Finite Group Databasebeta

← #11025043

Subgroups — #11025043

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×#5698C211yesmaximal, characteristic, complemented
2C2×#5477C211yesmaximal, characteristic, complemented
3C3×#2851C211yesmaximal, characteristic, complemented
4#83067C211yesmaximal, characteristic, complemented
5#82065C211yesmaximal, characteristic, complemented
6#76218C212yesmaximal, complemented
7#76007C212yesmaximal, complemented
8#75002C214yesmaximal, complemented
9#74992C214yesmaximal, complemented
10#74530C214yesmaximal, complemented
11#74527C214yesmaximal, complemented
12#74244C211yesmaximal, characteristic, complemented
13#74152C211yesmaximal, characteristic, complemented
14#74015C212yesmaximal, complemented
15#73877C212yesmaximal, complemented
16#37701—31nomaximal, Sylow
17C22×#235C2212yes—
18C2×C3×#125C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C2×#930C2214yes—
21C2×C4×#67C2211yescharacteristic
22#5698—28no—
23C2×#898C2214yes—
24C2×#897C2212yes—
25C2×#895C2212yes—
26C2×#890C2214yes—
27C2×#887C2212yes—
28C2×#887C2212yes—
29#5477—28no—
30C2×#875C2212yes—
31C2×#874C2211yescharacteristic
32C2×C4×D12C2214yes—
33C2×#872C2211yescharacteristic
34C2×#869C2211yescharacteristic
35C3×#389C2212yes—
36C3×#387C2214yes—
37C3×#385C2214yes—
38C3×#382C2211yescharacteristic
39C3×#380C2212yes—
40#5213C2212yes—
41#5210C2212yes—
42C4×#217C2211yescharacteristic
43#4994C2214yes—
44#4993C2214yes—
45#4991C2214yes—
46#4991C2214yes—
47#4990C2214yes—
48#4988C2212yes—
49#4988C2212yes—
50#4988C2214yes—
51#4988C2212yes—
52#4973C2212yes—
53#4961C2214yes—
54#4960C2214yes—
55#4957C2214yes—
56#4957C2214yes—
57#4940C2212yes—
58C4×(C2×D3)⋊C4C2211yescharacteristic
59#4933C2212yes—
60#4932C2211yescharacteristic
61C4×Dic3⋊C4C2211yescharacteristic
62#4675C2214yes—
63#4672C2214yes—
64#4672C2214yes—
65#4671C2214yes—
66#4670C2214yes—
67#4669C2214yes—
68#4668C2214yes—
69#4667C2214yes—
70#4666C2214yes—
71#4665C2214yes—
72#4654C2214yes—
73#4654C2214yes—
74C2×#541—31no—
75C2×#532—31no—
76#2851D311yescharacteristic, complemented
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—
89C23×C3×C4C2311yescharacteristic
90C23×C3×C4C2312yes—
91C22×#67C2314yes—
92C22×#67C2311yescharacteristic
93C2×#235—28no—
94C2×#235—28no—
95C2×#235—44no—
96C22×D12C2312yes—
97C22×D12C2314yes—
98C22×D12C2314yes—
99C22×C4×D3C2311yescharacteristic
100C22×C4×D3C2314yes—
101C22×Dic6C2312yes—
102C3×#125—24no—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C4⋊C4C2314yes—
107C2×C3×C22⋊C4C2314yes—
108C2×C3×C42—24no—
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2312yes—
111C2×C3×C42C2311yescharacteristic
112C2×#217C2311yescharacteristic
113C2×#217C2312yes—
114#930—216no—
115C4×#67—24no—
116C2×(C2×D3)⋊C4C2311yescharacteristic
117C2×(C2×D3)⋊C4C2314yes—
118C2×(C2×D3)⋊C4C2314yes—
119C2×(C2×D3)⋊C4C2314yes—
120C2×(C2×D3)⋊C4C2312yes—
121C2×(C2×D3)⋊C4C2314yes—
122C2×(C2×D3)⋊C4C2314yes—
123C2×(C2×D3)⋊C4C2314yes—
124C2×(C2×D3)⋊C4C2314yes—
125C2×(C2×D3)⋊C4C2314yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2312yes—
129C2×C4⋊Dic3C2314yes—
130C2×C4⋊Dic3C2314yes—
131C2×Dic3⋊C4C2314yes—
132C2×Dic3⋊C4C2311yescharacteristic
133C2×Dic3⋊C4C2312yes—
134C2×Dic3⋊C4C2314yes—
135C2×Dic3⋊C4C2312yes—
136C2×C4×Dic3C2314yes—
137C2×C4×Dic3C2311yescharacteristic
138C2×C4×Dic3C2311yescharacteristic
139C2×C4×Dic3C2311yescharacteristic
140C2×C4×Dic3C2314yes—
141#898—216no—
142#897—28no—
143#895—44no—
144#890—216no—
145#887—28no—
146#887—44no—
147#875—28no—
148#874—24no—
149C4×D12—216no—
150#872—24no—
151#869D418yescomplemented
152C3×#95C2314yes—
153C3×#95C2314yes—
154C3×#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2312yes—
158C3⋊#95C2312yes—
159C3⋊#95C2312yes—
160C3⋊#95C2312yes—
161C3⋊#95C2312yes—
162C22×#41—32no—
163#541—68no—
164#532—68no—
165C2×C4⋊Q8—31no—
166C2×#127—31no—
167C2×#126—34no—
168C2×#125—34no—
169C2×#125C2×D311yescharacteristic
170C2×#124—32no—
171C2×C22⋊Q8—32no—
172C2×#121—34no—
173C2×#121—32no—
174C2×C4×Q8—32no—
175C2×C4×Q8—32no—
176C2×C4×D4—31no—
177C2×C4×D4—34no—
178C2×#117—31no—
179C22×C42C2×D311yescharacteristic
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—32no—
193#390—32no—
194#390—34no—
195#390—34no—
196#390—34no—
197#389C2×D312yes—
198#388—34no—
199#388—32no—
200#388—34no—
201#388—34no—
202#387—32no—
203#387—34no—
204#387—34no—
205#387C2×D314yes—
206#386—32no—
207#386—34no—
208#385—32no—
209#385—34no—
210#385C2×D314yes—
211#382C2×D311yescharacteristic
212#380—34no—
213#380C2×D312yes—
214C4×C4⋊C4—31no—
215C4×C22⋊C4—31no—
216C4×C22⋊C4—31no—
217C4×C22⋊C4—34no—
218#376—31no—
219C24×C3C2411yescharacteristic
220C23×D3C2411yescharacteristic
221C23×D3C2414yes—
222C22×C3×C4—22no—
223C22×C3×C4—22no—
224C22×C3×C4—24no—
225C22×C3×C4—42no—
226C22×C3×C4—42no—
227C22×C3×C4#4114yes—
228C22×C3×C4C2414yes—
229C22×C3×C4C2412yes—
230C22×C3×C4C2414yes—
231C22×C3×C4C2412yes—
232C22×C3×C4C2411yescharacteristic
233C22×C3×C4C2411yescharacteristic
234C2×#67—22no—
235C2×#67—28no—
236C2×#67—28no—
237C2×#67—44no—
238C2×#67—42no—
239C22×Dic3C2411yescharacteristic
240C22×Dic3C2411yescharacteristic
241C22×Dic3C2412yes—
242C22×Dic3C2412yes—
243C22×Dic3C2411yescharacteristic
244C22×Dic3C2414yes—
245#235—432no—
246C2×D12—28no—
247C2×D12—24no—
248C2×D12—24no—
249C2×D12—24no—
250C2×D12—44no—
251C2×D12—44no—
252C2×D12—44no—
253C2×D12—44no—
254C2×D12—44no—
255C2×C4×D3—22no—
256C2×C4×D3—22no—
257C2×C4×D3—28no—
258C2×C4×D3—22no—
259C2×C4×D3—28no—
260C2×C4×D3—44no—
261C2×Dic6—24no—
262C2×Dic6—24no—
263C2×Dic6C2×D418yes—
264C3×C4⋊C4—24no—
265C3×C4⋊C4—28no—
266C3×C4⋊C4—24no—
267C3×C4⋊C4—44no—
268C3×C22⋊C4—44no—
269C3×C42—24no—
270C3×C42—22no—
271C3×C42C2×D414yes—
272#217—22no—
273#217—42no—
274(C2×D3)⋊C4—24no—
275(C2×D3)⋊C4—22no—
276(C2×D3)⋊C4—28no—
277(C2×D3)⋊C4—28no—
278(C2×D3)⋊C4—28no—
279(C2×D3)⋊C4—44no—
280(C2×D3)⋊C4—44no—
281(C2×D3)⋊C4—44no—
282(C2×D3)⋊C4—42no—
283(C2×D3)⋊C4—44no—
284(C2×D3)⋊C4—44no—
285(C2×D3)⋊C4—44no—
286(C2×D3)⋊C4—44no—
287C4⋊Dic3—28no—
288C4⋊Dic3C2×D4116yes—
289Dic3⋊C4—24no—
290Dic3⋊C4—22no—
291Dic3⋊C4—28no—
292Dic3⋊C4—44no—
293Dic3⋊C4—42no—
294C4×Dic3—22no—
295C4×Dic3—22no—
296C4×Dic3—22no—
297C4×Dic3—28no—
298C4×Dic3—44no—
299C2×#41—68no—
300C2×#41—68no—
301C2×#41—124no—
302C22×Q8—32no—
303C22×D4—34no—
304C22×D4—31no—
305C22×D4—32no—
306C22×D4—34no—
307C22×D4—34no—
308C23×C4—31no—
309C23×C4—34no—
310C23×C4C22×D311yescharacteristic
311C23×C4C22×D312yes—
312C4⋊Q8—38no—
313#127—64no—
314#126—616no—
315#125—24no—
316#125—616no—
317#124—68no—
318C22⋊Q8—68no—
319#121—616no—
320#121—124no—
321C4×Q8—68no—
322C4×Q8—124no—
323C4×D4—616no—
324C4×D4—64no—
325#117—64no—
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—34no—
341C2×C22⋊C4—31no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—32no—
344C2×C22⋊C4—34no—
345C2×C22⋊C4—32no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—34no—
350C2×C22⋊C4—34no—
351C2×C22⋊C4—34no—
352C2×C22⋊C4C22×D314yes—
353C2×C42—24no—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—31no—
357C2×C42—34no—
358C2×C42—34no—
359C2×C42C22×D312yes—
360C2×C42C22×D311yescharacteristic
361C2×C42C22×D311yescharacteristic
362#95—32no—
363#95—32no—
364#95—32no—
365#95—32no—
366#95—32no—
367#95—34no—
368#95—34no—
369#95C22×D314yes—
370#95C22×D314yes—
371#95C22×D314yes—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—22no—
375C23×C3—22no—
376C23×C3C2×#4112yes—
377C23×C3C2511yescharacteristic
378C22×D3—21no—
379C22×D3—24no—
380C22×D3—21no—
381C22×D3—24no—
382C22×D3—24no—
383C22×D3—21no—
384C22×D3—24no—
385C22×D3—24no—
386C22×D3—24no—
387C22×D3—24no—
388C22×D3—22no—
389C22×D3—22no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—28no—
395C2×C3×C4—24no—
396C2×C3×C4—22no—
397C2×C3×C4—24no—
398C2×C3×C4—22no—
399C2×C3×C4—24no—
400C2×C3×C4—24no—
401C2×C3×C4—24no—
402C2×C3×C4—24no—
403C2×C3×C4—22no—
404C2×C3×C4—48no—
405C2×C3×C4C22×D414yes—
406C2×C3×C4C22×D412yes—
407C2×C3×C4C2×#4112yes—
408C2×C3×C4C2×#4112yes—
409C2×C3×C4H(5,𝔽2)12yes—
410#67—416no—
411#67—44no—
412C2×Dic3—24no—
413C2×Dic3—24no—
414C2×Dic3—22no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—22no—
420C2×Dic3—21no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3C22×D418yes—
428D12—416no—
429D12—48no—
430D12—88no—
431C4×D3—416no—
432C4×D3—44no—
433Dic6—216no—
434C24—31no—
435C24—34no—
436C24C23×D311yescharacteristic
437#41—1232no—
438C2×Q8—38no—
439C2×Q8—64no—
440C2×Q8—64no—
441C2×D4—68no—
442C2×D4—62no—
443C2×D4—68no—
444C2×D4—68no—
445C2×D4—64no—
446C2×D4—64no—
447C2×D4—64no—
448C2×D4—124no—
449C2×D4—124no—
450C2×D4—124no—
451C2×D4—124no—
452C2×D4—124no—
453C2×D4—124no—
454C2×D4—122no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—34no—
459C22×C4—31no—
460C22×C4—31no—
461C22×C4—32no—
462C22×C4—32no—
463C22×C4—31no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—68no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—68no—
470C22×C4—62no—
471C22×C4—124no—
472C22×C4#23914yes—
473C22×C4C23×D314yes—
474C22×C4C23×D312yes—
475C22×C4C23×D314yes—
476C22×C4C23×D312yes—
477C22×C4C23×D311yescharacteristic
478C22×C4C23×D311yescharacteristic
479C4⋊C4—24no—
480C4⋊C4—28no—
481C4⋊C4—24no—
482C4⋊C4—316no—
483C4⋊C4—44no—
484C4⋊C4—68no—
485C4⋊C4—68no—
486C4⋊C4—64no—
487C4⋊C4—62no—
488C4⋊C4—124no—
489C4⋊C4—122no—
490C22⋊C4—44no—
491C22⋊C4—68no—
492C22⋊C4—68no—
493C22⋊C4—68no—
494C22⋊C4—62no—
495C22⋊C4—64no—
496C22⋊C4—62no—
497C22⋊C4—124no—
498C22⋊C4—124no—
499C22⋊C4—124no—
500C22⋊C4—124no—
501C22⋊C4—124no—
502C22⋊C4—124no—
503C22⋊C4—124no—
504C22⋊C4—122no—
505C22⋊C4—122no—
506C42—24no—
507C42—22no—
508C42—68no—
509C42—62no—
510C42—62no—
511C42—62no—
512C42—124no—
513C42D3×D414yes—
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)11yescharacteristic
522C22×C3#58512yes—
523C22×C3#58512yes—
524C2×D3—41no—
525C2×D3—44no—
526C2×D3—44no—
527C2×D3—41no—
528C2×D3—44no—
529C2×D3—44no—
530C2×D3—42no—
531C2×D3—41no—
532C2×D3—44no—
533C2×D3—42no—
534C2×D3—44no—
535C2×D3—44no—
536C3×C4—22no—
537C3×C4—24no—
538C3×C4—44no—
539C3×C4—42no—
540C3×C4—44no—
541C3×C4#54614yes—
542Dic3—28no—
543Dic3—41no—
544Dic3—41no—
545Dic3—42no—
546Dic3—41no—
547Dic3—42no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—22no—
552C23—61no—
553C23—64no—
554C23—61no—
555C23—64no—
556C23—64no—
557C23—61no—
558C23—64no—
559C23—64no—
560C23—64no—
561C23—64no—
562C23—62no—
563C23—62no—
564C23C2×#23912yes—
565C23C24×D311yescharacteristic
566Q8—616no—
567D4—1216no—
568D4—1216no—
569D4—124no—
570D4—128no—
571D4—248no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—38no—
587C2×C4—48no—
588C2×C4—64no—
589C2×C4—64no—
590C2×C4—62no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—62no—
596C2×C4—61no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—1216no—
604C2×C4—124no—
605C2×C4C2×D3×D414yes—
606C2×C4C2×D3×D412yes—
607C2×C4C2×#23912yes—
608C2×C4C2×#23912yes—
609C2×C4#100912yes—
610C2×C3—42no—
611C2×C3#345511yescharacteristic
612C2×C3C2×#54611yescharacteristic
613C2×C3#348712yes—
614C2×C3#351211yescharacteristic
615C2×C3#356212yes—
616D3—81no—
617D3—84no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—42no—
622C22—42no—
623C22—121no—
624C22—124no—
625C22—124no—
626C22—124no—
627C22—122no—
628C22—121no—
629C22—124no—
630C22—122no—
631C22—124no—
632C22—124no—
633C22—124no—
634C22—121no—
635C22C22×D3×D411yescharacteristic
636C22C22×#23911yescharacteristic
637C22C2×#100911yescharacteristic
638C22#596412yes—
639C22#596412yes—
640C4—24no—
641C4—22no—
642C4—42no—
643C4—44no—
644C4—44no—
645C4—68no—
646C4—121no—
647C4—122no—
648C4—121no—
649C4—122no—
650C4—121no—
651C4#563014yes—
652C3#3770111yescharacteristic, complemented, Sylow
653C2—42no—
654C2—241no—
655C2—244no—
656C2C2×#563011yescharacteristic
657C2#8602312yes—
658C2#8608711yescharacteristic
659C2#8618012yes—
660C2#8627311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2851C2
hypercenter#2851D3
hypocenterC3#37701
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2851

Complements

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

NH ≅ G/Ntype
1. C2×#5698C2semidirect
2. C2×#5477C2semidirect
3. C3×#2851C2semidirect
4. #83067C2semidirect
5. #82065C2semidirect
6. #76218C2semidirect
7. #76007C2semidirect
8. #75002C2semidirect
9. #74992C2semidirect
10. #74530C2semidirect
11. #74527C2semidirect
12. #74244C2semidirect
13. #74152C2semidirect
14. #74015C2semidirect
15. #73877C2semidirect
76. #2851D3semidirect
151. #869D4semidirect
652. C3#37701semidirect

Maximal subgroups

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

subgroupindex
C2×#56982
C2×#54772
C3×#28512
#830672
#820652
#762182
#760072
#750022
#749922
#745302
#745272
#742442
#741522
#740152
#738772
#377013