Finite Group Databasebeta

← #10828830

Subgroups — #10828830

Order 768.

1080 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5639C211yesmaximal, characteristic
2C2×D3×#106C211yesmaximal, characteristic, complemented
3C2×#4811C211yesmaximal, characteristic, complemented
4#73912C211yesmaximal, characteristic
5C3×#2035C211yesmaximal, characteristic, complemented
6#71694C211yesmaximal, characteristic, complemented
7#70566C211yesmaximal, characteristic, complemented
8#70270C211yesmaximal, characteristic, complemented
9#70238C211yesmaximal, characteristic, complemented
10#70237C211yesmaximal, characteristic, complemented
11#69957C211yesmaximal, characteristic
12#69929C211yesmaximal, characteristic, complemented
13#69506C211yesmaximal, characteristic, complemented
14#68871C211yesmaximal, characteristic, complemented
15#68829C211yesmaximal, characteristic, complemented
16#24095—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×C4×#67C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C22×D3×C8C2211yescharacteristic
23#5639—24no—
24#5639—24no—
25C2×#892—21no—
26C2×D3×C4⋊C4C2211yescharacteristic
27C2×D3×C4⋊C4C2211yescharacteristic
28C2×#890C2211yescharacteristic
29C2×#889—21no—
30C2×#883—21no—
31C2×#879—21no—
32C2×C3×#106C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×#103C2211yescharacteristic
35C2×C3×C22⋊C8C2211yescharacteristic
36C3×#380C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#835C2211yescharacteristic
39#5212C2211yescharacteristic
40C2×#830C2211yescharacteristic
41C2×(C2×D3)⋊C8C2211yescharacteristic
42C2×#818C2211yescharacteristic
43C2×#817C2211yescharacteristic
44#4987C2211yescharacteristic
45#4984C2211yescharacteristic
46#4972C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4956—21no—
51#4949—21no—
52D3×#106—24no—
53#4811—24no—
54#4668—21no—
55#4658—21no—
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4529C2211yescharacteristic
64C2×#552—31no—
65C2×#484—31no—
66C22×#106—31no—
67#2487—31no—
68#2059—31no—
69#2059—31no—
70#2035—31no—
71#2035D311yescharacteristic, complemented
72#1892—31no—
73#1869—31no—
74#1865—31no—
75#1865—31no—
76#1854—31no—
77#1854—31no—
78#1823—31no—
79C22×C3×Q8C2311yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C22×#67D411yescharacteristic
82C22×C4×D3—21no—
83C22×C4×D3—21no—
84C22×C4×D3C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217D411yescharacteristic
98#946—22no—
99#946D414yes—
100Q8×Dic3—22no—
101Q8×Dic3—22no—
102C4×#67—22no—
103C4×#67—22no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4D411yescharacteristic
107#926—22no—
108#926—22no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4D411yescharacteristic
122C2×C4×Dic3—21no—
123C2×C4×Dic3—21no—
124C2×C4×Dic3C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C2×D3×C8—22no—
128C2×D3×C8—22no—
129C2×D3×C8—22no—
130#892—28no—
131D3×C4⋊C4—22no—
132D3×C4⋊C4—22no—
133D3×C4⋊C4—22no—
134D3×C4⋊C4D414yes—
135#890—22no—
136#890—22no—
137#889—28no—
138#883—28no—
139#879—28no—
140C3×#106—22no—
141C3×#103—22no—
142C3×#103—22no—
143C3×C22⋊C8—22no—
144C3×#95—21no—
145#835—22no—
146#835—22no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150C3⋊#95—21no—
151#830—22no—
152(C2×D3)⋊C8—22no—
153#818—22no—
154#817—22no—
155#808—22no—
156#808—22no—
157#807—22no—
158C23×C8—31no—
159#552—64no—
160#552—64no—
161C2×#126—61no—
162C2×#125—31no—
163C2×#125—61no—
164C2×#123—61no—
165C2×C22⋊Q8—31no—
166C2×C22⋊Q8C2×D311yescharacteristic
167C2×C4×Q8—31no—
168C2×C4×D4—31no—
169C2×#117—31no—
170C2×#117—61no—
171C22×C4⋊C4—31no—
172C22×C4⋊C4—31no—
173#484—64no—
174C2×#107—31no—
175C2×#107—31no—
176C2×#106—31no—
177C2×#106—31no—
178C2×#106—31no—
179C2×#106—64no—
180C2×#106C2×D311yescharacteristic
181C2×#103—31no—
182C2×#103—31no—
183C2×#103C2×D311yescharacteristic
184C2×#103C2×D311yescharacteristic
185C2×C22⋊C8—31no—
186C2×C22⋊C8—31no—
187C2×C22⋊C8C2×D311yescharacteristic
188#400—61no—
189#399—61no—
190#389—31no—
191#389—31no—
192#386—61no—
193#385—61no—
194#380—31no—
195#380—31no—
196#380C2×D311yescharacteristic
197#340—31no—
198#340—31no—
199#340—31no—
200#340—31no—
201#340—31no—
202#340C2×D311yescharacteristic
203#336—31no—
204#336—31no—
205C24×C3C2×D411yescharacteristic
206C23×D3C2×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—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2411yescharacteristic
225C2×#67—22no—
226C2×#67—22no—
227C2×#67—22no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3—21no—
232C22×Dic3C2×D411yescharacteristic
233C22×Dic3C2×D411yescharacteristic
234C22×Dic3C2×D411yescharacteristic
235C2×C4×D3—21no—
236C2×C4×D3—21no—
237C2×C4×D3—21no—
238C2×C4×D3—21no—
239C2×C4×D3—24no—
240C2×C4×D3—42no—
241C2×C4×D3—42no—
242C2×C4×D3—42no—
243C2×C4×D3—81no—
244C2×C4×D3—81no—
245C2×C4×D3C2×D412yes—
246C2×C4×D3C2×D412yes—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8#4112yes—
253C3×C4⋊C4—24no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4C2×D412yes—
260C3×C22⋊C4—24no—
261C3×C22⋊C4—81no—
262#217—24no—
263#217—22no—
264#217—81no—
265(C2×D3)⋊C4—24no—
266(C2×D3)⋊C4—22no—
267(C2×D3)⋊C4—42no—
268C4⋊Dic3—24no—
269C4⋊Dic3—24no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—81no—
277C4⋊Dic3C2×D412yes—
278C4⋊Dic3C2×D412yes—
279Dic3⋊C4—24no—
280Dic3⋊C4—24no—
281Dic3⋊C4—24no—
282Dic3⋊C4—22no—
283Dic3⋊C4—42no—
284Dic3⋊C4—81no—
285C4×Dic3—24no—
286C4×Dic3—24no—
287C4×Dic3—21no—
288C4×Dic3—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8#4112yes—
295D3×C8—28no—
296C22×Q8C22×D311yescharacteristic
297C22×D4—31no—
298C23×C4—31no—
299C23×C4—61no—
300C23×C4—61no—
301C23×C4C22×D311yescharacteristic
302C22×C8—31no—
303C22×C8—31no—
304C22×C8—62no—
305C22×C8—62no—
306C22×C8—62no—
307C22×C8C22×D311yescharacteristic
308C22×C8C22×D311yescharacteristic
309#126—68no—
310#125—62no—
311#125—68no—
312#125—62no—
313#123—68no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—34no—
317C22⋊Q8—62no—
318C4×Q8—62no—
319C4×Q8—62no—
320C4×D4—62no—
321C4×D4—62no—
322#117—68no—
323#117—62no—
324#117—62no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—34no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—62no—
333C2×C4⋊C4—62no—
334C2×C4⋊C4—62no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4C22×D311yescharacteristic
344C2×C4⋊C4C22×D311yescharacteristic
345C2×C4⋊C4C22×D311yescharacteristic
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—31no—
349C2×C22⋊C4—31no—
350C2×C22⋊C4—61no—
351C2×C22⋊C4—61no—
352C2×C22⋊C4—61no—
353C2×C22⋊C4—61no—
354C2×C42—31no—
355C2×C42—61no—
356C2×C42—61no—
357#107—62no—
358#107—62no—
359#106—22no—
360#106—62no—
361#106—62no—
362#106—62no—
363#103—22no—
364#103—22no—
365#103—62no—
366#103—62no—
367C22⋊C8—22no—
368C22⋊C8—62no—
369C22⋊C8—62no—
370#95—21no—
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—21no—
382C23×C3C22×D411yescharacteristic
383C22×D3—21no—
384C22×D3—21no—
385C22×D3—21no—
386C22×D3—21no—
387C22×D3#12012yes—
388C22×D3#13512yes—
389C22×D3#13712yes—
390C3×Q8—22no—
391C3×Q8—22no—
392C3×Q8—41no—
393C3×Q8—41no—
394C2×C3×C4—21no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4C22×D411yescharacteristic
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4C2×#4111yescharacteristic
416C2×C3×C4C2×#4111yescharacteristic
417C2×C3×C42−1+411yescharacteristic
418C2×C3×C42−1+411yescharacteristic
419#67—44no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—22no—
424C2×Dic3—21no—
425C2×Dic3—22no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439C2×Dic3—41no—
440C2×Dic3#12012yes—
441C4×D3—22no—
442C4×D3—22no—
443C4×D3—22no—
444C4×D3—22no—
445C4×D3—48no—
446C4×D3—84no—
447C3×C8—22no—
448C3×C8—41no—
449C3⋊C8—22no—
450C3⋊C8—41no—
451C24—31no—
452C24D3×D411yescharacteristic
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8D3×D412yes—
459C2×D4—62no—
460C2×D4—62no—
461C2×D4—62no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—31no—
473C22×C4—32no—
474C22×C4—32no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—64no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—61no—
486C22×C4—122no—
487C22×C4—122no—
488C22×C4—122no—
489C22×C4—241no—
490C22×C4—241no—
491C22×C4D3×D411yescharacteristic
492C22×C4C23×D311yescharacteristic
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—32no—
499C2×C8—68no—
500C2×C8—61no—
501C2×C8—61no—
502C2×C8—61no—
503C2×C8—61no—
504C2×C8—61no—
505C2×C8#23712yes—
506C4⋊C4—24no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—32no—
513C4⋊C4—32no—
514C4⋊C4—64no—
515C4⋊C4—62no—
516C4⋊C4—64no—
517C4⋊C4—64no—
518C4⋊C4—64no—
519C4⋊C4—64no—
520C4⋊C4—61no—
521C4⋊C4—61no—
522C4⋊C4—61no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—122no—
527C4⋊C4—241no—
528C4⋊C4—241no—
529C4⋊C4D3×D412yes—
530C22⋊C4—24no—
531C22⋊C4—62no—
532C22⋊C4—64no—
533C22⋊C4—62no—
534C22⋊C4—64no—
535C22⋊C4—81no—
536C22⋊C4—122no—
537C22⋊C4—241no—
538C42—64no—
539C42—64no—
540C42—61no—
541C42—61no—
542C22×C3—41no—
543C22×C3—41no—
544C22×C3—41no—
545C22×C3—41no—
546C22×C3—41no—
547C22×C3—41no—
548C22×C3—41no—
549C22×C3C2×#12011yescharacteristic
550C22×C3#54711yescharacteristic
551C22×C3#54711yescharacteristic
552C22×C3C2×#13511yescharacteristic
553C22×C3C2×#13711yescharacteristic
554C22×C3#57711yescharacteristic
555C22×C3#57711yescharacteristic
556C2×D3—22no—
557C2×D3—22no—
558C2×D3—22no—
559C2×D3—22no—
560C2×D3—22no—
561C2×D3—22no—
562C2×D3#45214yes—
563C3×C4—41no—
564C3×C4—42no—
565C3×C4—42no—
566C3×C4—81no—
567C3×C4—81no—
568C3×C4#52511yescharacteristic
569C3×C4#53611yescharacteristic
570C3×C4#54711yescharacteristic
571C3×C4#54711yescharacteristic
572Dic3—22no—
573Dic3—42no—
574Dic3—41no—
575Dic3—41no—
576Dic3—42no—
577Dic3—81no—
578Dic3—81no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—32no—
587C23—32no—
588C23—32no—
589C23—61no—
590C23—61no—
591C23—61no—
592C23—61no—
593C23C2×D3×D411yescharacteristic
594Q8—22no—
595Q8—22no—
596Q8—41no—
597Q8—41no—
598D4—124no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—32no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—62no—
622C2×C4—62no—
623C2×C4—62no—
624C2×C4—62no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—62no—
631C2×C4—62no—
632C2×C4—61no—
633C2×C4—128no—
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—244no—
645C2×C4C2×D3×D411yescharacteristic
646C2×C4C2×D3×D411yescharacteristic
647C2×C4C2×#23711yescharacteristic
648C2×C4D3×#4111yescharacteristic
649C2×C4#101011yescharacteristic
650C2×C4#101011yescharacteristic
651C8—22no—
652C8—41no—
653C8—62no—
654C8—121no—
655C2×C3—81no—
656C2×C3#240611yescharacteristic
657C2×C3C2×#45211yescharacteristic
658C2×C3#301011yescharacteristic
659C2×C3#333711yescharacteristic
660C2×C3#333711yescharacteristic
661C2×C3#334411yescharacteristic
662C2×C3#334411yescharacteristic
663D3—24no—
664C22—34no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—62no—
673C22—62no—
674C22—62no—
675C22—62no—
676C22—62no—
677C22—62no—
678C22D3×#12011yescharacteristic
679C22#561111yescharacteristic
680C22#565511yescharacteristic
681C22#576211yescharacteristic
682C22D3×#13511yescharacteristic
683C22D3×#13711yescharacteristic
684C22#577811yescharacteristic
685C4—41no—
686C4—42no—
687C4—42no—
688C4—62no—
689C4—81no—
690C4—81no—
691C4—121no—
692C4—122no—
693C4—121no—
694C4—122no—
695C4—241no—
696C4—241no—
697C4#560611yescharacteristic
698C4#561111yescharacteristic
699C4#565511yescharacteristic
700C4#567511yescharacteristic
701C3#2409511yescharacteristic, complemented, Sylow
702C2—64no—
703C2—81no—
704C2#7710711yescharacteristic
705C2#8004111yescharacteristic
706C2D3×#45211yescharacteristic
707C2#8067311yescharacteristic
708C2#8091911yescharacteristic
709C2#8143811yescharacteristic
710C2#8153111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2035C2
socleC23×C3C22×D4
prime radicalC22×#67D4
hypercenter#2035D3
hypocenterC3#24095
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
2. C2×D3×#106C2semidirect
3. C2×#4811C2semidirect
5. C3×#2035C2semidirect
6. #71694C2semidirect
7. #70566C2semidirect
8. #70270C2semidirect
9. #70238C2semidirect
10. #70237C2semidirect
12. #69929C2semidirect
13. #69506C2semidirect
14. #68871C2semidirect
15. #68829C2semidirect
71. #2035D3semidirect
701. C3#24095semidirect

Maximal subgroups

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

subgroupindex
C2×#56392
C2×D3×#1062
C2×#48112
#739122
C3×#20352
#716942
#705662
#702702
#702382
#702372
#699572
#699292
#695062
#688712
#688292
#240953