Finite Group Databasebeta

← #11166380

Subgroups — #11166380

Order 768.

1980 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5541C211yesmaximal, characteristic
2C2×#5519C211yesmaximal, characteristic, complemented
3C3×#2395C211yesmaximal, characteristic
4#82373C211yesmaximal, characteristic
5#82263C211yesmaximal, characteristic, complemented
6#78318C212yesmaximal, complemented
7#78047C212yesmaximal, complemented
8#77837C212yesmaximal, complemented
9#77578C214yesmaximal, complemented
10#77538C212yesmaximal, complemented
11#77004C214yesmaximal
12#73716C212yesmaximal, complemented
13#73674C212yesmaximal, complemented
14#73517C211yesmaximal, characteristic, complemented
15#73505C212yesmaximal, complemented
16#73493C212yesmaximal, complemented
17#73459C211yesmaximal, characteristic, complemented
18#42964—31nomaximal, Sylow
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×#117C2211yescharacteristic
21C2×#935C2212yes—
22C2×#935C2212yes—
23C2×#926C2211yescharacteristic
24C22×C4⋊Dic3C2211yescharacteristic
25C22×C4⋊Dic3C2211yescharacteristic
26C2×#924C2211yescharacteristic
27C2×#924C2211yescharacteristic
28#5541—28no—
29#5519—28no—
30C2×#890C2212yes—
31C2×#889C2212yes—
32C2×#888C2212yes—
33C2×#879C2212yes—
34C2×#878C2214yes—
35C2×#878C2214yes—
36C2×#877C2212yes—
37C2×#870C2212yes—
38C2×#870C2211yescharacteristic
39C2×#869C2212yes—
40C2×C4×Dic6C2211yescharacteristic
41C3×#400C2211yescharacteristic
42C3×#399C2211yescharacteristic
43C3×#398C2211yescharacteristic
44C3×#397C2212yes—
45C3×#397C2211yescharacteristic
46C3×#394C2212yes—
47C3×C4×C4⋊C4C2212yes—
48C3×C4×C22⋊C4C2212yes—
49C3×C4×C22⋊C4C2211yescharacteristic
50#5212C2212yes—
51#5212C2211yescharacteristic
52#4985C2212yes—
53#4983C2212yes—
54#4981C2212yes—
55#4980C2214yes—
56#4979C2214yes—
57#4975C2214yes—
58#4972C2214yes—
59#4953C2212yes—
60#4952C2214yes—
61#4952C2214yes—
62#4950C2214yes—
63#4950C2214yes—
64#4944C2214yes—
65#4944C2214yes—
66#4943C2212yes—
67#4935C2212yes—
68C4×C4⋊Dic3C2212yes—
69#4929C2212yes—
70#4929C2212yes—
71#4928C2212yes—
72#4928C2212yes—
73#4928C2212yes—
74#4661C2214yes—
75#4661C2212yes—
76#4661C2212yes—
77#4658C2214yes—
78#4657C2214yes—
79#4657C2214yes—
80#4657C2214yes—
81#4656C2214yes—
82#4650C2214yes—
83#4650C2212yes—
84#4648C2212yes—
85#4646C2214yes—
86C2×#552—31no—
87C2×#544—31no—
88#2822—32no—
89#2698—34no—
90#2680—34no—
91#2660—32no—
92#2625—31no—
93#2621—32no—
94#2611—32no—
95#2552—32no—
96#2502—31no—
97#2489—32no—
98#2486—31no—
99#2451—32no—
100#2440—32no—
101#2395D311yescharacteristic
102#2346—31no—
103C22×C3×D4C2311yescharacteristic
104C23×C3×C4C2311yescharacteristic
105C23×C3×C4C2311yescharacteristic
106C23×Dic3C2312yes—
107C23×Dic3C2312yes—
108C22×Dic6C2311yescharacteristic
109C22×Dic6C2314yes—
110C3×C4×D4—24no—
111C3×#117—24no—
112C2×C3×C4⋊C4C2312yes—
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C4⋊C4C2311yescharacteristic
115C2×C3×C4⋊C4C2311yescharacteristic
116C2×C3×C4⋊C4C2312yes—
117C2×C3×C22⋊C4C2311yescharacteristic
118C2×C3×C22⋊C4C2312yes—
119C2×C3×C22⋊C4C2312yes—
120C2×C3×C22⋊C4C2312yes—
121C2×C3×C22⋊C4C2312yes—
122C2×C3×C22⋊C4C2311yescharacteristic
123C2×C3×C42C2311yescharacteristic
124C2×C3×C42C2312yes—
125C2×C3×C42C2312yes—
126C2×#217C2312yes—
127C2×#217C2314yes—
128C2×#217C2312yes—
129C2×#217C2312yes—
130C2×#217C2312yes—
131#935—28no—
132#935—44no—
133#926D418yes—
134C2×C4⋊Dic3—24no—
135C2×C4⋊Dic3—24no—
136C2×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2314yes—
138C2×C4⋊Dic3C2312yes—
139C2×C4⋊Dic3C2312yes—
140C2×C4⋊Dic3C2311yescharacteristic
141C2×C4⋊Dic3C2312yes—
142C2×C4⋊Dic3C2314yes—
143C2×C4⋊Dic3C2314yes—
144C2×C4⋊Dic3C2311yescharacteristic
145C2×C4⋊Dic3C2311yescharacteristic
146#924—24no—
147#924—24no—
148C2×Dic3⋊C4C2312yes—
149C2×Dic3⋊C4C2314yes—
150C2×Dic3⋊C4C2312yes—
151C2×Dic3⋊C4C2312yes—
152C2×Dic3⋊C4C2312yes—
153C2×Dic3⋊C4C2314yes—
154C2×Dic3⋊C4C2312yes—
155C2×Dic3⋊C4C2314yes—
156C2×Dic3⋊C4C2314yes—
157C2×Dic3⋊C4C2314yes—
158C2×C4×Dic3C2312yes—
159C2×C4×Dic3C2314yes—
160C2×C4×Dic3C2312yes—
161#890—28no—
162#889—28no—
163#888—28no—
164#879—28no—
165#878—216no—
166#878—48no—
167#877—28no—
168#870—28no—
169#870—24no—
170#869—28no—
171C4×Dic6—24no—
172C3×#95C2312yes—
173C3×#95C2311yescharacteristic
174C3×#95C2312yes—
175C3×#95C2311yescharacteristic
176C3×#95C2312yes—
177C3×#95C2312yes—
178C3⋊#95C2314yes—
179C3⋊#95C2314yes—
180C3⋊#95C2314yes—
181C3⋊#95C2312yes—
182C3⋊#95C2314yes—
183C3⋊#95C2314yes—
184C3⋊#95C2314yes—
185C3⋊#95C2314yes—
186C3⋊#95C2314yes—
187C3⋊#95C2314yes—
188#552—68no—
189#544—68no—
190C2×C4⋊Q8—32no—
191C2×C4⋊Q8—32no—
192C2×#126—32no—
193C2×#125—32no—
194C2×#125—32no—
195C2×#125—32no—
196C2×#125—31no—
197C2×#123—32no—
198C2×#123—32no—
199C2×C22⋊Q8—31no—
200C2×C22⋊Q8—31no—
201C2×C22⋊Q8—34no—
202C2×C22⋊Q8—34no—
203C2×C4×Q8—31no—
204C2×C4×D4C2×D311yescharacteristic
205C2×#117—31no—
206C2×#117—32no—
207C2×#117C2×D311yescharacteristic
208C22×C4⋊C4—31no—
209C22×C4⋊C4—31no—
210#400—32no—
211#400—34no—
212#400C2×D311yescharacteristic
213#399C2×D311yescharacteristic
214#398—32no—
215#398—32no—
216#398—34no—
217#398—34no—
218#398—32no—
219#398—34no—
220#398—34no—
221#398—32no—
222#398C2×D311yescharacteristic
223#397—32no—
224#397—34no—
225#397—34no—
226#397C2×D311yescharacteristic
227#397C2×D312yes—
228#395—34no—
229#395—34no—
230#394C2×D312yes—
231#391—32no—
232#391—32no—
233#391—34no—
234#389—34no—
235#389—32no—
236#389—32no—
237#389—32no—
238#388—34no—
239#388—34no—
240#387—34no—
241#385—34no—
242#385—34no—
243#384—34no—
244#384—32no—
245#382—32no—
246#382—34no—
247#381—32no—
248#380—31no—
249#380—32no—
250C4×C4⋊C4—32no—
251C4×C4⋊C4C2×D312yes—
252C4×C22⋊C4C2×D311yescharacteristic
253C4×C22⋊C4C2×D312yes—
254#376—32no—
255C24×C3C2411yescharacteristic
256C24×C3C2411yescharacteristic
257C2×C3×D4—22no—
258C2×C3×D4—22no—
259C2×C3×D4—41no—
260C22×C3×C4—24no—
261C22×C3×C4—22no—
262C22×C3×C4—24no—
263C22×C3×C4C2×D414yes—
264C22×C3×C4C2412yes—
265C22×C3×C4C2411yescharacteristic
266C22×C3×C4C2411yescharacteristic
267C22×C3×C4C2412yes—
268C22×C3×C4C2412yes—
269C22×C3×C4C2411yescharacteristic
270C22×C3×C4C2411yescharacteristic
271C22×C3×C4C2412yes—
272C22×C3×C4C2411yescharacteristic
273C22×Dic3—24no—
274C22×Dic3—24no—
275C22×Dic3—24no—
276C22×Dic3—42no—
277C22×Dic3—42no—
278C22×Dic3—42no—
279C22×Dic3C2414yes—
280C22×Dic3C2412yes—
281C22×Dic3C2414yes—
282C22×Dic3C2412yes—
283C22×Dic3C2412yes—
284C22×Dic3C2412yes—
285C2×Dic6—22no—
286C2×Dic6—24no—
287C2×Dic6—28no—
288C2×Dic6—44no—
289C2×Dic6—44no—
290C3×C4⋊C4—22no—
291C3×C4⋊C4—22no—
292C3×C4⋊C4—22no—
293C3×C4⋊C4—24no—
294C3×C4⋊C4—42no—
295C3×C22⋊C4—24no—
296C3×C22⋊C4—22no—
297C3×C22⋊C4—24no—
298C3×C22⋊C4—22no—
299C3×C22⋊C4—24no—
300C3×C22⋊C4—24no—
301C3×C42—22no—
302C3×C42—24no—
303C3×C42—24no—
304#217—24no—
305#217—28no—
306#217—24no—
307#217—42no—
308#217C2×D418yes—
309C4⋊Dic3—24no—
310C4⋊Dic3—22no—
311C4⋊Dic3—24no—
312C4⋊Dic3—24no—
313C4⋊Dic3—28no—
314C4⋊Dic3—24no—
315C4⋊Dic3—44no—
316C4⋊Dic3—44no—
317C4⋊Dic3C2×D414yes—
318C4⋊Dic3C2×D414yes—
319Dic3⋊C4—24no—
320Dic3⋊C4—24no—
321Dic3⋊C4—24no—
322Dic3⋊C4—28no—
323Dic3⋊C4—28no—
324Dic3⋊C4—24no—
325Dic3⋊C4—24no—
326Dic3⋊C4—44no—
327Dic3⋊C4—44no—
328Dic3⋊C4—44no—
329C4×Dic3—28no—
330C4×Dic3—24no—
331C4×Dic3C2×D418yes—
332C22×Q8—31no—
333C22×Q8—34no—
334C22×D4C22×D311yescharacteristic
335C23×C4—32no—
336C23×C4—32no—
337C23×C4C22×D311yescharacteristic
338C23×C4C22×D311yescharacteristic
339C4⋊Q8—68no—
340C4⋊Q8—68no—
341#126—68no—
342#125—68no—
343#125—64no—
344#125—68no—
345#125—68no—
346#123—68no—
347#123—124no—
348C22⋊Q8—616no—
349C22⋊Q8—64no—
350C22⋊Q8—64no—
351C22⋊Q8—128no—
352C4×Q8—64no—
353C4×D4—24no—
354#117—24no—
355#117—38no—
356#117—68no—
357C2×C4⋊C4—31no—
358C2×C4⋊C4—31no—
359C2×C4⋊C4—31no—
360C2×C4⋊C4—32no—
361C2×C4⋊C4—32no—
362C2×C4⋊C4—32no—
363C2×C4⋊C4—32no—
364C2×C4⋊C4—32no—
365C2×C4⋊C4—32no—
366C2×C4⋊C4—34no—
367C2×C4⋊C4—32no—
368C2×C4⋊C4—34no—
369C2×C4⋊C4—34no—
370C2×C4⋊C4—32no—
371C2×C4⋊C4—32no—
372C2×C4⋊C4—34no—
373C2×C4⋊C4—34no—
374C2×C4⋊C4—34no—
375C2×C4⋊C4—34no—
376C2×C4⋊C4—34no—
377C2×C4⋊C4—64no—
378C2×C4⋊C4—64no—
379C2×C4⋊C4C22×D311yescharacteristic
380C2×C4⋊C4C22×D311yescharacteristic
381C2×C4⋊C4C22×D311yescharacteristic
382C2×C4⋊C4C22×D312yes—
383C2×C4⋊C4C22×D312yes—
384C2×C22⋊C4—32no—
385C2×C22⋊C4—32no—
386C2×C22⋊C4—32no—
387C2×C22⋊C4—34no—
388C2×C22⋊C4—32no—
389C2×C22⋊C4C22×D311yescharacteristic
390C2×C22⋊C4C22×D312yes—
391C2×C22⋊C4C22×D311yescharacteristic
392C2×C22⋊C4C22×D312yes—
393C2×C22⋊C4C22×D312yes—
394C2×C22⋊C4C22×D312yes—
395C2×C42—32no—
396C2×C42—32no—
397C2×C42—34no—
398C2×C42C22×D312yes—
399C2×C42C22×D311yescharacteristic
400C2×C42C22×D312yes—
401#95—32no—
402#95—34no—
403#95—34no—
404#95—34no—
405#95—34no—
406#95—34no—
407#95—34no—
408#95—34no—
409#95—34no—
410#95—34no—
411#95C22×D311yescharacteristic
412#95C22×D312yes—
413#95C22×D311yescharacteristic
414#95C22×D312yes—
415#95C22×D312yes—
416#95C22×D312yes—
417C23×C3—21no—
418C23×C3—21no—
419C23×C3—22no—
420C23×C3—22no—
421C23×C3—22no—
422C23×C3—22no—
423C23×C3—21no—
424C23×C3—21no—
425C23×C3C22×D412yes—
426C23×C3H(5,𝔽2)12yes—
427C23×C3C2511yescharacteristic
428C3×D4—44no—
429C2×C3×C4—22no—
430C2×C3×C4—22no—
431C2×C3×C4—22no—
432C2×C3×C4—28no—
433C2×C3×C4—21no—
434C2×C3×C4—21no—
435C2×C3×C4—22no—
436C2×C3×C4—21no—
437C2×C3×C4—22no—
438C2×C3×C4—21no—
439C2×C3×C4—22no—
440C2×C3×C4—22no—
441C2×C3×C4—22no—
442C2×C3×C4—22no—
443C2×C3×C4—22no—
444C2×C3×C4—22no—
445C2×C3×C4—22no—
446C2×C3×C4—22no—
447C2×C3×C4—21no—
448C2×C3×C4—22no—
449C2×C3×C4—44no—
450C2×C3×C4C22×D412yes—
451C2×C3×C4C22×D412yes—
452C2×C3×C42−1+414yes—
453C2×C3×C42−1+414yes—
454C2×Dic3—22no—
455C2×Dic3—24no—
456C2×Dic3—22no—
457C2×Dic3—22no—
458C2×Dic3—22no—
459C2×Dic3—24no—
460C2×Dic3—24no—
461C2×Dic3—22no—
462C2×Dic3—24no—
463C2×Dic3—24no—
464C2×Dic3—24no—
465C2×Dic3—24no—
466C2×Dic3—22no—
467C2×Dic3—24no—
468C2×Dic3—48no—
469C2×Dic3—84no—
470C2×Dic3C22×D414yes—
471C2×Dic3C22×D414yes—
472Dic6—44no—
473Dic6—88no—
474C24C23×D311yescharacteristic
475C24C23×D311yescharacteristic
476C2×Q8—68no—
477C2×Q8—62no—
478C2×Q8—64no—
479C2×Q8—124no—
480C2×Q8—124no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—41no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—24no—
487C22×C4—32no—
488C22×C4—32no—
489C22×C4—34no—
490C22×C4—32no—
491C22×C4—34no—
492C22×C4—32no—
493C22×C4—64no—
494C22×C4—64no—
495C22×C4—64no—
496C22×C4—122no—
497C22×C4—122no—
498C22×C4—122no—
499C22×C4D3×D414yes—
500C22×C4C23×D312yes—
501C22×C4C23×D311yescharacteristic
502C22×C4C23×D311yescharacteristic
503C22×C4C23×D312yes—
504C22×C4C23×D312yes—
505C22×C4C23×D311yescharacteristic
506C22×C4C23×D311yescharacteristic
507C22×C4C23×D312yes—
508C22×C4C23×D311yescharacteristic
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—34no—
514C4⋊C4—34no—
515C4⋊C4—42no—
516C4⋊C4—64no—
517C4⋊C4—64no—
518C4⋊C4—64no—
519C4⋊C4—68no—
520C4⋊C4—64no—
521C4⋊C4—68no—
522C4⋊C4—68no—
523C4⋊C4—64no—
524C4⋊C4—62no—
525C4⋊C4—64no—
526C4⋊C4—64no—
527C4⋊C4—64no—
528C4⋊C4—64no—
529C4⋊C4—124no—
530C4⋊C4—124no—
531C4⋊C4—124no—
532C4⋊C4—124no—
533C4⋊C4—124no—
534C22⋊C4—24no—
535C22⋊C4—22no—
536C22⋊C4—24no—
537C22⋊C4—22no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—38no—
541C22⋊C4—64no—
542C22⋊C4—64no—
543C22⋊C4—68no—
544C22⋊C4—122no—
545C42—24no—
546C42—22no—
547C42—24no—
548C42—38no—
549C42—64no—
550C42—68no—
551C22×C3—22no—
552C22×C3—22no—
553C22×C3—22no—
554C22×C3—22no—
555C22×C3—24no—
556C22×C3—41no—
557C22×C3—41no—
558C22×C3—42no—
559C22×C3—42no—
560C22×C3—42no—
561C22×C3C23×D411yescharacteristic
562C22×C3C2×H(5,𝔽2)11yescharacteristic
563C22×C3C2×2−1+412yes—
564C22×C3C2×2−1+412yes—
565C22×C3#58511yescharacteristic
566C3×C4—22no—
567C3×C4—24no—
568C3×C4—41no—
569C3×C4—42no—
570C3×C4—41no—
571C3×C4—41no—
572C3×C4—42no—
573C3×C4—42no—
574C3×C4#53614yes—
575Dic3—24no—
576Dic3—24no—
577Dic3—42no—
578Dic3—44no—
579Dic3—42no—
580Dic3—44no—
581C23—21no—
582C23—21no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—21no—
588C23—21no—
589C23C2×D3×D412yes—
590C23#100912yes—
591C23C24×D311yescharacteristic
592Q8—124no—
593Q8—248no—
594D4—44no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—28no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—34no—
616C2×C4—34no—
617C2×C4—44no—
618C2×C4—62no—
619C2×C4—64no—
620C2×C4—62no—
621C2×C4—62no—
622C2×C4—62no—
623C2×C4—64no—
624C2×C4—64no—
625C2×C4—64no—
626C2×C4—62no—
627C2×C4—64no—
628C2×C4—64no—
629C2×C4—64no—
630C2×C4—62no—
631C2×C4—64no—
632C2×C4—128no—
633C2×C4—244no—
634C2×C4C2×D3×D412yes—
635C2×C4C2×D3×D412yes—
636C2×C4#101014yes—
637C2×C4#101014yes—
638C2×C3—42no—
639C2×C3—42no—
640C2×C3C2×#53611yescharacteristic
641C2×C3#348811yescharacteristic
642C2×C3#349911yescharacteristic
643C2×C3#354412yes—
644C2×C3#356312yes—
645C22—22no—
646C22—22no—
647C22—24no—
648C22—22no—
649C22—22no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—41no—
654C22—41no—
655C22C22×D3×D411yescharacteristic
656C22C2×#100911yescharacteristic
657C22C2×#101012yes—
658C22C2×#101012yes—
659C22#596411yescharacteristic
660C4—24no—
661C4—22no—
662C4—41no—
663C4—42no—
664C4—42no—
665C4—41no—
666C4—42no—
667C4—41no—
668C4—64no—
669C4—64no—
670C4—122no—
671C4—124no—
672C4—124no—
673C4—122no—
674C4#564814yes—
675C3#4296411yescharacteristic, complemented, Sylow
676C2—42no—
677C2—42no—
678C2C2×#564811yescharacteristic
679C2#8604111yescharacteristic
680C2#8613312yes—
681C2#8614212yes—
682C2#8624311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2395C2
prime radicalC22×C3×D4C23
hypercenter#2395D3
hypocenterC3#42964
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2395

Complements

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

NH ≅ G/Ntype
2. C2×#5519C2semidirect
5. #82263C2semidirect
6. #78318C2semidirect
7. #78047C2semidirect
8. #77837C2semidirect
9. #77578C2semidirect
10. #77538C2semidirect
12. #73716C2semidirect
13. #73674C2semidirect
14. #73517C2semidirect
15. #73505C2semidirect
16. #73493C2semidirect
17. #73459C2semidirect
675. C3#42964semidirect

Maximal subgroups

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

subgroupindex
C2×#55412
C2×#55192
C3×#23952
#823732
#822632
#783182
#780472
#778372
#775782
#775382
#770042
#737162
#736742
#735172
#735052
#734932
#734592
#429643