Finite Group Databasebeta

← #10972892

Subgroups — #10972892

Order 768.

2230 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5578C211yesmaximal, characteristic
2C2×C4×D3×Q8C211yesmaximal, characteristic
3C3×#2581C211yesmaximal, characteristic, complemented
4#82481C211yesmaximal, characteristic, complemented
5#82480C211yesmaximal, characteristic, complemented
6#78210C212yesmaximal, complemented
7#78152C212yesmaximal
8#77204C212yesmaximal, complemented
9#77173C212yesmaximal
10#76663C212yesmaximal
11#76545C212yesmaximal, complemented
12#76287C212yesmaximal
13#76060C212yesmaximal, complemented
14#75335C214yesmaximal
15#74948C214yesmaximal, complemented
16#74191C211yesmaximal, characteristic, complemented
17#74147C211yesmaximal, characteristic
18#35248—31nomaximal, Sylow
19C22×D3×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×#946C2211yescharacteristic
23C2×Q8×Dic3C2211yescharacteristic
24C2×#944C2211yescharacteristic
25#5578—28no—
26C4×D3×Q8—28no—
27C2×#897C2211yescharacteristic
28C2×#896C2212yes—
29C2×D3×C4⋊C4C2212yes—
30C2×D3×C4⋊C4C2211yescharacteristic
31C2×#889C2212yes—
32C2×#888C2211yescharacteristic
33C2×#888C2214yes—
34C2×#887C2211yescharacteristic
35C2×#887C2212yes—
36C2×#872C2212yes—
37C2×C42×D3C2212yes—
38C2×C42×D3C2211yescharacteristic
39C2×C4×Dic6C2212yes—
40C2×C4×Dic6C2211yescharacteristic
41C2×C4×Dic6C2211yescharacteristic
42C2×C4×Dic6C2212yes—
43C3×#400C2212yes—
44C3×#395C2212yes—
45C3×#391C2212yes—
46C3×#389C2212yes—
47C3×#387C2214yes—
48C3×#382C2211yescharacteristic
49#5230C2212yes—
50#5229C2212yes—
51#4989C2214yes—
52#4987C2212yes—
53#4986C2212yes—
54#4985C2212yes—
55#4984C2212yes—
56#4982C2214yes—
57#4980C2214yes—
58#4979C2212yes—
59#4978C2214yes—
60Dic3×C4⋊C4C2212yes—
61#4973C2212yes—
62#4972C2212yes—
63#4936C2211yescharacteristic
64#4933C2214yes—
65#4929C2212yes—
66#4928C2212yes—
67#4668C2214yes—
68#4666C2212yes—
69#4664C2212yes—
70#4664C2214yes—
71#4664C2214yes—
72#4660C2214yes—
73#4659C2212yes—
74#4658C2214yes—
75#4657C2214yes—
76#4656C2212yes—
77#4655C2214yes—
78#4654C2214yes—
79#4653C2214yes—
80#4652C2214yes—
81#4651C2212yes—
82#4650C2214yes—
83#4648C2212yes—
84#4646C2214yes—
85#4646C2214yes—
86C2×#533—31no—
87C22×C4×Q8—31no—
88#2855—32no—
89#2855—32no—
90#2634—32no—
91#2634—32no—
92#2581—31no—
93#2581—31no—
94#2581—31no—
95#2581D311yescharacteristic, complemented
96#2568—32no—
97#2546—32no—
98#2503—32no—
99#2484—34no—
100#2434—32no—
101#2327—34no—
102#2283—31no—
103C22×C3×Q8C2311yescharacteristic
104C22×C3×Q8C2311yescharacteristic
105C2×D3×Q8—24no—
106C2×D3×Q8—28no—
107C22×C4×D3C2312yes—
108C22×C4×D3C2312yes—
109C22×C4×D3C2311yescharacteristic
110C22×C4×D3C2311yescharacteristic
111C22×C4×D3C2311yescharacteristic
112C22×Dic6C2312yes—
113C22×Dic6C2311yescharacteristic
114C22×Dic6C2311yescharacteristic
115C22×Dic6C2312yes—
116C3×C4×Q8—24no—
117C3×C4×Q8—24no—
118C2×C3×C4⋊C4C2311yescharacteristic
119C2×C3×C4⋊C4C2312yes—
120C2×C3×C4⋊C4C2312yes—
121C2×C3×C4⋊C4C2312yes—
122C2×C3×C4⋊C4C2314yes—
123C2×C3×C4⋊C4C2311yescharacteristic
124C2×C3×C42C2312yes—
125C2×C3×C42C2312yes—
126C2×C3×C42C2311yescharacteristic
127#946—24no—
128Q8×Dic3—24no—
129#944—24no—
130C2×(C2×D3)⋊C4C2314yes—
131C2×(C2×D3)⋊C4C2312yes—
132C2×(C2×D3)⋊C4C2312yes—
133C2×C4⋊Dic3C2314yes—
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2311yescharacteristic
136C2×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2311yescharacteristic
138C2×C4⋊Dic3C2312yes—
139C2×Dic3⋊C4C2312yes—
140C2×Dic3⋊C4C2312yes—
141C2×Dic3⋊C4C2312yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2312yes—
144C2×Dic3⋊C4C2314yes—
145C2×Dic3⋊C4C2312yes—
146C2×Dic3⋊C4C2312yes—
147C2×Dic3⋊C4C2312yes—
148C2×Dic3⋊C4C2312yes—
149C2×C4×Dic3C2314yes—
150C2×C4×Dic3C2312yes—
151C2×C4×Dic3C2312yes—
152C2×C4×Dic3C2312yes—
153C2×C4×Dic3C2312yes—
154C2×C4×Dic3C2311yescharacteristic
155C2×C4×Dic3C2311yescharacteristic
156C2×C4×Dic3C2311yescharacteristic
157#897—24no—
158#896—28no—
159D3×C4⋊C4—28no—
160D3×C4⋊C4—24no—
161#889—28no—
162#888—216no—
163#888—24no—
164#887—24no—
165#887—28no—
166#872—28no—
167C42×D3—28no—
168C42×D3D418yes—
169C4×Dic6—24no—
170C4×Dic6—24no—
171C4×Dic6—28no—
172C4×Dic6—28no—
173C3×#95C2314yes—
174C3×#95C2314yes—
175C3×#95C2312yes—
176C3×#95C2314yes—
177C3×#95C2312yes—
178C3⋊#95C2314yes—
179C3⋊#95C2314yes—
180C3⋊#95C2314yes—
181C3⋊#95C2314yes—
182C3⋊#95C2314yes—
183C3⋊#95C2314yes—
184C3⋊#95C2314yes—
185C3⋊#95C2312yes—
186C3⋊#95C2314yes—
187C3⋊#95C2314yes—
188C3⋊#95C2314yes—
189C3⋊#95C2314yes—
190C3⋊#95C2312yes—
191C23×Q8—31no—
192#533—68no—
193C2×C4⋊Q8—31no—
194C2×C4⋊Q8—34no—
195C2×C4⋊Q8—31no—
196C2×#125—32no—
197C2×C22⋊Q8—31no—
198C2×C22⋊Q8—31no—
199C2×C22⋊Q8—32no—
200C2×C4×Q8—31no—
201C2×C4×Q8—32no—
202C2×C4×Q8—31no—
203C2×C4×Q8—32no—
204C2×C4×Q8—31no—
205C2×C4×Q8—31no—
206C2×C4×Q8—32no—
207C2×C4×Q8—68no—
208C2×C4×Q8C2×D311yescharacteristic
209C2×C4×Q8C2×D311yescharacteristic
210C2×#117—32no—
211C22×C4⋊C4—32no—
212C22×C4⋊C4—31no—
213C22×C42—31no—
214C22×C42—32no—
215#400—32no—
216#400—32no—
217#400—34no—
218#400—34no—
219#400—32no—
220#400C2×D312yes—
221#398—34no—
222#398—34no—
223#395—32no—
224#395—32no—
225#395—32no—
226#395C2×D312yes—
227#391—32no—
228#391—32no—
229#391—32no—
230#391—34no—
231#391—34no—
232#391C2×D312yes—
233#389—32no—
234#389—32no—
235#389—32no—
236#389C2×D312yes—
237#387—34no—
238#387—34no—
239#387—34no—
240#387—34no—
241#387—34no—
242#387—34no—
243#387—34no—
244#387C2×D314yes—
245#385—34no—
246#385—34no—
247#383—32no—
248#382—31no—
249#382—34no—
250#382C2×D311yescharacteristic
251#381—34no—
252#381—34no—
253#381—32no—
254#380—32no—
255C4×C4⋊C4—32no—
256C4×C22⋊C4—32no—
257#376—32no—
258C23×D3C2411yescharacteristic
259C2×C3×Q8—22no—
260C2×C3×Q8—22no—
261C2×C3×Q8—24no—
262C2×C3×Q8—22no—
263C2×C3×Q8—22no—
264C22×C3×C4C2414yes—
265C22×C3×C4C2412yes—
266C22×C3×C4C2412yes—
267C22×C3×C4C2412yes—
268C22×C3×C4C2411yescharacteristic
269C22×C3×C4C2411yescharacteristic
270C22×C3×C4C2412yes—
271C22×C3×C4C2411yescharacteristic
272C22×Dic3C2414yes—
273C22×Dic3C2412yes—
274C22×Dic3C2412yes—
275C22×Dic3C2412yes—
276C22×Dic3C2411yescharacteristic
277C22×Dic3C2411yescharacteristic
278C22×Dic3C2412yes—
279C22×Dic3C2411yescharacteristic
280D3×Q8—232no—
281C2×C4×D3—24no—
282C2×C4×D3—24no—
283C2×C4×D3—24no—
284C2×C4×D3—22no—
285C2×C4×D3—22no—
286C2×C4×D3—24no—
287C2×C4×D3—24no—
288C2×C4×D3—24no—
289C2×C4×D3—28no—
290C2×C4×D3C2×D414yes—
291C2×C4×D3C2×D414yes—
292C2×C4×D3C2×D414yes—
293C2×Dic6—22no—
294C2×Dic6—24no—
295C2×Dic6—24no—
296C2×Dic6—24no—
297C2×Dic6—22no—
298C2×Dic6—22no—
299C2×Dic6—22no—
300C2×Dic6—24no—
301C2×Dic6—28no—
302C2×Dic6#4118yes—
303C3×C4⋊C4—24no—
304C3×C4⋊C4—22no—
305C3×C4⋊C4—22no—
306C3×C4⋊C4—24no—
307C3×C4⋊C4—28no—
308C3×C4⋊C4—24no—
309C3×C42—24no—
310C3×C42—24no—
311C3×C42C2×D414yes—
312(C2×D3)⋊C4—24no—
313(C2×D3)⋊C4—24no—
314(C2×D3)⋊C4—28no—
315C4⋊Dic3—22no—
316C4⋊Dic3—22no—
317C4⋊Dic3—24no—
318C4⋊Dic3—24no—
319C4⋊Dic3—24no—
320C4⋊Dic3—28no—
321Dic3⋊C4—24no—
322Dic3⋊C4—24no—
323Dic3⋊C4—28no—
324Dic3⋊C4—24no—
325Dic3⋊C4—24no—
326Dic3⋊C4—24no—
327Dic3⋊C4—24no—
328Dic3⋊C4—24no—
329Dic3⋊C4—24no—
330Dic3⋊C4—28no—
331C4×Dic3—24no—
332C4×Dic3—24no—
333C4×Dic3—24no—
334C4×Dic3—24no—
335C4×Dic3—28no—
336C4×Dic3C2×D414yes—
337C4×Dic3C2×D414yes—
338C4×Dic3C2×D414yes—
339C22×Q8—32no—
340C22×Q8—31no—
341C22×Q8—32no—
342C22×Q8—31no—
343C22×Q8—64no—
344C22×Q8—68no—
345C22×Q8C22×D311yescharacteristic
346C22×Q8C22×D311yescharacteristic
347C23×C4—31no—
348C23×C4—31no—
349C23×C4—31no—
350C23×C4—32no—
351C23×C4—32no—
352C4⋊Q8—64no—
353C4⋊Q8—64no—
354C4⋊Q8—616no—
355#125—68no—
356C22⋊Q8—68no—
357C22⋊Q8—64no—
358C22⋊Q8—64no—
359C4×Q8—24no—
360C4×Q8—24no—
361C4×Q8—68no—
362C4×Q8—68no—
363C4×Q8—68no—
364C4×Q8—64no—
365C4×Q8—64no—
366C4×Q8—64no—
367C4×Q8—64no—
368#117—68no—
369C2×C4⋊C4—32no—
370C2×C4⋊C4—32no—
371C2×C4⋊C4—32no—
372C2×C4⋊C4—32no—
373C2×C4⋊C4—31no—
374C2×C4⋊C4—32no—
375C2×C4⋊C4—32no—
376C2×C4⋊C4—32no—
377C2×C4⋊C4—32no—
378C2×C4⋊C4—32no—
379C2×C4⋊C4—32no—
380C2×C4⋊C4—31no—
381C2×C4⋊C4—34no—
382C2×C4⋊C4—34no—
383C2×C4⋊C4—34no—
384C2×C4⋊C4—32no—
385C2×C4⋊C4—64no—
386C2×C4⋊C4—68no—
387C2×C4⋊C4C22×D312yes—
388C2×C4⋊C4C22×D311yescharacteristic
389C2×C4⋊C4C22×D312yes—
390C2×C4⋊C4C22×D311yescharacteristic
391C2×C4⋊C4C22×D314yes—
392C2×C4⋊C4C22×D312yes—
393C2×C22⋊C4—32no—
394C2×C22⋊C4—32no—
395C2×C22⋊C4—34no—
396C2×C42—31no—
397C2×C42—31no—
398C2×C42—31no—
399C2×C42—38no—
400C2×C42—32no—
401C2×C42—32no—
402C2×C42—32no—
403C2×C42—32no—
404C2×C42—34no—
405C2×C42—68no—
406C2×C42C22×D312yes—
407C2×C42C22×D312yes—
408C2×C42C22×D311yescharacteristic
409#95—34no—
410#95—32no—
411#95—32no—
412#95—34no—
413#95—34no—
414#95—34no—
415#95—34no—
416#95—34no—
417#95—34no—
418#95—34no—
419#95—34no—
420#95—34no—
421#95—34no—
422#95C22×D312yes—
423#95C22×D312yes—
424#95C22×D314yes—
425#95C22×D314yes—
426#95C22×D314yes—
427C23×C3C2511yescharacteristic
428C22×D3—22no—
429C22×D3—21no—
430C22×D3—21no—
431C22×D3C22×D412yes—
432C22×D32−1+414yes—
433C3×Q8—28no—
434C3×Q8—44no—
435C2×C3×C4—22no—
436C2×C3×C4—22no—
437C2×C3×C4—22no—
438C2×C3×C4—22no—
439C2×C3×C4—22no—
440C2×C3×C4—22no—
441C2×C3×C4—22no—
442C2×C3×C4—24no—
443C2×C3×C4—24no—
444C2×C3×C4—24no—
445C2×C3×C4—24no—
446C2×C3×C4—24no—
447C2×C3×C4—22no—
448C2×C3×C4—22no—
449C2×C3×C4C22×D412yes—
450C2×C3×C4C22×D412yes—
451C2×C3×C4C22×D412yes—
452C2×C3×C4C2×#4114yes—
453C2×C3×C42−1+412yes—
454C2×C3×C42−1+412yes—
455C2×Dic3—22no—
456C2×Dic3—22no—
457C2×Dic3—21no—
458C2×Dic3—21no—
459C2×Dic3—22no—
460C2×Dic3—22no—
461C2×Dic3—22no—
462C2×Dic3—22no—
463C2×Dic3—24no—
464C2×Dic3—24no—
465C2×Dic3—24no—
466C2×Dic3—24no—
467C2×Dic3—22no—
468C2×Dic3—22no—
469C2×Dic3—24no—
470C2×Dic3C22×D412yes—
471C2×Dic3C22×D412yes—
472C2×Dic3C22×D412yes—
473C2×Dic3C2×#4114yes—
474C2×Dic3C2×#4114yes—
475C4×D3—216no—
476C4×D3—28no—
477C4×D3—28no—
478C4×D3—28no—
479C4×D3—48no—
480Dic6—28no—
481Dic6—216no—
482Dic6—44no—
483Dic6—48no—
484C24—31no—
485C2×Q8—22no—
486C2×Q8—22no—
487C2×Q8—22no—
488C2×Q8—24no—
489C2×Q8—22no—
490C2×Q8—38no—
491C2×Q8—64no—
492C2×Q8—64no—
493C2×Q8—64no—
494C2×Q8—64no—
495C2×Q8—68no—
496C2×Q8—632no—
497C2×Q8—62no—
498C2×Q8—62no—
499C2×Q8—62no—
500C2×Q8—62no—
501C22×C4—31no—
502C22×C4—31no—
503C22×C4—31no—
504C22×C4—32no—
505C22×C4—34no—
506C22×C4—34no—
507C22×C4—34no—
508C22×C4—34no—
509C22×C4—32no—
510C22×C4—32no—
511C22×C4—32no—
512C22×C4—64no—
513C22×C4—64no—
514C22×C4—64no—
515C22×C4—64no—
516C22×C4—64no—
517C22×C4—68no—
518C22×C4—62no—
519C22×C4—64no—
520C22×C4—62no—
521C22×C4C23×D314yes—
522C22×C4C23×D312yes—
523C22×C4C23×D312yes—
524C22×C4C23×D312yes—
525C22×C4C23×D311yescharacteristic
526C22×C4C23×D311yescharacteristic
527C22×C4C23×D312yes—
528C22×C4C23×D311yescharacteristic
529C4⋊C4—24no—
530C4⋊C4—22no—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—28no—
534C4⋊C4—24no—
535C4⋊C4—64no—
536C4⋊C4—64no—
537C4⋊C4—64no—
538C4⋊C4—64no—
539C4⋊C4—64no—
540C4⋊C4—64no—
541C4⋊C4—64no—
542C4⋊C4—64no—
543C4⋊C4—68no—
544C4⋊C4—68no—
545C4⋊C4—64no—
546C4⋊C4—62no—
547C4⋊C4—64no—
548C4⋊C4—62no—
549C4⋊C4—64no—
550C4⋊C4—68no—
551C22⋊C4—64no—
552C22⋊C4—64no—
553C22⋊C4—68no—
554C42—24no—
555C42—24no—
556C42—34no—
557C42—34no—
558C42—34no—
559C42—64no—
560C42—64no—
561C42—68no—
562C42—64no—
563C42—64no—
564C42D3×D414yes—
565C22×C3C23×D411yescharacteristic
566C22×C3C22×#4112yes—
567C22×C3C2×2−1+411yescharacteristic
568C22×C3C2×2−1+411yescharacteristic
569C22×C3C2×2−1+412yes—
570C2×D3—22no—
571C2×D3—22no—
572C2×D3—24no—
573C2×D3—24no—
574C2×D3#53614yes—
575C3×C4—22no—
576C3×C4—24no—
577C3×C4—22no—
578C3×C4—42no—
579C3×C4—42no—
580C3×C4—42no—
581C3×C4—44no—
582C3×C4#53614yes—
583Dic3—24no—
584Dic3—22no—
585Dic3—22no—
586Dic3—42no—
587Dic3—42no—
588Dic3—42no—
589Dic3—44no—
590Dic3#52514yes—
591C23—32no—
592C23—34no—
593C23—61no—
594C23—62no—
595C23—61no—
596C23C24×D311yescharacteristic
597Q8—28no—
598Q8—44no—
599Q8—616no—
600Q8—68no—
601Q8—128no—
602Q8—124no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—32no—
618C2×C4—34no—
619C2×C4—32no—
620C2×C4—32no—
621C2×C4—34no—
622C2×C4—68no—
623C2×C4—68no—
624C2×C4—616no—
625C2×C4—62no—
626C2×C4—61no—
627C2×C4—62no—
628C2×C4—61no—
629C2×C4—62no—
630C2×C4—68no—
631C2×C4—62no—
632C2×C4—62no—
633C2×C4—62no—
634C2×C4—64no—
635C2×C4—62no—
636C2×C4—64no—
637C2×C4—64no—
638C2×C4—62no—
639C2×C4—64no—
640C2×C4—64no—
641C2×C4—128no—
642C2×C4C2×D3×D412yes—
643C2×C4C2×D3×D412yes—
644C2×C4C2×D3×D412yes—
645C2×C4#100712yes—
646C2×C4D3×#4114yes—
647C2×C4#101012yes—
648C2×C3C2×#52511yescharacteristic
649C2×C3C2×#53611yescharacteristic
650C2×C3C2×#53611yescharacteristic
651C2×C3#352312yes—
652C2×C3#352312yes—
653D3—24no—
654C22—34no—
655C22—62no—
656C22—64no—
657C22—62no—
658C22—64no—
659C22C22×D3×D411yescharacteristic
660C22C2×#100711yescharacteristic
661C22C2×D3×#4112yes—
662C22C2×#101011yescharacteristic
663C22D3×2−1+412yes—
664C4—22no—
665C4—24no—
666C4—22no—
667C4—34no—
668C4—42no—
669C4—42no—
670C4—44no—
671C4—42no—
672C4—64no—
673C4—62no—
674C4—62no—
675C4—122no—
676C4—122no—
677C4—122no—
678C4—124no—
679C4#562814yes—
680C3#3524811yescharacteristic, complemented, Sylow
681C2—64no—
682C2C2×#562811yescharacteristic
683C2D3×#52511yescharacteristic
684C2D3×#53611yescharacteristic
685C2#8699312yes—
686C2#8702512yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2581C2
prime radicalC23×D3C24
hypercenter#2581D3
hypocenterC3#35248
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2581

Complements

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

NH ≅ G/Ntype
3. C3×#2581C2semidirect
4. #82481C2semidirect
5. #82480C2semidirect
6. #78210C2semidirect
8. #77204C2semidirect
11. #76545C2semidirect
13. #76060C2semidirect
15. #74948C2semidirect
16. #74191C2semidirect
95. #2581D3semidirect
680. C3#35248semidirect

Maximal subgroups

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

subgroupindex
C2×#55782
C2×C4×D3×Q82
C3×#25812
#824812
#824802
#782102
#781522
#772042
#771732
#766632
#765452
#762872
#760602
#753352
#749482
#741912
#741472
#352483