Finite Group Databasebeta

← #11069610

Subgroups — #11069610

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×#5698C211yesmaximal, characteristic, complemented
2C2×#5697C211yesmaximal, characteristic, complemented
3C3×C4×#125C211yesmaximal, characteristic, complemented
4#76219C212yesmaximal, complemented
5#76218C212yesmaximal, complemented
6#75160C214yesmaximal, complemented
7#75151C214yesmaximal, complemented
8#74244C211yesmaximal, characteristic, complemented
9#74193C211yesmaximal, characteristic, complemented
10#73803C214yesmaximal, complemented
11#73798C212yesmaximal, complemented
12#73787C214yesmaximal, complemented
13#73654C212yesmaximal, complemented
14#73313C211yesmaximal, characteristic, complemented
15#73299C211yesmaximal, characteristic, complemented
16#39382—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18#5698—28no—
19#5697—28no—
20C2×#898C2214yes—
21C2×#897C2212yes—
22C2×#897C2212yes—
23C2×#895C2214yes—
24C2×#895C2212yes—
25C2×#895C2212yes—
26C2×#894C2214yes—
27C2×#893C2212yes—
28C2×#890C2214yes—
29C2×#887C2212yes—
30C2×#875C2212yes—
31C2×#875C2212yes—
32C2×#874C2211yescharacteristic
33C2×C4×D12C2211yescharacteristic
34C2×#872C2211yescharacteristic
35C2×#872C2211yescharacteristic
36C2×#869C2211yescharacteristic
37C3×#389C2212yes—
38C3×#387C2214yes—
39C3×#382C2211yescharacteristic
40C3×C4×C4⋊C4C2214yes—
41C3×C4×C4⋊C4C2212yes—
42C3×C43C2211yescharacteristic
43#4994C2214yes—
44#4991C2214yes—
45#4988C2212yes—
46#4988C2214yes—
47#4987C2214yes—
48#4973C2212yes—
49#4940C2212yes—
50#4940C2214yes—
51#4940C2212yes—
52#4939C2214yes—
53#4939C2214yes—
54#4939C2214yes—
55#4939C2212yes—
56#4939C2214yes—
57C4×(C2×D3)⋊C4C2214yes—
58C4×(C2×D3)⋊C4C2211yescharacteristic
59C4×(C2×D3)⋊C4C2211yescharacteristic
60C4×C4⋊Dic3C2211yescharacteristic
61#4928C2212yes—
62#4677C2214yes—
63#4674C2214yes—
64#4674C2214yes—
65#4672C2214yes—
66#4672C2214yes—
67#4671C2214yes—
68#4671C2214yes—
69#4670C2214yes—
70#4670C2214yes—
71#4668C2214yes—
72#4665C2214yes—
73#4661C2214yes—
74C2×#541—31no—
75C2×#540—31no—
76#2858—31no—
77#2714—34no—
78#2694—34no—
79#2621—31no—
80#2619—31no—
81#2554—34no—
82#2540—32no—
83#2490—34no—
84#2427—32no—
85#2426—32no—
86#2367—32no—
87C4×#125D311yescharacteristic, complemented
88#2293—31no—
89C22×D12C2312yes—
90C22×D12C2311yescharacteristic
91C22×D12C2314yes—
92C22×D12C2314yes—
93C22×D12C2314yes—
94C22×C4×D3C2311yescharacteristic
95C22×C4×D3C2311yescharacteristic
96C22×C4×D3C2314yes—
97C22×Dic6C2312yes—
98C3×#125D418yescomplemented
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2314yes—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2314yes—
103C2×C3×C42C2311yescharacteristic
104C2×C3×C42C2312yes—
105C2×C3×C42C2314yes—
106C2×C3×C42C2312yes—
107C2×C3×C42C2311yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C2×(C2×D3)⋊C4C2314yes—
110C2×(C2×D3)⋊C4C2314yes—
111C2×(C2×D3)⋊C4C2314yes—
112C2×(C2×D3)⋊C4C2314yes—
113C2×(C2×D3)⋊C4C2312yes—
114C2×(C2×D3)⋊C4C2312yes—
115C2×(C2×D3)⋊C4C2314yes—
116C2×(C2×D3)⋊C4C2311yescharacteristic
117C2×(C2×D3)⋊C4C2311yescharacteristic
118C2×(C2×D3)⋊C4C2312yes—
119C2×(C2×D3)⋊C4C2314yes—
120C2×(C2×D3)⋊C4C2314yes—
121C2×(C2×D3)⋊C4C2314yes—
122C2×(C2×D3)⋊C4C2314yes—
123C2×(C2×D3)⋊C4C2314yes—
124C2×(C2×D3)⋊C4C2314yes—
125C2×(C2×D3)⋊C4C2312yes—
126C2×C4⋊Dic3C2314yes—
127C2×C4⋊Dic3C2311yescharacteristic
128C2×C4⋊Dic3C2314yes—
129C2×C4⋊Dic3C2314yes—
130C2×C4⋊Dic3C2312yes—
131C2×Dic3⋊C4C2312yes—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2312yes—
134C2×C4×Dic3C2314yes—
135C2×C4×Dic3C2311yescharacteristic
136C2×C4×Dic3C2311yescharacteristic
137#898—216no—
138#897—28no—
139#897—44no—
140#895—216no—
141#895—28no—
142#895—44no—
143#894—216no—
144#893—28no—
145#890—216no—
146#887—28no—
147#875—28no—
148#875—44no—
149#874—24no—
150C4×D12—24no—
151#872—24no—
152#872—24no—
153#869—24no—
154C3×#95C2314yes—
155C3×#95C2314yes—
156C3×#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2312yes—
160C3⋊#95C2312yes—
161C3⋊#95C2312yes—
162#541—68no—
163#540—68no—
164C2×C4⋊Q8—31no—
165C2×#127—31no—
166C2×#126—34no—
167C2×#125—34no—
168C2×#125C2×D311yescharacteristic
169C2×#124—32no—
170C2×#124—32no—
171C2×#123—34no—
172C2×C22⋊Q8—32no—
173C2×C22⋊Q8—32no—
174C2×#121—32no—
175C2×#121—34no—
176C2×#121—32no—
177C2×C4×Q8—32no—
178C2×C4×D4—32no—
179C2×C4×D4—31no—
180C2×#117—31no—
181C2×#117—31no—
182#399—34no—
183#398—34no—
184#397—34no—
185#397—34no—
186#396—34no—
187#394—34no—
188#394—34no—
189#394—34no—
190#392—34no—
191#392—34no—
192#391—32no—
193#390—32no—
194#390—34no—
195#390—34no—
196#390—34no—
197#390—32no—
198#390—34no—
199#390—34no—
200#389—32no—
201#389C2×D312yes—
202#388—34no—
203#388—32no—
204#388—32no—
205#388—34no—
206#388—34no—
207#387C2×D314yes—
208#386—34no—
209#385—34no—
210#382C2×D311yescharacteristic
211#380—34no—
212C4×C4⋊C4—31no—
213C4×C4⋊C4C2×D312yes—
214C4×C4⋊C4C2×D314yes—
215C4×C22⋊C4—34no—
216C4×C22⋊C4—31no—
217C4×C22⋊C4—31no—
218C43C2×D311yescharacteristic
219C23×D3C2411yescharacteristic
220C23×D3C2414yes—
221C23×D3C2411yescharacteristic
222C22×C3×C4C2414yes—
223C22×C3×C4C2412yes—
224C22×C3×C4C2414yes—
225C22×C3×C4C2412yes—
226C22×C3×C4C2411yescharacteristic
227C22×C3×C4C2411yescharacteristic
228C22×C3×C4C2411yescharacteristic
229C22×Dic3C2412yes—
230C22×Dic3C2412yes—
231C22×Dic3C2414yes—
232C22×Dic3C2411yescharacteristic
233C22×Dic3C2411yescharacteristic
234C2×D12—24no—
235C2×D12—22no—
236C2×D12—24no—
237C2×D12—24no—
238C2×D12—28no—
239C2×D12—28no—
240C2×D12—42no—
241C2×D12—44no—
242C2×D12—44no—
243C2×D12—44no—
244C2×D12—44no—
245C2×D12—44no—
246C2×D12—44no—
247C2×D12—44no—
248C2×C4×D3—22no—
249C2×C4×D3—22no—
250C2×C4×D3—22no—
251C2×C4×D3—22no—
252C2×C4×D3—22no—
253C2×C4×D3—22no—
254C2×C4×D3—28no—
255C2×C4×D3—44no—
256C2×C4×D3—44no—
257C2×Dic6—24no—
258C2×Dic6—24no—
259C2×Dic6—24no—
260C3×C4⋊C4—24no—
261C3×C4⋊C4—44no—
262C3×C4⋊C4C2×D418yes—
263C3×C4⋊C4C2×D4116yes—
264C3×C42—22no—
265C3×C42—42no—
266C3×C42—44no—
267C3×C42—42no—
268C3×C42C2×D414yes—
269C3×C42#4114yes—
270(C2×D3)⋊C4—24no—
271(C2×D3)⋊C4—24no—
272(C2×D3)⋊C4—28no—
273(C2×D3)⋊C4—28no—
274(C2×D3)⋊C4—28no—
275(C2×D3)⋊C4—22no—
276(C2×D3)⋊C4—22no—
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—44no—
283(C2×D3)⋊C4—44no—
284(C2×D3)⋊C4—42no—
285(C2×D3)⋊C4—44no—
286(C2×D3)⋊C4—42no—
287C4⋊Dic3—28no—
288C4⋊Dic3—28no—
289C4⋊Dic3—22no—
290C4⋊Dic3—44no—
291C4⋊Dic3—42no—
292Dic3⋊C4—24no—
293Dic3⋊C4—24no—
294Dic3⋊C4—28no—
295C4×Dic3—22no—
296C4×Dic3—22no—
297C4×Dic3—28no—
298C22×Q8—32no—
299C22×D4—32no—
300C22×D4—31no—
301C22×D4—34no—
302C22×D4—34no—
303C22×D4—34no—
304C23×C4—31no—
305C23×C4—31no—
306C23×C4—34no—
307C4⋊Q8—64no—
308#127—64no—
309#126—616no—
310#125—616no—
311#125D1218yescomplemented
312#124—68no—
313#124—124no—
314#123—616no—
315C22⋊Q8—68no—
316C22⋊Q8—124no—
317#121—616no—
318#121—68no—
319#121—124no—
320C4×Q8—68no—
321C4×D4—68no—
322C4×D4—64no—
323#117—64no—
324#117—64no—
325C2×C4⋊C4—32no—
326C2×C4⋊C4—32no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—34no—
329C2×C4⋊C4—34no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—32no—
333C2×C4⋊C4C22×D312yes—
334C2×C4⋊C4C22×D314yes—
335C2×C4⋊C4C22×D312yes—
336C2×C4⋊C4C22×D314yes—
337C2×C22⋊C4—32no—
338C2×C22⋊C4—32no—
339C2×C22⋊C4—34no—
340C2×C22⋊C4—31no—
341C2×C22⋊C4—31no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—34no—
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—32no—
351C2×C22⋊C4—32no—
352C2×C22⋊C4—34no—
353C2×C22⋊C4—34no—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—34no—
357C2×C42C22×D311yescharacteristic
358C2×C42C22×D312yes—
359C2×C42C22×D312yes—
360C2×C42C22×D314yes—
361C2×C42C22×D311yescharacteristic
362C2×C42C22×D311yescharacteristic
363#95—32no—
364#95—32no—
365#95—34no—
366#95—32no—
367#95—34no—
368#95C22×D314yes—
369#95C22×D314yes—
370#95C22×D314yes—
371C23×C3C2511yescharacteristic
372C22×D3—21no—
373C22×D3—21no—
374C22×D3—24no—
375C22×D3—24no—
376C22×D3—21no—
377C22×D3—21no—
378C22×D3—21no—
379C22×D3—24no—
380C22×D3—21no—
381C22×D3—24no—
382C22×D3—24no—
383C22×D3—22no—
384C22×D3—22no—
385C22×D3—24no—
386C22×D3—22no—
387C22×D3—22no—
388C22×D3—24no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—24no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—24no—
395C2×C3×C4—24no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—24no—
399C2×C3×C4—24no—
400C2×C3×C4—22no—
401C2×C3×C4C22×D418yes—
402C2×C3×C4C22×D414yes—
403C2×C3×C4C22×D412yes—
404C2×C3×C4C2×#4112yes—
405C2×C3×C4C2×#4112yes—
406C2×C3×C4C2×#4112yes—
407C2×C3×C4H(5,𝔽2)12yes—
408C2×Dic3—22no—
409C2×Dic3—24no—
410C2×Dic3—22no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—24no—
416C2×Dic3—21no—
417C2×Dic3—24no—
418C2×Dic3—21no—
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423D12—44no—
424D12—48no—
425D12—88no—
426D12—88no—
427D12—88no—
428C4×D3—44no—
429C4×D3—44no—
430C4×D3—88no—
431Dic6—48no—
432C24—31no—
433C24—34no—
434C24—31no—
435C2×Q8—64no—
436C2×Q8—64no—
437C2×Q8—64no—
438C2×D4—64no—
439C2×D4—68no—
440C2×D4—68no—
441C2×D4—62no—
442C2×D4—64no—
443C2×D4—64no—
444C2×D4—122no—
445C2×D4—124no—
446C2×D4—124no—
447C2×D4—124no—
448C2×D4—124no—
449C2×D4—124no—
450C2×D4—124no—
451C2×D4—124no—
452C22×C4—32no—
453C22×C4—32no—
454C22×C4—34no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—68no—
464C22×C4—124no—
465C22×C4—124no—
466C22×C4C23×D314yes—
467C22×C4C23×D312yes—
468C22×C4C23×D314yes—
469C22×C4C23×D312yes—
470C22×C4C23×D311yescharacteristic
471C22×C4C23×D311yescharacteristic
472C22×C4C23×D311yescharacteristic
473C4⋊C4—24no—
474C4⋊C4—44no—
475C4⋊C4—64no—
476C4⋊C4—64no—
477C4⋊C4—68no—
478C4⋊C4—68no—
479C4⋊C4—68no—
480C4⋊C4—62no—
481C4⋊C4—124no—
482C4⋊C4—122no—
483C4⋊C4C2×D1218yes—
484C4⋊C4C2×D12116yes—
485C22⋊C4—64no—
486C22⋊C4—64no—
487C22⋊C4—68no—
488C22⋊C4—68no—
489C22⋊C4—68no—
490C22⋊C4—62no—
491C22⋊C4—62no—
492C22⋊C4—68no—
493C22⋊C4—68no—
494C22⋊C4—124no—
495C22⋊C4—124no—
496C22⋊C4—124no—
497C22⋊C4—124no—
498C22⋊C4—122no—
499C22⋊C4—122no—
500C22⋊C4—124no—
501C22⋊C4—124no—
502C42—22no—
503C42—44no—
504C42—42no—
505C42—42no—
506C42—62no—
507C42—62no—
508C42—68no—
509C42C2×D1214yes—
510C42#23914yes—
511C22×C3C23×D411yescharacteristic
512C22×C3C22×#4111yescharacteristic
513C22×C3C2×H(5,𝔽2)11yescharacteristic
514C22×C3#58512yes—
515C22×C3#58512yes—
516C2×D3—41no—
517C2×D3—41no—
518C2×D3—44no—
519C2×D3—41no—
520C2×D3—41no—
521C2×D3—44no—
522C2×D3—44no—
523C2×D3—42no—
524C2×D3—44no—
525C2×D3—42no—
526C2×D3—44no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3—42no—
530C2×D3—44no—
531C2×D3—42no—
532C2×D3—44no—
533C3×C4—22no—
534C3×C4—28no—
535C3×C4—24no—
536C3×C4—22no—
537C3×C4—44no—
538C3×C4—42no—
539C3×C4#55014yes—
540Dic3—44no—
541Dic3—41no—
542Dic3—41no—
543Dic3—42no—
544Dic3—42no—
545C23—61no—
546C23—61no—
547C23—64no—
548C23—64no—
549C23—61no—
550C23—61no—
551C23—61no—
552C23—64no—
553C23—61no—
554C23—64no—
555C23—64no—
556C23—62no—
557C23—62no—
558C23—64no—
559C23—62no—
560C23—62no—
561C23—64no—
562C23C24×D311yescharacteristic
563Q8—128no—
564D4—124no—
565D4—128no—
566D4—248no—
567D4—248no—
568D4—248no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—64no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—64no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—64no—
591C2×C4—61no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—124no—
597C2×C4—124no—
598C2×C4—248no—
599C2×C4C22×D1218yes—
600C2×C4C22×D1214yes—
601C2×C4C22×D1212yes—
602C2×C4C2×#23912yes—
603C2×C4C2×#23912yes—
604C2×C4C2×#23912yes—
605C2×C4#100912yes—
606C2×C3#345511yescharacteristic
607C2×C3C2×#55011yescharacteristic
608C2×C3#350012yes—
609C2×C3#351211yescharacteristic
610C2×C3#356212yes—
611D3—81no—
612D3—81no—
613D3—84no—
614C22—121no—
615C22—121no—
616C22—124no—
617C22—121no—
618C22—121no—
619C22—124no—
620C22—124no—
621C22—121no—
622C22—121no—
623C22—124no—
624C22—122no—
625C22—124no—
626C22—122no—
627C22—122no—
628C22—124no—
629C22—122no—
630C22—124no—
631C22C23×D1211yescharacteristic
632C22C22×#23911yescharacteristic
633C22C2×#100911yescharacteristic
634C22#596412yes—
635C22#596412yes—
636C4—22no—
637C4—22no—
638C4—28no—
639C4—24no—
640C4—44no—
641C4—42no—
642C4—121no—
643C4—121no—
644C4—124no—
645C4—122no—
646C4—122no—
647C4#557714yes—
648C3#3938211yescharacteristic, complemented, Sylow
649C2—241no—
650C2—241no—
651C2—244no—
652C2C2×#557711yescharacteristic
653C2#8584712yes—
654C2#8608711yescharacteristic
655C2#8618012yes—
656C2#8627311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×C4×#125C2
hypercenterC4×#125D3
hypocenterC3#39382
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×#125

Complements

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

NH ≅ G/Ntype
1. C2×#5698C2semidirect
2. C2×#5697C2semidirect
3. C3×C4×#125C2semidirect
4. #76219C2semidirect
5. #76218C2semidirect
6. #75160C2semidirect
7. #75151C2semidirect
8. #74244C2semidirect
9. #74193C2semidirect
10. #73803C2semidirect
11. #73798C2semidirect
12. #73787C2semidirect
13. #73654C2semidirect
14. #73313C2semidirect
15. #73299C2semidirect
87. C4×#125D3semidirect
98. C3×#125D4semidirect
311. #125D12semidirect
648. C3#39382semidirect

Maximal subgroups

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

subgroupindex
C2×#56982
C2×#56972
C3×C4×#1252
#762192
#762182
#751602
#751512
#742442
#741932
#738032
#737982
#737872
#736542
#733132
#732992
#393823