Finite Group Databasebeta

← #10955871

Subgroups — #10955871

Order 768.

2094 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5578C211yesmaximal, characteristic
2C3×#2276C211yesmaximal, characteristic, complemented
3#82465C212yesmaximal, complemented
4#82464C212yesmaximal
5#82454C211yesmaximal, characteristic, complemented
6#73764C214yesmaximal
7#73657C212yesmaximal, complemented
8#73647C212yesmaximal
9#73643C212yesmaximal, complemented
10#73635C212yesmaximal
11C4×#889C214yesmaximal, complemented
12#73616C211yesmaximal, characteristic, complemented
13C4×#892C211yesmaximal, characteristic
14#73413C211yesmaximal, characteristic
15#73393C212yesmaximal, complemented
16#73392C212yesmaximal
17#73386C211yesmaximal, characteristic, complemented
18#34231—31nomaximal, Sylow
19C2×C3×C4×Q8C2212yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×#946C2211yescharacteristic
22C2×#944C2211yescharacteristic
23#5578C4116yescomplemented
24C2×#897C2211yescharacteristic
25C2×#896C2212yes—
26C2×#892C2212yes—
27C2×#889C2212yes—
28C2×#888C2211yescharacteristic
29C2×#887C2212yes—
30C2×#872C2212yes—
31C2×C42×D3C2211yescharacteristic
32C2×C4×Dic6C2211yescharacteristic
33C2×C4×Dic6C2212yes—
34C3×C4×C4⋊C4C2214yes—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×C4×C4⋊C4C2212yes—
37C3×C4×C4⋊C4C2212yes—
38C3×#376C2211yescharacteristic
39C3×#376C2212yes—
40#5229C2212yes—
41#4990C2214yes—
42#4984C2212yes—
43#4980C2214yes—
44#4978C2214yes—
45#4978C2214yes—
46#4976C2214yes—
47Dic3×C4⋊C4C2212yes—
48Dic3×C4⋊C4C2212yes—
49#4940C2214yes—
50#4940C2214yes—
51C4×(C2×D3)⋊C4C2212yes—
52C4×(C2×D3)⋊C4C2212yes—
53#4937C2212yes—
54#4936C2212yes—
55C4×C4⋊Dic3C2211yescharacteristic
56C4×C4⋊Dic3C2214yes—
57#4933C2214yes—
58#4932C2211yescharacteristic
59C4×Dic3⋊C4C2214yes—
60C4×Dic3⋊C4C2214yes—
61C4×Dic3⋊C4C2212yes—
62C4×Dic3⋊C4C2214yes—
63C4×Dic3⋊C4C2214yes—
64C42×Dic3C2212yes—
65#4929C2212yes—
66#4929C2214yes—
67#4928C2212yes—
68#4670C2214yes—
69#4667C2211yescharacteristic
70#4667C2211yescharacteristic
71#4666C2212yes—
72#4664C2212yes—
73#4653C2214yes—
74#4652C2214yes—
75#4651C2212yes—
76#4651C2211yescharacteristic
77#4651C2211yescharacteristic
78#4650C2212yes—
79#4650C2212yes—
80#4649C2212yes—
81#4648C2212yes—
82#4648C2212yes—
83#4647C2214yes—
84#4646C2214yes—
85C2×#533—31no—
86#2369—32no—
87#2365—32no—
88#2362—32no—
89#2362—32no—
90#2356—32no—
91#2356—32no—
92#2347—32no—
93#2346—32no—
94#2332—34no—
95#2320—31no—
96#2320—31no—
97C4×#125—34no—
98#2283—31no—
99#2276—31no—
100#2276D311yescharacteristic, complemented
101C4×#117—31no—
102C22×C3×Q8C2311yescharacteristic
103C22×C4×D3C2311yescharacteristic
104C22×C4×D3C2311yescharacteristic
105C22×C4×D3C2311yescharacteristic
106C22×Dic6C2311yescharacteristic
107C22×Dic6C2312yes—
108C3×C4×Q8—28no—
109C3×C4×Q8C2×C418yes—
110C2×C3×C4⋊C4C2314yes—
111C2×C3×C4⋊C4C2312yes—
112C2×C3×C4⋊C4C2311yescharacteristic
113C2×C3×C4⋊C4C2312yes—
114C2×C3×C42C2312yes—
115C2×C3×C42C2314yes—
116C2×C3×C42C2314yes—
117C2×C3×C42C2312yes—
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2312yes—
120#946C2×C418yes—
121#944C2×C418yes—
122C2×(C2×D3)⋊C4C2314yes—
123C2×(C2×D3)⋊C4C2312yes—
124C2×(C2×D3)⋊C4C2312yes—
125C2×(C2×D3)⋊C4C2312yes—
126C2×(C2×D3)⋊C4C2312yes—
127C2×C4⋊Dic3C2312yes—
128C2×C4⋊Dic3C2314yes—
129C2×C4⋊Dic3C2312yes—
130C2×C4⋊Dic3C2311yescharacteristic
131C2×Dic3⋊C4C2312yes—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2312yes—
134C2×Dic3⋊C4C2314yes—
135C2×Dic3⋊C4C2312yes—
136C2×Dic3⋊C4C2312yes—
137C2×Dic3⋊C4C2312yes—
138C2×Dic3⋊C4C2312yes—
139C2×Dic3⋊C4C2312yes—
140C2×C4×Dic3C2314yes—
141C2×C4×Dic3C2314yes—
142C2×C4×Dic3C2314yes—
143C2×C4×Dic3C2312yes—
144C2×C4×Dic3C2314yes—
145C2×C4×Dic3C2312yes—
146C2×C4×Dic3C2311yescharacteristic
147C2×C4×Dic3C2312yes—
148C2×C4×Dic3C2311yescharacteristic
149C2×C4×Dic3C2311yescharacteristic
150#897C2×C418yes—
151#896C2×C4116yes—
152#892—28no—
153#889C2×C4116yes—
154#888C2×C418yes—
155#887—28no—
156#872C2×C4116yes—
157C42×D3C2×C418yes—
158C4×Dic6C2×C4116yes—
159C4×Dic6C2×C418yes—
160C3×#95C2314yes—
161C3×#95C2312yes—
162C3×#95C2311yescharacteristic
163C3×#95C2312yes—
164C3×#95C2311yescharacteristic
165C3⋊#95C2312yes—
166C3⋊#95C2314yes—
167C3⋊#95C2314yes—
168C3⋊#95C2314yes—
169C3⋊#95C2311yescharacteristic
170C3⋊#95C2312yes—
171C3⋊#95C2311yescharacteristic
172C3⋊#95C2312yes—
173C3⋊#95C2312yes—
174C3⋊#95C2314yes—
175C3⋊#95C2312yes—
176C3⋊#95C2314yes—
177C3⋊#95C2314yes—
178C3⋊#95C2312yes—
179C3⋊#95C2314yes—
180C3⋊#95C2314yes—
181#533—316no—
182C2×C4⋊Q8—31no—
183C2×C4⋊Q8—31no—
184C2×#125—32no—
185C2×C22⋊Q8—31no—
186C2×C22⋊Q8—32no—
187C2×C22⋊Q8—31no—
188C2×C4×Q8—32no—
189C2×C4×Q8—31no—
190C2×C4×Q8—32no—
191C2×C4×Q8C2×D312yes—
192C2×C4×Q8C2×D311yescharacteristic
193C2×#117—32no—
194C2×#117—32no—
195C22×C42—31no—
196#391—32no—
197#391—32no—
198#391—34no—
199#391—34no—
200#389—34no—
201#389—32no—
202#389—32no—
203#389—34no—
204#388—34no—
205#388—34no—
206#388—34no—
207#388—34no—
208#387—34no—
209#387—34no—
210#387—34no—
211#387—34no—
212#387—34no—
213#387—34no—
214#383—32no—
215#383—32no—
216#383—31no—
217#383—31no—
218#382—32no—
219#382—32no—
220#382—32no—
221#381—32no—
222#380—31no—
223#380—31no—
224C4×C4⋊C4—32no—
225C4×C4⋊C4—32no—
226C4×C4⋊C4—34no—
227C4×C4⋊C4—34no—
228C4×C4⋊C4—34no—
229C4×C4⋊C4—34no—
230C4×C4⋊C4—34no—
231C4×C4⋊C4—31no—
232C4×C4⋊C4—32no—
233C4×C4⋊C4—32no—
234C4×C4⋊C4C2×D314yes—
235C4×C4⋊C4C2×D312yes—
236C4×C4⋊C4C2×D312yes—
237C4×C4⋊C4C2×D311yescharacteristic
238C4×C22⋊C4—32no—
239C4×C22⋊C4—32no—
240C4×C22⋊C4—32no—
241#376—31no—
242#376—32no—
243#376—32no—
244#376C2×D311yescharacteristic
245#376C2×D312yes—
246C43—32no—
247C23×D3C2411yescharacteristic
248C2×C3×Q8—24no—
249C2×C3×Q8C22×C414yes—
250C22×C3×C4C2412yes—
251C22×C3×C4C2412yes—
252C22×C3×C4C2411yescharacteristic
253C22×C3×C4C2412yes—
254C22×C3×C4C2411yescharacteristic
255C22×C3×C4C2412yes—
256C22×C3×C4C2414yes—
257C22×C3×C4C2411yescharacteristic
258C22×Dic3C2412yes—
259C22×Dic3C2412yes—
260C22×Dic3C2414yes—
261C22×Dic3C2411yescharacteristic
262C22×Dic3C2411yescharacteristic
263C22×Dic3C2412yes—
264C22×Dic3C2412yes—
265C22×Dic3C2411yescharacteristic
266C2×C4×D3—24no—
267C2×C4×D3—24no—
268C2×C4×D3—41no—
269C2×C4×D3—41no—
270C2×C4×D3C22×C414yes—
271C2×C4×D3C22×C414yes—
272C2×C4×D3C22×C414yes—
273C2×Dic6—24no—
274C2×Dic6—42no—
275C2×Dic6—42no—
276C2×Dic6C22×C414yes—
277C2×Dic6C22×C418yes—
278C3×C4⋊C4—28no—
279C3×C4⋊C4C22×C414yes—
280C3×C4⋊C4C22×C418yes—
281C3×C4⋊C4#4118yes—
282C3×C42—28no—
283C3×C42—24no—
284C3×C42—42no—
285C3×C42—44no—
286C3×C42C22×C414yes—
287C3×C42C22×C418yes—
288(C2×D3)⋊C4—28no—
289(C2×D3)⋊C4—42no—
290(C2×D3)⋊C4—42no—
291(C2×D3)⋊C4C22×C418yes—
292(C2×D3)⋊C4C22×C418yes—
293C4⋊Dic3—24no—
294C4⋊Dic3—44no—
295C4⋊Dic3C22×C414yes—
296C4⋊Dic3C22×C418yes—
297Dic3⋊C4—28no—
298Dic3⋊C4—24no—
299Dic3⋊C4—24no—
300Dic3⋊C4—44no—
301Dic3⋊C4C22×C418yes—
302Dic3⋊C4C22×C418yes—
303Dic3⋊C4C22×C418yes—
304Dic3⋊C4C22×C418yes—
305Dic3⋊C4C22×C418yes—
306C4×Dic3—28no—
307C4×Dic3—24no—
308C4×Dic3—28no—
309C4×Dic3—28no—
310C4×Dic3—44no—
311C4×Dic3—42no—
312C4×Dic3C22×C414yes—
313C4×Dic3C22×C418yes—
314C4×Dic3C22×C414yes—
315C4×Dic3C22×C414yes—
316C22×Q8—31no—
317C22×Q8—32no—
318C22×Q8C22×D311yescharacteristic
319C23×C4—31no—
320C23×C4—31no—
321C23×C4—31no—
322C4⋊Q8—38no—
323C4⋊Q8—38no—
324#125—316no—
325C22⋊Q8—38no—
326C22⋊Q8—316no—
327C22⋊Q8—38no—
328C4×Q8—28no—
329C4×Q8—316no—
330C4×Q8—38no—
331C4×Q8—68no—
332C4×Q8C4×D318yes—
333#117—316no—
334#117—68no—
335C2×C4⋊C4—32no—
336C2×C4⋊C4—32no—
337C2×C4⋊C4—32no—
338C2×C4⋊C4—32no—
339C2×C4⋊C4—32no—
340C2×C4⋊C4—31no—
341C2×C4⋊C4—32no—
342C2×C4⋊C4—32no—
343C2×C4⋊C4—34no—
344C2×C4⋊C4—32no—
345C2×C4⋊C4—32no—
346C2×C4⋊C4—34no—
347C2×C4⋊C4—34no—
348C2×C4⋊C4C22×D314yes—
349C2×C4⋊C4C22×D312yes—
350C2×C4⋊C4C22×D311yescharacteristic
351C2×C4⋊C4C22×D312yes—
352C2×C22⋊C4—32no—
353C2×C22⋊C4—32no—
354C2×C22⋊C4—34no—
355C2×C22⋊C4—32no—
356C2×C22⋊C4—32no—
357C2×C42—32no—
358C2×C42—31no—
359C2×C42—31no—
360C2×C42—31no—
361C2×C42—38no—
362C2×C42—34no—
363C2×C42—32no—
364C2×C42—34no—
365C2×C42—34no—
366C2×C42—32no—
367C2×C42—34no—
368C2×C42C22×D312yes—
369C2×C42C22×D314yes—
370C2×C42C22×D312yes—
371C2×C42C22×D314yes—
372C2×C42C22×D312yes—
373C2×C42C22×D311yescharacteristic
374#95—31no—
375#95—32no—
376#95—34no—
377#95—31no—
378#95—32no—
379#95—34no—
380#95—32no—
381#95—32no—
382#95—32no—
383#95—32no—
384#95—34no—
385#95—34no—
386#95—34no—
387#95—34no—
388#95—34no—
389#95—34no—
390#95C22×D312yes—
391#95C22×D311yescharacteristic
392#95C22×D311yescharacteristic
393#95C22×D312yes—
394#95C22×D314yes—
395C23×C3C2511yescharacteristic
396C22×D3—22no—
397C22×D3—22no—
398C22×D3—21no—
399C22×D3—21no—
400C22×D3C23×C412yes—
401C3×Q8—28no—
402C2×C3×C4—24no—
403C2×C3×C4—24no—
404C2×C3×C4—22no—
405C2×C3×C4—21no—
406C2×C3×C4—24no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—21no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4C23×C412yes—
417C2×C3×C4C23×C412yes—
418C2×C3×C4C23×C414yes—
419C2×C3×C4C23×C414yes—
420C2×C3×C4C23×C412yes—
421C2×C3×C4C2×#4118yes—
422C2×C3×C4C2×#4114yes—
423C2×Dic3—22no—
424C2×Dic3—24no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—24no—
432C2×Dic3—24no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic3—22no—
436C2×Dic3—24no—
437C2×Dic3—22no—
438C2×Dic3C23×C412yes—
439C2×Dic3C23×C412yes—
440C2×Dic3C23×C414yes—
441C2×Dic3C23×C414yes—
442C2×Dic3C23×C412yes—
443C2×Dic32−1+414yes—
444C4×D3—28no—
445C4×D3—28no—
446C4×D3—44no—
447Dic6—28no—
448Dic6—48no—
449C24—31no—
450C2×Q8—24no—
451C2×Q8—34no—
452C2×Q8—38no—
453C2×Q8—64no—
454C2×Q8—122no—
455C2×Q8—122no—
456C2×Q8C2×C4×D314yes—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—31no—
460C22×C4—32no—
461C22×C4—32no—
462C22×C4—34no—
463C22×C4—34no—
464C22×C4—34no—
465C22×C4—32no—
466C22×C4—34no—
467C22×C4—32no—
468C22×C4—64no—
469C22×C4—64no—
470C22×C4—121no—
471C22×C4—121no—
472C22×C4C23×D312yes—
473C22×C4C23×D312yes—
474C22×C4C23×D311yescharacteristic
475C22×C4C23×D312yes—
476C22×C4C23×D311yescharacteristic
477C22×C4C23×D312yes—
478C22×C4C23×D314yes—
479C22×C4C23×D311yescharacteristic
480C4⋊C4—28no—
481C4⋊C4—38no—
482C4⋊C4—38no—
483C4⋊C4—38no—
484C4⋊C4—38no—
485C4⋊C4—38no—
486C4⋊C4—38no—
487C4⋊C4—34no—
488C4⋊C4—64no—
489C4⋊C4—64no—
490C4⋊C4—64no—
491C4⋊C4—68no—
492C4⋊C4—124no—
493C4⋊C4—124no—
494C4⋊C4C2×C4×D318yes—
495C4⋊C4C2×C4×D314yes—
496C4⋊C4#23518yes—
497C22⋊C4—38no—
498C22⋊C4—38no—
499C22⋊C4—68no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C42—28no—
503C42—24no—
504C42—34no—
505C42—34no—
506C42—34no—
507C42—38no—
508C42—42no—
509C42—44no—
510C42—68no—
511C42—68no—
512C42—68no—
513C42—64no—
514C42—124no—
515C42—122no—
516C42C2×C4×D314yes—
517C42C2×C4×D318yes—
518C22×C3C24×C411yescharacteristic
519C22×C3C22×#4112yes—
520C22×C3C2×2−1+412yes—
521C22×C3#58511yescharacteristic
522C22×C3#58511yescharacteristic
523C2×D3—22no—
524C2×D3—22no—
525C2×D3—22no—
526C2×D3—42no—
527C2×D3—42no—
528C3×C4—24no—
529C3×C4—22no—
530C3×C4—24no—
531C3×C4—22no—
532C3×C4—28no—
533C3×C4—42no—
534C3×C4—42no—
535C3×C4C4×#4114yes—
536Dic3—24no—
537Dic3—22no—
538Dic3—22no—
539Dic3—24no—
540Dic3—42no—
541Dic3—42no—
542Dic3—44no—
543Dic3#51914yes—
544C23—32no—
545C23—62no—
546C23—61no—
547C23—62no—
548C23—61no—
549C23C24×D311yescharacteristic
550Q8—28no—
551Q8—68no—
552Q8—128no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—32no—
568C2×C4—34no—
569C2×C4—32no—
570C2×C4—32no—
571C2×C4—34no—
572C2×C4—34no—
573C2×C4—68no—
574C2×C4—62no—
575C2×C4—64no—
576C2×C4—62no—
577C2×C4—61no—
578C2×C4—62no—
579C2×C4—61no—
580C2×C4—68no—
581C2×C4—62no—
582C2×C4—64no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—64no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—64no—
589C2×C4—62no—
590C2×C4—124no—
591C2×C4C22×C4×D312yes—
592C2×C4C22×C4×D312yes—
593C2×C4C22×C4×D314yes—
594C2×C4C22×C4×D314yes—
595C2×C4C22×C4×D312yes—
596C2×C4C2×#23518yes—
597C2×C4C2×#23514yes—
598C2×C3C2×C4×#4111yescharacteristic
599C2×C3C2×#51911yescharacteristic
600C2×C3#342811yescharacteristic
601C2×C3#351512yes—
602C2×C3#351812yes—
603D3—42no—
604C22—62no—
605C22—62no—
606C22—62no—
607C22—122no—
608C22—122no—
609C22C23×C4×D311yescharacteristic
610C22C22×#23512yes—
611C22#596611yescharacteristic
612C22D3×2−1+412yes—
613C22#596811yescharacteristic
614C4—22no—
615C4—22no—
616C4—24no—
617C4—28no—
618C4—24no—
619C4—34no—
620C4—42no—
621C4—42no—
622C4—62no—
623C4—64no—
624C4—62no—
625C4—64no—
626C4—122no—
627C4—124no—
628C4—122no—
629C4C4×#23514yes—
630C3#3423111yescharacteristic, complemented, Sylow
631C2—122no—
632C2C2×C4×#23511yescharacteristic
633C2#8587812yes—
634C2D3×#51911yescharacteristic
635C2#8588511yescharacteristic
636C2#8589512yes—
subgroupG/H
center, FrattiniC23C24×D3
commutatorC22×C3C24×C4
FittingC3×#2276C2
socle, cosocleC23×C3C25
prime radicalC23×D3C24
hypercenter#2276D3
hypocenterC3#34231
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2276

Complements

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

NH ≅ G/Ntype
2. C3×#2276C2semidirect
3. #82465C2semidirect
5. #82454C2semidirect
7. #73657C2semidirect
9. #73643C2semidirect
11. C4×#889C2semidirect
12. #73616C2semidirect
15. #73393C2semidirect
17. #73386C2semidirect
23. #5578C4semidirect
100. #2276D3semidirect
630. C3#34231semidirect

Maximal subgroups

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

subgroupindex
C2×#55782
C3×#22762
#824652
#824642
#824542
#737642
#736572
#736472
#736432
#736352
C4×#8892
#736162
C4×#8922
#734132
#733932
#733922
#733862
#342313