Finite Group Databasebeta

← #10901403

Subgroups — #10901403

Order 768.

1056 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#529C211yesmaximal, characteristic
2C2×#5022C211yesmaximal, characteristic, complemented
3C2×#5009C211yesmaximal, characteristic, complemented
4#82074C211yesmaximal, characteristic, complemented
5#71739C211yesmaximal, characteristic, complemented
6#71739C211yesmaximal, characteristic, complemented
7#71722C211yesmaximal, characteristic, complemented
8#71722C211yesmaximal, characteristic, complemented
9#71695C211yesmaximal, characteristic, complemented
10#71694C211yesmaximal, characteristic, complemented
11#71680C211yesmaximal, characteristic
12#71679C211yesmaximal, characteristic
13#71627C211yesmaximal, characteristic, complemented
14#71605C211yesmaximal, characteristic, complemented
15#71530C211yesmaximal, characteristic, complemented
16#29693—31nomaximal, Sylow
17C3×#529—24no—
18C3×#529—24no—
19C2×C3×#126—21no—
20C2×C3×#125C2211yescharacteristic
21C2×C3×#124C2211yescharacteristic
22C2×C3×#123—21no—
23C2×C3×C22⋊Q8C2211yescharacteristic
24C2×C3×#121C2211yescharacteristic
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×D4—21no—
27C2×C3×C4×D4C2211yescharacteristic
28C2×C3×#117—21no—
29C22×C3×C42C2211yescharacteristic
30C2×#870C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#832C2211yescharacteristic
39#5212C2211yescharacteristic
40#5212C2211yescharacteristic
41#5210—21no—
42#5210—21no—
43C2×#830C2211yescharacteristic
44C2×#830C2211yescharacteristic
45#5022—24no—
46#5009—24no—
47C2×#808C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#807C2211yescharacteristic
50C2×#807C2211yescharacteristic
51#4936C2211yescharacteristic
52#4933—21no—
53#4933—21no—
54#4928C2211yescharacteristic
55#4928C2211yescharacteristic
56C2×C4⋊(C3⋊C8)C2211yescharacteristic
57C2×C4⋊(C3⋊C8)C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4524C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#529D311yescharacteristic
65C2×#479—31no—
66C2×#478—31no—
67#2851—31no—
68#2059—31no—
69#2059—31no—
70#2042—31no—
71#2042—31no—
72#2036—31no—
73#2035—31no—
74#2034—31no—
75#2033—31no—
76#1805—31no—
77#1804—31no—
78#1797—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C3×D4—21no—
81C22×C3×D4C2311yescharacteristic
82C23×C3×C4—21no—
83C23×C3×C4D411yescharacteristic
84C23×C3×C4D411yescharacteristic
85C23×C3×C4C2311yescharacteristic
86C3×#126—28no—
87C3×#125—22no—
88C3×#125—22no—
89C3×#124—22no—
90C3×#124—22no—
91C3×#123—28no—
92C3×C22⋊Q8—22no—
93C3×C22⋊Q8—22no—
94C3×#121—22no—
95C3×#121—22no—
96C3×C4×Q8—22no—
97C3×C4×Q8D414yes—
98C3×C4×D4—28no—
99C3×C4×D4—22no—
100C3×C4×D4D414yes—
101C3×#117—28no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C42—21no—
114C2×C3×C42—22no—
115C2×C3×C42—22no—
116C2×C3×C42D411yescharacteristic
117C2×C3×C42D411yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C2×#217—21no—
121C2×#217—21no—
122C2×C4⋊Dic3C2311yescharacteristic
123C2×C4⋊Dic3C2311yescharacteristic
124C2×C4⋊Dic3C2311yescharacteristic
125C2×C4⋊Dic3C2311yescharacteristic
126C2×Dic3⋊C4—21no—
127C2×Dic3⋊C4—21no—
128C2×Dic3⋊C4—21no—
129C2×Dic3⋊C4—21no—
130C22×C3⋊C8C2311yescharacteristic
131C22×C3⋊C8C2311yescharacteristic
132C22×C3⋊C8C2311yescharacteristic
133C22×C3⋊C8C2311yescharacteristic
134#870—22no—
135#870—22no—
136#835—22no—
137#835—22no—
138#835—41no—
139#835—41no—
140#832—22no—
141#832—22no—
142#832—41no—
143#832—41no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150#830—41no—
151#830—41no—
152#808—22no—
153#808—22no—
154#807—22no—
155#807—22no—
156C4⋊(C3⋊C8)—22no—
157C4⋊(C3⋊C8)—22no—
158#529—24no—
159#529—24no—
160C2×#126—21no—
161C2×#125—31no—
162C2×#125C2×D311yescharacteristic
163C2×#124C2×D311yescharacteristic
164C2×#123—21no—
165C2×C22⋊Q8C2×D311yescharacteristic
166C2×#121C2×D311yescharacteristic
167C2×C4×Q8C2×D311yescharacteristic
168C2×C4×D4—21no—
169C2×C4×D4C2×D311yescharacteristic
170C2×#117—21no—
171C22×C42C2×D311yescharacteristic
172#479—64no—
173#478—64no—
174C2×#107—31no—
175C2×#107—31no—
176C2×#106—31no—
177C2×#106—31no—
178C2×C4⋊C8—31no—
179C2×C4⋊C8—31no—
180C2×#103—31no—
181C2×#103—31no—
182C2×#103—31no—
183C2×#103—31no—
184C2×#102—31no—
185C2×#102—31no—
186C2×#102—31no—
187C2×#102—31no—
188C2×C22⋊C8—31no—
189C2×C22⋊C8—31no—
190#389—31no—
191#389—31no—
192#387—61no—
193#387—61no—
194#385—61no—
195#385—61no—
196#382—31no—
197#380—31no—
198#380—31no—
199#340—31no—
200#340—31no—
201#340—31no—
202#340—31no—
203#336—31no—
204#336—31no—
205C24×C3—21no—
206C24×C3C2×D411yescharacteristic
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8C2×D412yes—
213C2×C3×D4—24no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—42no—
220C2×C3×D4—42no—
221C2×C3×D4C2×D412yes—
222C22×C3×C4—22no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—22no—
227C22×C3×C4—22no—
228C22×C3×C4—24no—
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4—21no—
234C22×C3×C4—21no—
235C22×C3×C4—41no—
236C22×C3×C4—41no—
237C22×C3×C4—41no—
238C22×C3×C4—42no—
239C22×C3×C4—42no—
240C22×C3×C4C2×D411yescharacteristic
241C22×C3×C4C2×D411yescharacteristic
242C22×C3×C4C2×D411yescharacteristic
243C22×C3×C4C2×D411yescharacteristic
244C22×C3×C4C2×D411yescharacteristic
245C22×C3×C4C2411yescharacteristic
246C22×Dic3—21no—
247C22×Dic3—21no—
248C22×Dic3—21no—
249C22×Dic3—21no—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—24no—
252C3×C4⋊C4—24no—
253C3×C4⋊C4—24no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4C2×D412yes—
257C3×C4⋊C4C2×D412yes—
258C3×C22⋊C4—24no—
259C3×C22⋊C4—24no—
260C3×C22⋊C4—24no—
261C3×C22⋊C4—24no—
262C3×C22⋊C4—24no—
263C3×C42—22no—
264C3×C42—22no—
265C3×C42—24no—
266C3×C42—21no—
267C3×C42—21no—
268C3×C42C2×D412yes—
269C3×C42C2×D412yes—
270#217—42no—
271#217—42no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280Dic3⋊C4—42no—
281Dic3⋊C4—42no—
282Dic3⋊C4—42no—
283Dic3⋊C4—42no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C22×Q8C22×D311yescharacteristic
297C22×D4—21no—
298C22×D4C22×D311yescharacteristic
299C23×C4—21no—
300C23×C4#6711yescharacteristic
301C23×C4#6711yescharacteristic
302C23×C4C22×D311yescharacteristic
303C22×C8—31no—
304C22×C8—31no—
305C22×C8—31no—
306C22×C8—31no—
307#126—28no—
308#125—22no—
309#125—22no—
310#125—62no—
311#125—62no—
312#124—22no—
313#124—22no—
314#123—28no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317#121—22no—
318#121—22no—
319C4×Q8—22no—
320C4×Q8#6714yes—
321C4×D4—28no—
322C4×D4—22no—
323C4×D4#6714yes—
324#117—28no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4C22×D311yescharacteristic
338C2×C4⋊C4C22×D311yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—61no—
345C2×C22⋊C4—61no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42#6711yescharacteristic
350C2×C42#6711yescharacteristic
351C2×C42C22×D311yescharacteristic
352C2×C42C22×D311yescharacteristic
353#107—62no—
354#107—62no—
355#106—62no—
356#106—62no—
357C4⋊C8—62no—
358C4⋊C8—62no—
359#103—62no—
360#103—62no—
361#103—121no—
362#103—121no—
363#102—62no—
364#102—62no—
365#102—121no—
366#102—121no—
367C22⋊C8—121no—
368C22⋊C8—121no—
369#95—61no—
370#95—61no—
371#95—61no—
372#95—61no—
373#95—61no—
374#95—61no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—21no—
378C23×C3—21no—
379C23×C3—21no—
380C23×C3—21no—
381C23×C3—22no—
382C23×C3—41no—
383C23×C3—41no—
384C23×C3—41no—
385C23×C3—41no—
386C23×C3—41no—
387C23×C3—41no—
388C23×C3#13512yes—
389C23×C3C22×D411yescharacteristic
390C3×Q8—22no—
391C3×Q8—22no—
392C3×Q8—41no—
393C3×Q8—41no—
394C3×D4—22no—
395C3×D4—22no—
396C3×D4—48no—
397C3×D4—41no—
398C3×D4—41no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—44no—
416C2×C3×C4—44no—
417C2×C3×C4—48no—
418C2×C3×C4—41no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4—41no—
422C2×C3×C4—41no—
423C2×C3×C4—41no—
424C2×C3×C4—41no—
425C2×C3×C4—41no—
426C2×C3×C4—41no—
427C2×C3×C4—41no—
428C2×C3×C4#12012yes—
429C2×C3×C4#12012yes—
430C2×C3×C4C22×D411yescharacteristic
431C2×C3×C4C22×D411yescharacteristic
432C2×C3×C4H(5,𝔽2)11yescharacteristic
433C2×C3×C42−1+411yescharacteristic
434C2×C3×C42−1+411yescharacteristic
435C2×C3×C42−1+411yescharacteristic
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439C2×Dic3—41no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—41no—
445C2×Dic3—41no—
446C2×Dic3—41no—
447C2×Dic3—41no—
448C3⋊C8—41no—
449C3⋊C8—41no—
450C3⋊C8—41no—
451C3⋊C8—41no—
452C24—21no—
453C24C2×#6711yescharacteristic
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8C2×#6712yes—
460C2×D4—24no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4C2×#6712yes—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—61no—
488C22×C4—61no—
489C22×C4—61no—
490C22×C4—61no—
491C22×C4C2×#6711yescharacteristic
492C22×C4C2×#6711yescharacteristic
493C22×C4C2×#6711yescharacteristic
494C22×C4C2×#6711yescharacteristic
495C22×C4C2×#6711yescharacteristic
496C22×C4C23×D311yescharacteristic
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C2×C8—61no—
501C2×C8—61no—
502C2×C8—61no—
503C2×C8—61no—
504C2×C8—61no—
505C2×C8—61no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C4⋊C4—61no—
518C4⋊C4—61no—
519C4⋊C4—61no—
520C4⋊C4—61no—
521C4⋊C4—61no—
522C4⋊C4—61no—
523C4⋊C4—122no—
524C4⋊C4—122no—
525C4⋊C4—122no—
526C4⋊C4—122no—
527C4⋊C4C2×#6712yes—
528C4⋊C4C2×#6712yes—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—122no—
535C22⋊C4—122no—
536C42—22no—
537C42—22no—
538C42—24no—
539C42—21no—
540C42—21no—
541C42C2×#6712yes—
542C42C2×#6712yes—
543C22×C3—22no—
544C22×C3—22no—
545C22×C3—22no—
546C22×C3—41no—
547C22×C3—41no—
548C22×C3—41no—
549C22×C3—41no—
550C22×C3—42no—
551C22×C3—42no—
552C22×C3—42no—
553C22×C3—81no—
554C22×C3—81no—
555C22×C3—81no—
556C22×C3—81no—
557C22×C3C2×#12011yescharacteristic
558C22×C3#53611yescharacteristic
559C22×C3#53711yescharacteristic
560C22×C3C2×#13511yescharacteristic
561C22×C3#57511yescharacteristic
562C22×C3#57511yescharacteristic
563C22×C3#57511yescharacteristic
564C3×C4—22no—
565C3×C4—22no—
566C3×C4—41no—
567C3×C4—42no—
568C3×C4—41no—
569C3×C4—42no—
570C3×C4—41no—
571C3×C4—42no—
572C3×C4#53611yescharacteristic
573C3×C4#53611yescharacteristic
574C3×C4#53711yescharacteristic
575C3×C4#53711yescharacteristic
576Dic3—81no—
577Dic3—81no—
578Dic3—81no—
579Dic3—81no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—22no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23#95012yes—
594C23C22×#6711yescharacteristic
595Q8—22no—
596Q8—22no—
597Q8—41no—
598Q8—41no—
599D4—22no—
600D4—22no—
601D4—48no—
602D4—41no—
603D4—41no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—48no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4#95312yes—
646C2×C4#95312yes—
647C2×C4#100911yescharacteristic
648C2×C4#101011yescharacteristic
649C2×C4#101011yescharacteristic
650C2×C4#101011yescharacteristic
651C2×C4C22×#6711yescharacteristic
652C2×C4C22×#6711yescharacteristic
653C8—121no—
654C8—121no—
655C8—121no—
656C8—121no—
657C2×C3—42no—
658C2×C3—82no—
659C2×C3#261611yescharacteristic
660C2×C3#300211yescharacteristic
661C2×C3#300511yescharacteristic
662C2×C3#323211yescharacteristic
663C2×C3#324011yescharacteristic
664C2×C3#325011yescharacteristic
665C2×C3#325611yescharacteristic
666C22—22no—
667C22—22no—
668C22—22no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22C2×#95011yescharacteristic
681C22#582211yescharacteristic
682C22#582211yescharacteristic
683C22#582211yescharacteristic
684C22#582511yescharacteristic
685C22#583311yescharacteristic
686C22C2×#95311yescharacteristic
687C4—22no—
688C4—22no—
689C4—41no—
690C4—42no—
691C4—41no—
692C4—42no—
693C4—41no—
694C4—42no—
695C4—241no—
696C4—241no—
697C4—241no—
698C4—241no—
699C4#582511yescharacteristic
700C4#582511yescharacteristic
701C4#583311yescharacteristic
702C4#583311yescharacteristic
703C3#2969311yescharacteristic, complemented, Sylow
704C2—42no—
705C2—82no—
706C2#8312111yescharacteristic
707C2#8312511yescharacteristic
708C2#8314911yescharacteristic
709C2#8315111yescharacteristic
710C2#8408011yescharacteristic
711C2#8408111yescharacteristic
712C2#8409411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#529C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenterC2×#529D3
hypocenterC3#29693
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
2. C2×#5022C2semidirect
3. C2×#5009C2semidirect
4. #82074C2semidirect
5. #71739C2semidirect
6. #71739C2semidirect
7. #71722C2semidirect
8. #71722C2semidirect
9. #71695C2semidirect
10. #71694C2semidirect
13. #71627C2semidirect
14. #71605C2semidirect
15. #71530C2semidirect
703. C3#29693semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5292
C2×#50222
C2×#50092
#820742
#717392
#717222
#716952
#716942
#716802
#716792
#716272
#716052
#715302
#296933