2×C4#100912yes—
607C2×C4C22×#6718yes—
608C2×C4C22×#6714yes—
609C2×C4C22×#6712yes—
610C2×C3—81no—
611C2×C3#345511yescharacteristic
612C2×C3C2×#54611yescharacteristic
613C2×C3#348712yes—
614C2×C3#351211yescharacteristic
615C2×C3#356212yes—
616D3—42no—
617D3—84no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—62no—
624C22—62no—
625C22—62no—
626C22—122no—
627C22—122no—
628C22—124no—
629C22—124no—
630C22—124no—
631C22—124no—
632C22—124no—
633C22—124no—
634C22—124no—
635C22C2×D3×#4111yescharacteristic
636C22C2×#100911yescharacteristic
637C22#596812yes—
638C22#596812yes—
639C22C23×#6711yescharacteristic
640C4—28no—
641C4—24no—
642C4—41no—
643C4—42no—
644C4—44no—
645C4—62no—
646C4—121no—
647C4—121no—
648C4—124no—
649C4—122no—
650C4—122no—
651C4#582914yes—
652C3#3770111yescharacteristic, complemented, Sylow
653C2—81no—
654C2—122no—
655C2—244no—
656C2#8649511yescharacteristic
657C2#8649711yescharacteristic
658C2#8650412yes—
659C2C2×#582911yescharacteristic
660C2#8729012yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#541C2
hypercenterC2×#541D3
hypocenterC3#37701
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
1. C2×C3×#541C2semidirect
2. C2×#5725C2semidirect
3. #83227C2semidirect
4. #83117C2semidirect
5. #83106C2semidirect
6. #83072C2semidirect
7. #83070C2semidirect
8. #83067C2semidirect
9. #83059C2semidirect
10. #82812C2semidirect
11. #82755C2semidirect
12. #82469C2semidirect
13. #82457C2semidirect
14. #82286C2semidirect
15. #82268C2semidirect
74. C2×#541D3semidirect
102. C3×C4⋊Q8D4semidirect
312. C4⋊Q8#67semidirect
652. C3#37701semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C2×#57252
#832272
#831172
#831062
#830722
#830702
#830672
#830592
#828122
#827552
#824692
#824572
#822862
#822682
#377013