Finite Group Databasebeta

← #11025196

Subgroups — #11025196

Order 768.

2328 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×#533C211yesmaximal, characteristic, complemented
2C2×#5697C211yesmaximal, characteristic, complemented
3#83257C212yesmaximal, complemented
4#83236C212yesmaximal, complemented
5#83067C211yesmaximal, characteristic, complemented
6#83065C211yesmaximal, characteristic, complemented
7#83023C211yesmaximal, characteristic, complemented
8#82829C214yesmaximal, complemented
9#82827C214yesmaximal, complemented
10#82497C214yesmaximal, complemented
11#82495C214yesmaximal, complemented
12#82276C212yesmaximal, complemented
13#82244C212yesmaximal, complemented
14#82081C211yesmaximal, characteristic, complemented
15#82065C211yesmaximal, characteristic, complemented
16#37710—31nomaximal, Sylow
17C3×#533—28no—
18C2×C3×C4⋊Q8C2212yes—
19C2×C3×#125C2211yescharacteristic
20C2×C3×#125C2211yescharacteristic
21C2×C3×C22⋊Q8C2214yes—
22C2×C3×C4×Q8C2214yes—
23C2×C3×#117C2212yes—
24C22×C3×C42C2211yescharacteristic
25C2×#947C2214yes—
26C2×#946C2214yes—
27C2×#930C2212yes—
28C2×#930C2214yes—
29C2×C4×#67C2211yescharacteristic
30#5697—28no—
31C2×#895C2212yes—
32C2×#895C2214yes—
33C2×#894C2214yes—
34C2×#893C2212yes—
35C2×#874C2211yescharacteristic
36C2×#872C2211yescharacteristic
37#5231C2214yes—
38#5231C2214yes—
39#5213C2212yes—
40#5210C2212yes—
41C4×#217C2211yescharacteristic
42#4994C2214yes—
43#4994C2214yes—
44#4991C2214yes—
45#4991C2214yes—
46#4991C2214yes—
47#4990C2214yes—
48#4989C2214yes—
49#4988C2212yes—
50#4988C2212yes—
51#4987C2212yes—
52#4984C2212yes—
53#4984C2214yes—
54#4982C2214yes—
55#4980C2214yes—
56Dic3×C4⋊C4C2212yes—
57#4972C2212yes—
58#4961C2214yes—
59#4960C2214yes—
60#4957C2214yes—
61#4956C2214yes—
62#4940C2212yes—
63#4940C2214yes—
64#4940C2214yes—
65#4939C2212yes—
66#4939C2214yes—
67#4939C2214yes—
68C4×(C2×D3)⋊C4C2211yescharacteristic
69C4×(C2×D3)⋊C4C2214yes—
70#4935C2211yescharacteristic
71#4933C2212yes—
72#4932C2211yescharacteristic
73C4×Dic3⋊C4C2211yescharacteristic
74C2×#540—31no—
75C2×#533D311yescharacteristic, complemented
76#2856—31no—
77#2681—34no—
78#2676—34no—
79#2641—31no—
80#2629—31no—
81#2491—32no—
82#2458—34no—
83#2452—32no—
84#2426—34no—
85#2414—32no—
86#2363—32no—
87#2330—31no—
88#2290—31no—
89C22×C3×Q8C2314yes—
90C23×C3×C4C2312yes—
91C23×C3×C4C2311yescharacteristic
92C22×#67C2314yes—
93C22×#67C2311yescharacteristic
94C22×D12C2312yes—
95C22×D12C2314yes—
96C22×D12C2314yes—
97C22×C4×D3C2311yescharacteristic
98C22×C4×D3C2314yes—
99C3×C4⋊Q8—28no—
100C3×#125—24no—
101C3×#125D418yescomplemented
102C3×C22⋊Q8—216no—
103C3×C4×Q8—216no—
104C3×#117—28no—
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C4⋊C4C2314yes—
107C2×C3×C4⋊C4C2314yes—
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C4⋊C4C2314yes—
110C2×C3×C22⋊C4C2314yes—
111C2×C3×C42—24no—
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2312yes—
114C2×C3×C42C2314yes—
115C2×C3×C42C2311yescharacteristic
116C2×#217C2311yescharacteristic
117C2×#217C2312yes—
118#947—48no—
119#946—48no—
120#930—28no—
121#930—48no—
122C4×#67—24no—
123C2×(C2×D3)⋊C4C2314yes—
124C2×(C2×D3)⋊C4C2312yes—
125C2×(C2×D3)⋊C4C2311yescharacteristic
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2314yes—
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×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2314yes—
138C2×C4⋊Dic3C2314yes—
139C2×Dic3⋊C4C2312yes—
140C2×Dic3⋊C4C2311yescharacteristic
141C2×Dic3⋊C4C2312yes—
142C2×Dic3⋊C4C2314yes—
143C2×C4×Dic3C2311yescharacteristic
144C2×C4×Dic3C2311yescharacteristic
145C2×C4×Dic3C2311yescharacteristic
146C2×C4×Dic3C2314yes—
147#895—216no—
148#895—28no—
149#894—216no—
150#893—28no—
151#874—24no—
152#872—24no—
153C3⋊#95C2312yes—
154C3⋊#95C2312yes—
155C3⋊#95C2312yes—
156C3⋊#95C2312yes—
157C3⋊#95C2312yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162#540—68no—
163#533—28no—
164C2×C4⋊Q8C2×D312yes—
165C2×#127—31no—
166C2×#125C2×D311yescharacteristic
167C2×#125C2×D311yescharacteristic
168C2×#124—34no—
169C2×#123—34no—
170C2×C22⋊Q8—34no—
171C2×C22⋊Q8C2×D314yes—
172C2×#121—32no—
173C2×#121—34no—
174C2×#121—32no—
175C2×#121—34no—
176C2×C4×Q8C2×D314yes—
177C2×C4×D4—31no—
178C2×C4×D4—32no—
179C2×#117—31no—
180C2×#117C2×D312yes—
181C22×C42C2×D311yescharacteristic
182#398—34no—
183#396—34no—
184#396—34no—
185#394—34no—
186#394—34no—
187#394—34no—
188#394—34no—
189#393—34no—
190#393—34no—
191#392—34no—
192#390—32no—
193#390—32no—
194#390—34no—
195#390—34no—
196#390—32no—
197#390—34no—
198#389—32no—
199#389—34no—
200#389—32no—
201#388—34no—
202#388—34no—
203#388—34no—
204#388—32no—
205#387—34no—
206#387—32no—
207#386—32no—
208#386—34no—
209#385—32no—
210#385—34no—
211#384—31no—
212#380—32no—
213C4×C4⋊C4—32no—
214C4×C4⋊C4—31no—
215C4×C22⋊C4—31no—
216C4×C22⋊C4—34no—
217C4×C22⋊C4—31no—
218#376—31no—
219C24×C3C2411yescharacteristic
220C23×D3C2411yescharacteristic
221C23×D3C2414yes—
222C2×C3×Q8—28no—
223C2×C3×Q8—28no—
224C2×C3×Q8—44no—
225C22×C3×C4—22no—
226C22×C3×C4—22no—
227C22×C3×C4—24no—
228C22×C3×C4—24no—
229C22×C3×C4—42no—
230C22×C3×C4#4114yes—
231C22×C3×C4C2414yes—
232C22×C3×C4C2414yes—
233C22×C3×C4C2412yes—
234C22×C3×C4C2411yescharacteristic
235C22×C3×C4C2412yes—
236C22×C3×C4C2411yescharacteristic
237C2×#67—22no—
238C2×#67—28no—
239C2×#67—42no—
240C2×#67—44no—
241C2×#67—44no—
242C22×Dic3C2411yescharacteristic
243C22×Dic3C2412yes—
244C22×Dic3C2411yescharacteristic
245C22×Dic3C2412yes—
246C22×Dic3C2414yes—
247C22×Dic3C2411yescharacteristic
248C2×D12—24no—
249C2×D12—24no—
250C2×D12—24no—
251C2×D12—28no—
252C2×D12—28no—
253C2×D12—44no—
254C2×D12—44no—
255C2×D12—44no—
256C2×D12—44no—
257C2×C4×D3—22no—
258C2×C4×D3—22no—
259C2×C4×D3—22no—
260C2×C4×D3—28no—
261C2×C4×D3—44no—
262C2×C4×D3—44no—
263C3×C4⋊C4—24no—
264C3×C4⋊C4—28no—
265C3×C4⋊C4—28no—
266C3×C4⋊C4C2×D418yes—
267C3×C4⋊C4C2×D4116yes—
268C3×C22⋊C4—28no—
269C3×C42—24no—
270C3×C42—22no—
271C3×C42—28no—
272C3×C42C2×D414yes—
273#217—22no—
274#217—42no—
275(C2×D3)⋊C4—24no—
276(C2×D3)⋊C4—28no—
277(C2×D3)⋊C4—28no—
278(C2×D3)⋊C4—22no—
279(C2×D3)⋊C4—28no—
280(C2×D3)⋊C4—28no—
281(C2×D3)⋊C4—28no—
282(C2×D3)⋊C4—24no—
283(C2×D3)⋊C4—44no—
284(C2×D3)⋊C4—44no—
285(C2×D3)⋊C4—44no—
286(C2×D3)⋊C4—44no—
287(C2×D3)⋊C4—44no—
288C4⋊Dic3—24no—
289C4⋊Dic3—44no—
290C4⋊Dic3—44no—
291Dic3⋊C4—24no—
292Dic3⋊C4—22no—
293Dic3⋊C4—24no—
294Dic3⋊C4—44no—
295C4×Dic3—22no—
296C4×Dic3—22no—
297C4×Dic3—22no—
298C4×Dic3—44no—
299C22×Q8C22×D314yes—
300C22×D4—31no—
301C22×D4—34no—
302C22×D4—32no—
303C22×D4—34no—
304C22×D4—34no—
305C23×C4—31no—
306C23×C4—34no—
307C23×C4C22×D312yes—
308C23×C4C22×D311yescharacteristic
309C4⋊Q8—28no—
310#127—64no—
311#125—24no—
312#125#6718yescomplemented
313#124—128no—
314#123—616no—
315C22⋊Q8—216no—
316C22⋊Q8—128no—
317#121—68no—
318#121—616no—
319#121—68no—
320#121—128no—
321C4×Q8—216no—
322C4×D4—64no—
323C4×D4—68no—
324#117—28no—
325#117—64no—
326C2×C4⋊C4—32no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—32no—
329C2×C4⋊C4—32no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—34no—
333C2×C4⋊C4C22×D312yes—
334C2×C4⋊C4C22×D314yes—
335C2×C4⋊C4C22×D314yes—
336C2×C4⋊C4C22×D312yes—
337C2×C4⋊C4C22×D314yes—
338C2×C22⋊C4—31no—
339C2×C22⋊C4—32no—
340C2×C22⋊C4—34no—
341C2×C22⋊C4—31no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—34no—
344C2×C22⋊C4—34no—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—32no—
347C2×C22⋊C4—32no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—34no—
350C2×C22⋊C4—34no—
351C2×C22⋊C4—34no—
352C2×C22⋊C4—34no—
353C2×C22⋊C4C22×D314yes—
354C2×C42—24no—
355C2×C42—31no—
356C2×C42—31no—
357C2×C42—31no—
358C2×C42—34no—
359C2×C42C22×D314yes—
360C2×C42C22×D312yes—
361C2×C42C22×D311yescharacteristic
362C2×C42C22×D311yescharacteristic
363#95—32no—
364#95—32no—
365#95—32no—
366#95—32no—
367#95—34no—
368#95—32no—
369#95—34no—
370#95—34no—
371#95—34no—
372C23×C3—21no—
373C23×C3—22no—
374C23×C3—21no—
375C23×C3—22no—
376C23×C3C2×#4112yes—
377C23×C3C2511yescharacteristic
378C22×D3—21no—
379C22×D3—24no—
380C22×D3—21no—
381C22×D3—24no—
382C22×D3—22no—
383C22×D3—24no—
384C22×D3—24no—
385C22×D3—21no—
386C22×D3—24no—
387C22×D3—24no—
388C22×D3—24no—
389C22×D3—22no—
390C3×Q8—416no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—24no—
394C2×C3×C4—22no—
395C2×C3×C4—21no—
396C2×C3×C4—24no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—28no—
400C2×C3×C4—24no—
401C2×C3×C4—24no—
402C2×C3×C4—24no—
403C2×C3×C4—48no—
404C2×C3×C4C22×D414yes—
405C2×C3×C4C22×D418yes—
406C2×C3×C4C22×D412yes—
407C2×C3×C4C2×#4112yes—
408C2×C3×C4C2×#4112yes—
409C2×C3×C4H(5,𝔽2)14yes—
410C2×C3×C42−1+412yes—
411#67—44no—
412#67—88no—
413C2×Dic3—22no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—24no—
419C2×Dic3—22no—
420C2×Dic3—24no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—24no—
424C2×Dic3—22no—
425C2×Dic3—21no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429D12—48no—
430D12—88no—
431D12—88no—
432C4×D3—44no—
433C4×D3—88no—
434C24—31no—
435C24—34no—
436C24C23×D311yescharacteristic
437C2×Q8—28no—
438C2×Q8—28no—
439C2×Q8—44no—
440C2×D4—62no—
441C2×D4—68no—
442C2×D4—64no—
443C2×D4—64no—
444C2×D4—64no—
445C2×D4—68no—
446C2×D4—68no—
447C2×D4—124no—
448C2×D4—124no—
449C2×D4—124no—
450C2×D4—124no—
451C2×D4—122no—
452C2×D4—124no—
453C2×D4—124no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—31no—
459C22×C4—32no—
460C22×C4—31no—
461C22×C4—34no—
462C22×C4—32no—
463C22×C4—31no—
464C22×C4—42no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—68no—
469C22×C4—124no—
470C22×C4—124no—
471C22×C4#23914yes—
472C22×C4C23×D314yes—
473C22×C4C23×D311yescharacteristic
474C22×C4C23×D312yes—
475C22×C4C23×D314yes—
476C22×C4C23×D312yes—
477C22×C4C23×D311yescharacteristic
478C4⋊C4—24no—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—62no—
482C4⋊C4—64no—
483C4⋊C4—64no—
484C4⋊C4—64no—
485C4⋊C4—124no—
486C4⋊C4—124no—
487C4⋊C4—124no—
488C4⋊C4C2×#6718yes—
489C4⋊C4C2×#67116yes—
490C22⋊C4—28no—
491C22⋊C4—62no—
492C22⋊C4—62no—
493C22⋊C4—68no—
494C22⋊C4—68no—
495C22⋊C4—68no—
496C22⋊C4—64no—
497C22⋊C4—64no—
498C22⋊C4—68no—
499C22⋊C4—68no—
500C22⋊C4—124no—
501C22⋊C4—124no—
502C22⋊C4—124no—
503C22⋊C4—124no—
504C22⋊C4—124no—
505C22⋊C4—122no—
506C42—24no—
507C42—22no—
508C42—28no—
509C42—62no—
510C42—62no—
511C42—62no—
512C42—124no—
513C42C2×#6714yes—
514C22×C3—22no—
515C22×C3—22no—
516C22×C3—22no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3C23×D411yescharacteristic
520C22×C3C22×#4111yescharacteristic
521C22×C3C2×H(5,𝔽2)12yes—
522C22×C3C2×2−1+411yescharacteristic
523C22×C3#58512yes—
524C2×D3—41no—
525C2×D3—44no—
526C2×D3—41no—
527C2×D3—44no—
528C2×D3—44no—
529C2×D3—42no—
530C2×D3—44no—
531C2×D3—44no—
532C2×D3—41no—
533C2×D3—42no—
534C2×D3—44no—
535C2×D3—44no—
536C3×C4—22no—
537C3×C4—28no—
538C3×C4—24no—
539C3×C4—24no—
540C3×C4—44no—
541C3×C4#54814yes—
542Dic3—41no—
543Dic3—41no—
544Dic3—42no—
545Dic3—41no—
546Dic3—44no—
547Dic3—42no—
548C23—21no—
549C23—22no—
550C23—22no—
551C23—21no—
552C23—61no—
553C23—64no—
554C23—61no—
555C23—64no—
556C23—62no—
557C23—64no—
558C23—64no—
559C23—64no—
560C23—64no—
561C23—62no—
562C23—61no—
563C23—64no—
564C23C2×#23912yes—
565C23C24×D311yescharacteristic
566Q8—416no—
567D4—124no—
568D4—128no—
569D4—248no—
570D4—248no—
571D4—248no—
572C2×C4—21no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—48no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—64no—
589C2×C4—62no—
590C2×C4—61no—
591C2×C4—62no—
592C2×C4—64no—
593C2×C4—64no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—61no—
601C2×C4—124no—
602C2×C4—248no—
603C2×C4C2×#23912yes—
604C2×C4C2×#23912yes—
605C2×C4#100712yes—
606C2×C4#100914yes—
607C2×C4C22×#6714yes—
608C2×C4C22×#6718yes—
609C2×C4C22×#6712yes—
610C2×C3—42no—
611C2×C3#345511yescharacteristic
612C2×C3C2×#54811yescharacteristic
613C2×C3#348812yes—
614C2×C3#350811yescharacteristic
615C2×C3#354712yes—
616D3—81no—
617D3—84no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—42no—
622C22—42no—
623C22—121no—
624C22—124no—
625C22—122no—
626C22—124no—
627C22—124no—
628C22—121no—
629C22—124no—
630C22—124no—
631C22—122no—
632C22—124no—
633C22—124no—
634C22—121no—
635C22C22×#23911yescharacteristic
636C22C2×#100711yescharacteristic
637C22C2×#100912yes—
638C22#596412yes—
639C22C23×#6711yescharacteristic
640C4—22no—
641C4—24no—
642C4—24no—
643C4—28no—
644C4—44no—
645C4—121no—
646C4—122no—
647C4—121no—
648C4—122no—
649C4—124no—
650C4—121no—
651C4#581714yes—
652C3#3771011yescharacteristic, complemented, Sylow
653C2—42no—
654C2—241no—
655C2—244no—
656C2#8626111yescharacteristic
657C2#8627311yescharacteristic
658C2#8628712yes—
659C2C2×#581711yescharacteristic
660C2#8727512yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#533C2
hypercenterC2×#533D3
hypocenterC3#37710
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×C3×#533C2semidirect
2. C2×#5697C2semidirect
3. #83257C2semidirect
4. #83236C2semidirect
5. #83067C2semidirect
6. #83065C2semidirect
7. #83023C2semidirect
8. #82829C2semidirect
9. #82827C2semidirect
10. #82497C2semidirect
11. #82495C2semidirect
12. #82276C2semidirect
13. #82244C2semidirect
14. #82081C2semidirect
15. #82065C2semidirect
75. C2×#533D3semidirect
101. C3×#125D4semidirect
312. #125#67semidirect
652. C3#37710semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×#56972
#832572
#832362
#830672
#830652
#830232
#828292
#828272
#824972
#824952
#822762
#822442
#820812
#820652
#377103