Finite Group Databasebeta

← #10828819

Subgroups — #10828819

Order 768.

1100 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×#5639C211yesmaximal, characteristic
2C2×D3×#107C211yesmaximal, characteristic, complemented
3C2×#4805C211yesmaximal, characteristic, complemented
4#73911C211yesmaximal, characteristic
5C3×#2036C211yesmaximal, characteristic, complemented
6#71695C211yesmaximal, characteristic, complemented
7#70618C211yesmaximal, characteristic, complemented
8#70270C211yesmaximal, characteristic, complemented
9#70236C211yesmaximal, characteristic, complemented
10#70235C211yesmaximal, characteristic, complemented
11#69957C211yesmaximal, characteristic
12#69929C211yesmaximal, characteristic, complemented
13#69506C211yesmaximal, characteristic, complemented
14#68871C211yesmaximal, characteristic, complemented
15#68828C211yesmaximal, characteristic, complemented
16#24094—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×#883—21no—
32C2×#879—21no—
33C2×C3×#107C2211yescharacteristic
34C2×C3×#103C2211yescharacteristic
35C2×C3×#103C2211yescharacteristic
36C2×C3×C22⋊C8C2211yescharacteristic
37C3×#380C2211yescharacteristic
38C2×#835C2211yescharacteristic
39C2×#835C2211yescharacteristic
40#5212C2211yescharacteristic
41C2×#830C2211yescharacteristic
42C2×(C2×D3)⋊C8C2211yescharacteristic
43C2×#818C2211yescharacteristic
44C2×#817C2211yescharacteristic
45#4987C2211yescharacteristic
46#4984C2211yescharacteristic
47#4972C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#807C2211yescharacteristic
50C2×#807C2211yescharacteristic
51#4952—21no—
52D3×#107—24no—
53#4805—24no—
54#4664—21no—
55#4657—21no—
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4529C2211yescharacteristic
64C2×#552—31no—
65C2×#483—31no—
66C22×#107—31no—
67#2486—31no—
68#2059—31no—
69#2059—31no—
70#2036—31no—
71#2036D311yescharacteristic, complemented
72#1892—31no—
73#1869—31no—
74#1863—31no—
75#1863—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#883—44no—
140#879—28no—
141C3×#107—22no—
142C3×#103—22no—
143C3×#103—22no—
144C3×C22⋊C8—22no—
145C3×#95—21no—
146#835—22no—
147#835—22no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150C3⋊#95—21no—
151C3⋊#95—21no—
152#830—22no—
153(C2×D3)⋊C8—22no—
154#818—22no—
155#817—22no—
156#808—22no—
157#807—22no—
158#807—22no—
159C23×C8—31no—
160#552—64no—
161#552—64no—
162C2×#126—61no—
163C2×#125—31no—
164C2×#125—61no—
165C2×#123—61no—
166C2×#123—61no—
167C2×C22⋊Q8—31no—
168C2×C22⋊Q8C2×D311yescharacteristic
169C2×C4×Q8—31no—
170C2×C4×D4—31no—
171C2×#117—31no—
172C2×#117—61no—
173C22×C4⋊C4—31no—
174C22×C4⋊C4—31no—
175#483—64no—
176C2×#107—31no—
177C2×#107—31no—
178C2×#107—31no—
179C2×#107—64no—
180C2×#107C2×D311yescharacteristic
181C2×#106—31no—
182C2×#106—31no—
183C2×#103—31no—
184C2×#103—31no—
185C2×#103C2×D311yescharacteristic
186C2×#103C2×D311yescharacteristic
187C2×C22⋊C8—31no—
188C2×C22⋊C8—31no—
189C2×C22⋊C8C2×D311yescharacteristic
190#398—61no—
191#397—61no—
192#389—31no—
193#389—31no—
194#381—61no—
195#380—31no—
196#380—31no—
197#380C2×D311yescharacteristic
198#340—31no—
199#340—31no—
200#340—31no—
201#340—31no—
202#340—31no—
203#340C2×D311yescharacteristic
204#336—31no—
205#336—31no—
206C24×C3C2×D411yescharacteristic
207C23×D3C2×D411yescharacteristic
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8C2×D412yes—
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×C4—21no—
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C2×#67—22no—
227C2×#67—22no—
228C2×#67—22no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3—21no—
232C22×Dic3—21no—
233C22×Dic3C2×D411yescharacteristic
234C22×Dic3C2×D411yescharacteristic
235C22×Dic3C2×D411yescharacteristic
236C2×C4×D3—21no—
237C2×C4×D3—21no—
238C2×C4×D3—21no—
239C2×C4×D3—21no—
240C2×C4×D3—24no—
241C2×C4×D3—42no—
242C2×C4×D3—42no—
243C2×C4×D3—42no—
244C2×C4×D3—42no—
245C2×C4×D3—42no—
246C2×C4×D3C2×D412yes—
247C2×C4×D3C2×D412yes—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8#4112yes—
254C3×C4⋊C4—24no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4C2×D412yes—
261C3×C22⋊C4—24no—
262C3×C22⋊C4—42no—
263#217—24no—
264#217—22no—
265#217—42no—
266(C2×D3)⋊C4—24no—
267(C2×D3)⋊C4—22no—
268(C2×D3)⋊C4—42no—
269C4⋊Dic3—24no—
270C4⋊Dic3—24no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—42no—
278C4⋊Dic3C2×D412yes—
279C4⋊Dic3C2×D412yes—
280Dic3⋊C4—24no—
281Dic3⋊C4—24no—
282Dic3⋊C4—24no—
283Dic3⋊C4—22no—
284Dic3⋊C4—42no—
285Dic3⋊C4—42no—
286C4×Dic3—24no—
287C4×Dic3—24no—
288C4×Dic3—21no—
289C4×Dic3—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8#4112yes—
296D3×C8—28no—
297C22×Q8C22×D311yescharacteristic
298C22×D4—31no—
299C23×C4—31no—
300C23×C4—61no—
301C23×C4—61no—
302C23×C4C22×D311yescharacteristic
303C22×C8—31no—
304C22×C8—31no—
305C22×C8—62no—
306C22×C8—62no—
307C22×C8—62no—
308C22×C8C22×D311yescharacteristic
309C22×C8C22×D311yescharacteristic
310#126—68no—
311#125—62no—
312#125—68no—
313#125—62no—
314#123—68no—
315#123—124no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C22⋊Q8—34no—
319C22⋊Q8—62no—
320C4×Q8—62no—
321C4×Q8—62no—
322C4×D4—62no—
323C4×D4—62no—
324#117—68no—
325#117—62no—
326#117—62no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—62no—
335C2×C4⋊C4—62no—
336C2×C4⋊C4—62no—
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⋊C4—61no—
344C2×C4⋊C4—61no—
345C2×C4⋊C4C22×D311yescharacteristic
346C2×C4⋊C4C22×D311yescharacteristic
347C2×C4⋊C4C22×D311yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—31no—
351C2×C22⋊C4—31no—
352C2×C22⋊C4—61no—
353C2×C22⋊C4—61no—
354C2×C22⋊C4—61no—
355C2×C22⋊C4—61no—
356C2×C42—31no—
357C2×C42—61no—
358C2×C42—61no—
359#107—22no—
360#107—62no—
361#107—62no—
362#107—62no—
363#106—62no—
364#106—62no—
365#103—22no—
366#103—22no—
367#103—62no—
368#103—62no—
369C22⋊C8—22no—
370C22⋊C8—62no—
371C22⋊C8—62no—
372#95—21no—
373#95—61no—
374#95—61no—
375#95—61no—
376#95—61no—
377C23×C3—21no—
378C23×C3—21no—
379C23×C3—21no—
380C23×C3—21no—
381C23×C3—21no—
382C23×C3—21no—
383C23×C3—21no—
384C23×C3C22×D411yescharacteristic
385C22×D3—21no—
386C22×D3—21no—
387C22×D3—21no—
388C22×D3—21no—
389C22×D3#12012yes—
390C22×D3#13512yes—
391C22×D3AGL(1,ℤ/8)12yes—
392C3×Q8—22no—
393C3×Q8—22no—
394C3×Q8—41no—
395C3×Q8—41no—
396C2×C3×C4—21no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
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×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4C22×D411yescharacteristic
416C2×C3×C4C22×D411yescharacteristic
417C2×C3×C4C2×#4111yescharacteristic
418C2×C3×C4C2×#4111yescharacteristic
419C2×C3×C42−1+411yescharacteristic
420C2×C3×C42−1+411yescharacteristic
421#67—44no—
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—21no—
425C2×Dic3—22no—
426C2×Dic3—21no—
427C2×Dic3—22no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
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—41no—
441C2×Dic3—41no—
442C2×Dic3#12012yes—
443C4×D3—22no—
444C4×D3—22no—
445C4×D3—22no—
446C4×D3—22no—
447C4×D3—48no—
448C4×D3—84no—
449C3×C8—22no—
450C3×C8—41no—
451C3⋊C8—22no—
452C3⋊C8—41no—
453C24—31no—
454C24D3×D411yescharacteristic
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8D3×D412yes—
461C2×D4—62no—
462C2×D4—62no—
463C2×D4—62no—
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—21no—
473C22×C4—21no—
474C22×C4—31no—
475C22×C4—32no—
476C22×C4—32no—
477C22×C4—31no—
478C22×C4—31no—
479C22×C4—64no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—61no—
486C22×C4—61no—
487C22×C4—61no—
488C22×C4—122no—
489C22×C4—122no—
490C22×C4—122no—
491C22×C4—122no—
492C22×C4—122no—
493C22×C4D3×D411yescharacteristic
494C22×C4C23×D311yescharacteristic
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—32no—
501C2×C8—68no—
502C2×C8—61no—
503C2×C8—61no—
504C2×C8—61no—
505C2×C8—61no—
506C2×C8—61no—
507C2×C8#23712yes—
508C4⋊C4—24no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—32no—
515C4⋊C4—32no—
516C4⋊C4—64no—
517C4⋊C4—62no—
518C4⋊C4—64no—
519C4⋊C4—64no—
520C4⋊C4—64no—
521C4⋊C4—64no—
522C4⋊C4—61no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—61no—
528C4⋊C4—122no—
529C4⋊C4—122no—
530C4⋊C4—122no—
531C4⋊C4D3×D412yes—
532C22⋊C4—24no—
533C22⋊C4—42no—
534C22⋊C4—62no—
535C22⋊C4—64no—
536C22⋊C4—62no—
537C22⋊C4—64no—
538C22⋊C4—122no—
539C22⋊C4—122no—
540C42—64no—
541C42—64no—
542C42—61no—
543C42—61no—
544C22×C3—41no—
545C22×C3—41no—
546C22×C3—41no—
547C22×C3—41no—
548C22×C3—41no—
549C22×C3—41no—
550C22×C3—41no—
551C22×C3C2×#12011yescharacteristic
552C22×C3#54711yescharacteristic
553C22×C3#54711yescharacteristic
554C22×C3C2×#13511yescharacteristic
555C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
556C22×C3#57811yescharacteristic
557C22×C3#57811yescharacteristic
558C2×D3—22no—
559C2×D3—22no—
560C2×D3—22no—
561C2×D3—22no—
562C2×D3—22no—
563C2×D3—22no—
564C2×D3#44914yes—
565C3×C4—41no—
566C3×C4—42no—
567C3×C4—42no—
568C3×C4—81no—
569C3×C4—81no—
570C3×C4#52511yescharacteristic
571C3×C4#53611yescharacteristic
572C3×C4#54711yescharacteristic
573C3×C4#54711yescharacteristic
574Dic3—22no—
575Dic3—42no—
576Dic3—41no—
577Dic3—41no—
578Dic3—42no—
579Dic3—81no—
580Dic3—81no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—32no—
589C23—32no—
590C23—32no—
591C23—61no—
592C23—61no—
593C23—61no—
594C23—61no—
595C23C2×D3×D411yescharacteristic
596Q8—22no—
597Q8—22no—
598Q8—41no—
599Q8—41no—
600D4—124no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—32no—
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—41no—
620C2×C4—41no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—62no—
624C2×C4—62no—
625C2×C4—62no—
626C2×C4—62no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—61no—
631C2×C4—61no—
632C2×C4—62no—
633C2×C4—62no—
634C2×C4—61no—
635C2×C4—128no—
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—121no—
646C2×C4—244no—
647C2×C4C2×D3×D411yescharacteristic
648C2×C4C2×D3×D411yescharacteristic
649C2×C4C2×#23711yescharacteristic
650C2×C4D3×#4111yescharacteristic
651C2×C4#101011yescharacteristic
652C2×C4#101011yescharacteristic
653C8—22no—
654C8—41no—
655C8—62no—
656C8—121no—
657C2×C3—81no—
658C2×C3#240611yescharacteristic
659C2×C3C2×#44911yescharacteristic
660C2×C3#301111yescharacteristic
661C2×C3#333611yescharacteristic
662C2×C3#333611yescharacteristic
663C2×C3#334311yescharacteristic
664C2×C3#334311yescharacteristic
665D3—24no—
666C22—34no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—62no—
675C22—62no—
676C22—62no—
677C22—62no—
678C22—62no—
679C22—62no—
680C22D3×#12011yescharacteristic
681C22#561111yescharacteristic
682C22#565511yescharacteristic
683C22#576611yescharacteristic
684C22D3×#13511yescharacteristic
685C22D3×AGL(1,ℤ/8)11yescharacteristic
686C22#577911yescharacteristic
687C4—41no—
688C4—42no—
689C4—42no—
690C4—62no—
691C4—81no—
692C4—81no—
693C4—121no—
694C4—122no—
695C4—121no—
696C4—122no—
697C4—241no—
698C4—241no—
699C4#560611yescharacteristic
700C4#561111yescharacteristic
701C4#565511yescharacteristic
702C4#567511yescharacteristic
703C3#2409411yescharacteristic, complemented, Sylow
704C2—64no—
705C2—81no—
706C2#7710711yescharacteristic
707C2D3×#44911yescharacteristic
708C2#8018511yescharacteristic
709C2#8077011yescharacteristic
710C2#8082911yescharacteristic
711C2#8147711yescharacteristic
712C2#8153011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2036C2
socleC23×C3C22×D4
prime radicalC22×#67D4
hypercenter#2036D3
hypocenterC3#24094
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2036

Complements

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

NH ≅ G/Ntype
2. C2×D3×#107C2semidirect
3. C2×#4805C2semidirect
5. C3×#2036C2semidirect
6. #71695C2semidirect
7. #70618C2semidirect
8. #70270C2semidirect
9. #70236C2semidirect
10. #70235C2semidirect
12. #69929C2semidirect
13. #69506C2semidirect
14. #68871C2semidirect
15. #68828C2semidirect
71. #2036D3semidirect
703. C3#24094semidirect

Maximal subgroups

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

subgroupindex
C2×#56392
C2×D3×#1072
C2×#48052
#739112
C3×#20362
#716952
#706182
#702702
#702362
#702352
#699572
#699292
#695062
#688712
#688282
#240943