Finite Group Databasebeta

← #11069617

Subgroups — #11069617

Order 768.

2206 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5731C211yesmaximal, characteristic, complemented
2C2×#5535C211yesmaximal, characteristic, complemented
3C3×#2621C211yesmaximal, characteristic, complemented
4#83251C211yesmaximal, characteristic, complemented
5#82779C212yesmaximal, complemented
6#82755C212yesmaximal, complemented
7#82225C211yesmaximal, characteristic, complemented
8#77848C214yesmaximal, complemented
9#77789C214yesmaximal, complemented
10#77585C214yesmaximal, complemented
11#77487C214yesmaximal, complemented
12#76624C212yesmaximal, complemented
13#76493C212yesmaximal, complemented
14#74244C211yesmaximal, characteristic, complemented
15#74150C211yesmaximal, characteristic, complemented
16#39382—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×#117C2211yescharacteristic
20C2×#947C2212yes—
21C2×#947C2212yes—
22C2×#946C2212yes—
23C2×Q8×Dic3C2212yes—
24C2×#930C2212yes—
25C2×C4×#67C2211yescharacteristic
26#5731—28no—
27#5535—28no—
28C2×#898C2214yes—
29C2×#895C2212yes—
30C2×#894C2214yes—
31C2×#889C2214yes—
32C2×#884C2214yes—
33C2×#874C2211yescharacteristic
34C2×C4×D12C2212yes—
35C2×#872C2211yescharacteristic
36C2×#870C2211yescharacteristic
37C3×#398C2214yes—
38C3×#397C2214yes—
39C3×#391C2212yes—
40C3×#382C2211yescharacteristic
41C4×#217C2211yescharacteristic
42#4994C2214yes—
43#4993C2214yes—
44#4990C2214yes—
45#4990C2212yes—
46#4988C2212yes—
47#4988C2214yes—
48#4988C2214yes—
49#4988C2214yes—
50#4976C2212yes—
51Dic3×C4⋊C4C2212yes—
52#4972C2212yes—
53#4961C2214yes—
54#4960C2214yes—
55#4957C2212yes—
56#4957C2214yes—
57#4957C2214yes—
58#4945C2212yes—
59C4×(C2×D3)⋊C4C2211yescharacteristic
60C4×Dic3⋊C4C2211yescharacteristic
61C42×Dic3C2211yescharacteristic
62#4675C2214yes—
63#4672C2214yes—
64#4672C2214yes—
65#4671C2214yes—
66#4670C2214yes—
67#4669C2214yes—
68#4668C2214yes—
69#4667C2214yes—
70#4666C2214yes—
71#4665C2214yes—
72#4652C2214yes—
73#4649C2214yes—
74C2×#541—31no—
75C2×#540—31no—
76#2858—31no—
77#2714—34no—
78#2694—34no—
79#2621D311yescharacteristic, complemented
80#2619—31no—
81#2554—34no—
82#2540—32no—
83#2490—34no—
84#2427—32no—
85#2426—32no—
86#2367—32no—
87C4×#125—31no—
88#2293—31no—
89C22×C3×Q8C2312yes—
90C23×C3×C4C2311yescharacteristic
91C22×#67C2311yescharacteristic
92C22×#67C2314yes—
93C22×D12C2312yes—
94C22×D12C2314yes—
95C22×D12C2314yes—
96C22×C4×D3C2311yescharacteristic
97C22×C4×D3C2314yes—
98C3×C4⋊Q8—24no—
99C3×C22⋊Q8—44no—
100C3×#117—24no—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2314yes—
104C2×C3×C4⋊C4C2314yes—
105C2×C3×C22⋊C4C2312yes—
106C2×C3×C22⋊C4C2314yes—
107C2×C3×C42C2311yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C2×#217C2311yescharacteristic
110C2×#217C2312yes—
111#947—28no—
112#947—44no—
113#946—28no—
114Q8×Dic3—28no—
115#930—28no—
116C4×#67—24no—
117C2×(C2×D3)⋊C4C2314yes—
118C2×(C2×D3)⋊C4C2314yes—
119C2×(C2×D3)⋊C4C2314yes—
120C2×(C2×D3)⋊C4C2314yes—
121C2×(C2×D3)⋊C4C2312yes—
122C2×(C2×D3)⋊C4C2311yescharacteristic
123C2×(C2×D3)⋊C4C2314yes—
124C2×(C2×D3)⋊C4C2312yes—
125C2×(C2×D3)⋊C4C2314yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×C4⋊Dic3C2314yes—
131C2×C4⋊Dic3C2312yes—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2312yes—
134C2×Dic3⋊C4C2311yescharacteristic
135C2×Dic3⋊C4C2314yes—
136C2×Dic3⋊C4C2312yes—
137C2×Dic3⋊C4C2314yes—
138C2×C4×Dic3C2314yes—
139C2×C4×Dic3C2311yescharacteristic
140C2×C4×Dic3C2311yescharacteristic
141C2×C4×Dic3C2314yes—
142C2×C4×Dic3C2312yes—
143C2×C4×Dic3C2312yes—
144C2×C4×Dic3C2311yescharacteristic
145#898—216no—
146#895—44no—
147#894—216no—
148#889—216no—
149#884—216no—
150#874—24no—
151C4×D12—28no—
152#872—24no—
153#870D418yescomplemented
154C3×#95C2314yes—
155C3×#95C2314yes—
156C3×#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2312yes—
160C3⋊#95C2312yes—
161C3⋊#95C2312yes—
162#541—68no—
163#540—68no—
164C2×C4⋊Q8C2×D311yescharacteristic
165C2×#127—31no—
166C2×#126—34no—
167C2×#125—34no—
168C2×#125—31no—
169C2×#124—32no—
170C2×#124—32no—
171C2×#123—34no—
172C2×C22⋊Q8—32no—
173C2×C22⋊Q8C2×D312yes—
174C2×#121—32no—
175C2×#121—32no—
176C2×#121—34no—
177C2×C4×Q8—32no—
178C2×C4×D4—31no—
179C2×C4×D4—32no—
180C2×#117—31no—
181C2×#117C2×D311yescharacteristic
182#399—34no—
183#398C2×D314yes—
184#397—34no—
185#397C2×D314yes—
186#396—34no—
187#394—34no—
188#394—34no—
189#394—34no—
190#392—34no—
191#392—34no—
192#391C2×D312yes—
193#390—34no—
194#390—34no—
195#390—32no—
196#390—34no—
197#390—34no—
198#390—32no—
199#390—34no—
200#389—32no—
201#389—32no—
202#388—34no—
203#388—32no—
204#388—32no—
205#388—34no—
206#388—34no—
207#387—34no—
208#386—34no—
209#385—34no—
210#382C2×D311yescharacteristic
211#380—34no—
212C4×C4⋊C4—32no—
213C4×C4⋊C4—31no—
214C4×C4⋊C4—34no—
215C4×C22⋊C4—31no—
216C4×C22⋊C4—31no—
217C4×C22⋊C4—34no—
218C43—31no—
219C24×C3C2411yescharacteristic
220C23×D3C2411yescharacteristic
221C23×D3C2414yes—
222C2×C3×Q8—24no—
223C2×C3×Q8—24no—
224C2×C3×Q8—24no—
225C22×C3×C4—22no—
226C22×C3×C4—22no—
227C22×C3×C4—22no—
228C22×C3×C4C2414yes—
229C22×C3×C4C2412yes—
230C22×C3×C4C2411yescharacteristic
231C22×C3×C4C2414yes—
232C22×C3×C4C2412yes—
233C22×C3×C4C2411yescharacteristic
234C2×#67—28no—
235C2×#67—22no—
236C2×#67—42no—
237C2×#67—44no—
238C2×#67—44no—
239C22×Dic3C2412yes—
240C22×Dic3C2412yes—
241C22×Dic3C2411yescharacteristic
242C22×Dic3C2414yes—
243C22×Dic3C2411yescharacteristic
244C22×Dic3C2411yescharacteristic
245C2×D12—24no—
246C2×D12—24no—
247C2×D12—24no—
248C2×D12—28no—
249C2×D12—44no—
250C2×D12—44no—
251C2×D12—44no—
252C2×D12—44no—
253C2×D12—44no—
254C2×C4×D3—22no—
255C2×C4×D3—22no—
256C2×C4×D3—22no—
257C2×C4×D3—28no—
258C2×C4×D3—44no—
259C2×C4×D3—44no—
260C3×C4⋊C4—28no—
261C3×C4⋊C4—24no—
262C3×C4⋊C4—24no—
263C3×C4⋊C4—44no—
264C3×C22⋊C4—24no—
265C3×C22⋊C4—44no—
266C3×C42—22no—
267C3×C42C2×D414yes—
268#217—22no—
269#217—42no—
270(C2×D3)⋊C4—24no—
271(C2×D3)⋊C4—28no—
272(C2×D3)⋊C4—28no—
273(C2×D3)⋊C4—28no—
274(C2×D3)⋊C4—22no—
275(C2×D3)⋊C4—28no—
276(C2×D3)⋊C4—28no—
277(C2×D3)⋊C4—44no—
278(C2×D3)⋊C4—42no—
279(C2×D3)⋊C4—44no—
280(C2×D3)⋊C4—44no—
281(C2×D3)⋊C4—44no—
282(C2×D3)⋊C4—44no—
283C4⋊Dic3—28no—
284C4⋊Dic3C2×D418yes—
285Dic3⋊C4—24no—
286Dic3⋊C4—28no—
287Dic3⋊C4—22no—
288Dic3⋊C4—42no—
289Dic3⋊C4—44no—
290Dic3⋊C4C2×D4116yes—
291C4×Dic3—22no—
292C4×Dic3—22no—
293C4×Dic3—28no—
294C4×Dic3—44no—
295C4×Dic3—42no—
296C4×Dic3—42no—
297C4×Dic3#4114yes—
298C22×Q8C22×D312yes—
299C22×D4—31no—
300C22×D4—34no—
301C22×D4—32no—
302C22×D4—34no—
303C22×D4—34no—
304C23×C4—31no—
305C23×C4—34no—
306C23×C4C22×D311yescharacteristic
307C4⋊Q8—24no—
308#127—64no—
309#126—616no—
310#125—38no—
311#125—616no—
312#124—68no—
313#124—124no—
314#123—616no—
315C22⋊Q8—44no—
316C22⋊Q8—68no—
317#121—616no—
318#121—68no—
319#121—124no—
320C4×Q8—68no—
321C4×D4—68no—
322C4×D4—64no—
323#117—24no—
324#117—64no—
325C2×C4⋊C4—32no—
326C2×C4⋊C4—34no—
327C2×C4⋊C4—32no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—34no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—32no—
333C2×C4⋊C4C22×D312yes—
334C2×C4⋊C4C22×D312yes—
335C2×C4⋊C4C22×D314yes—
336C2×C4⋊C4C22×D314yes—
337C2×C22⋊C4—31no—
338C2×C22⋊C4—32no—
339C2×C22⋊C4—34no—
340C2×C22⋊C4—31no—
341C2×C22⋊C4—34no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—34no—
344C2×C22⋊C4—34no—
345C2×C22⋊C4—32no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—32no—
350C2×C22⋊C4—34no—
351C2×C22⋊C4—34no—
352C2×C22⋊C4C22×D312yes—
353C2×C22⋊C4C22×D314yes—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—34no—
357C2×C42—31no—
358C2×C42—32no—
359C2×C42—32no—
360C2×C42—34no—
361C2×C42C22×D311yescharacteristic
362C2×C42C22×D311yescharacteristic
363#95—32no—
364#95—32no—
365#95—32no—
366#95—34no—
367#95—34no—
368#95C22×D314yes—
369#95C22×D314yes—
370#95C22×D314yes—
371C23×C3—21no—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—22no—
375C23×C3—22no—
376C23×C3C2511yescharacteristic
377C22×D3—21no—
378C22×D3—24no—
379C22×D3—24no—
380C22×D3—21no—
381C22×D3—24no—
382C22×D3—21no—
383C22×D3—24no—
384C22×D3—22no—
385C22×D3—24no—
386C22×D3—24no—
387C22×D3—22no—
388C22×D3—24no—
389C3×Q8—48no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—24no—
393C2×C3×C4—21no—
394C2×C3×C4—22no—
395C2×C3×C4—21no—
396C2×C3×C4—24no—
397C2×C3×C4—24no—
398C2×C3×C4—24no—
399C2×C3×C4—21no—
400C2×C3×C4—24no—
401C2×C3×C4—24no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—44no—
405C2×C3×C4C22×D414yes—
406C2×C3×C4C22×D412yes—
407C2×C3×C4C2×#4112yes—
408C2×C3×C4H(5,𝔽2)12yes—
409#67—44no—
410#67—88no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—24no—
414C2×Dic3—24no—
415C2×Dic3—22no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—22no—
419C2×Dic3—21no—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3C22×D418yes—
425C2×Dic3C2×#4112yes—
426C2×Dic3C2×#4112yes—
427D12—48no—
428D12—88no—
429D12—88no—
430C4×D3—44no—
431C4×D3—88no—
432C24—31no—
433C24—34no—
434C24C23×D311yescharacteristic
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—24no—
438C2×D4—62no—
439C2×D4—68no—
440C2×D4—64no—
441C2×D4—68no—
442C2×D4—64no—
443C2×D4—64no—
444C2×D4—124no—
445C2×D4—124no—
446C2×D4—124no—
447C2×D4—124no—
448C2×D4—124no—
449C2×D4—122no—
450C2×D4—124no—
451C2×D4—124no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—34no—
458C22×C4—32no—
459C22×C4—32no—
460C22×C4—31no—
461C22×C4—68no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—124no—
466C22×C4—124no—
467C22×C4C23×D314yes—
468C22×C4C23×D312yes—
469C22×C4C23×D311yescharacteristic
470C22×C4C23×D314yes—
471C22×C4C23×D312yes—
472C22×C4C23×D311yescharacteristic
473C4⋊C4—24no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—38no—
477C4⋊C4—316no—
478C4⋊C4—44no—
479C4⋊C4—62no—
480C4⋊C4—68no—
481C4⋊C4—68no—
482C4⋊C4—64no—
483C4⋊C4—124no—
484C4⋊C4—122no—
485C22⋊C4—24no—
486C22⋊C4—44no—
487C22⋊C4—68no—
488C22⋊C4—62no—
489C22⋊C4—68no—
490C22⋊C4—68no—
491C22⋊C4—68no—
492C22⋊C4—68no—
493C22⋊C4—62no—
494C22⋊C4—64no—
495C22⋊C4—124no—
496C22⋊C4—124no—
497C22⋊C4—124no—
498C22⋊C4—124no—
499C22⋊C4—124no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C42—22no—
503C42—34no—
504C42—68no—
505C42—62no—
506C42—62no—
507C42—122no—
508C42—122no—
509C42—124no—
510C42D3×D414yes—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3—42no—
514C22×C3—41no—
515C22×C3—42no—
516C22×C3C23×D411yescharacteristic
517C22×C3C22×#4111yescharacteristic
518C22×C3C2×H(5,𝔽2)11yescharacteristic
519C22×C3#58512yes—
520C22×C3#58512yes—
521C2×D3—41no—
522C2×D3—44no—
523C2×D3—41no—
524C2×D3—44no—
525C2×D3—44no—
526C2×D3—44no—
527C2×D3—42no—
528C2×D3—44no—
529C2×D3—41no—
530C2×D3—42no—
531C2×D3—44no—
532C2×D3—44no—
533C3×C4—24no—
534C3×C4—41no—
535C3×C4—44no—
536C3×C4—42no—
537C3×C4—44no—
538C3×C4#55014yes—
539Dic3—22no—
540Dic3—28no—
541Dic3—22no—
542Dic3—41no—
543Dic3—42no—
544Dic3—42no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—22no—
549C23—22no—
550C23—61no—
551C23—64no—
552C23—64no—
553C23—61no—
554C23—64no—
555C23—61no—
556C23—64no—
557C23—64no—
558C23—62no—
559C23—64no—
560C23—64no—
561C23—62no—
562C23C24×D311yescharacteristic
563Q8—48no—
564D4—124no—
565D4—128no—
566D4—248no—
567D4—248no—
568D4—248no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—21no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—21no—
577C2×C4—24no—
578C2×C4—21no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—38no—
584C2×C4—32no—
585C2×C4—32no—
586C2×C4—44no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—64no—
590C2×C4—64no—
591C2×C4—62no—
592C2×C4—61no—
593C2×C4—62no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—124no—
601C2×C4—248no—
602C2×C4C2×D3×D414yes—
603C2×C4C2×D3×D412yes—
604C2×C4#100412yes—
605C2×C4D3×#4112yes—
606C2×C3—81no—
607C2×C3#345511yescharacteristic
608C2×C3C2×#55011yescharacteristic
609C2×C3#350012yes—
610C2×C3#351211yescharacteristic
611C2×C3#356212yes—
612D3—81no—
613D3—84no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—124no—
620C22—124no—
621C22—122no—
622C22—122no—
623C22—124no—
624C22—124no—
625C22—121no—
626C22—124no—
627C22—121no—
628C22—124no—
629C22—124no—
630C22—121no—
631C22C22×D3×D411yescharacteristic
632C22C2×#100411yescharacteristic
633C22C2×D3×#4111yescharacteristic
634C22#596812yes—
635C22#596812yes—
636C4—24no—
637C4—42no—
638C4—44no—
639C4—41no—
640C4—44no—
641C4—68no—
642C4—62no—
643C4—62no—
644C4—122no—
645C4—122no—
646C4—121no—
647C4#559914yes—
648C3#3938211yescharacteristic, complemented, Sylow
649C2—81no—
650C2—244no—
651C2—241no—
652C2C2×#559911yescharacteristic
653C2#8634112yes—
654C2#8635212yes—
655C2#8649511yescharacteristic
656C2#8649711yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2621C2
hypercenter#2621D3
hypocenterC3#39382
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2621

Complements

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

NH ≅ G/Ntype
1. C2×#5731C2semidirect
2. C2×#5535C2semidirect
3. C3×#2621C2semidirect
4. #83251C2semidirect
5. #82779C2semidirect
6. #82755C2semidirect
7. #82225C2semidirect
8. #77848C2semidirect
9. #77789C2semidirect
10. #77585C2semidirect
11. #77487C2semidirect
12. #76624C2semidirect
13. #76493C2semidirect
14. #74244C2semidirect
15. #74150C2semidirect
79. #2621D3semidirect
153. #870D4semidirect
648. C3#39382semidirect

Maximal subgroups

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

subgroupindex
C2×#57312
C2×#55352
C3×#26212
#832512
#827792
#827552
#822252
#778482
#777892
#775852
#774872
#766242
#764932
#742442
#741502
#393823