Finite Group Databasebeta

← #10901414

Subgroups — #10901414

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×#5691C211yesmaximal, characteristic
2C2×#4877C211yesmaximal, characteristic, complemented
3C2×#4871C211yesmaximal, characteristic, complemented
4#76285C211yesmaximal, characteristic, complemented
5C3×#2059C211yesmaximal, characteristic, complemented
6#71739C211yesmaximal, characteristic, complemented
7#70634C211yesmaximal, characteristic
8#70582C211yesmaximal, characteristic, complemented
9#69628C211yesmaximal, characteristic, complemented
10#68905C211yesmaximal, characteristic, complemented
11#68893C211yesmaximal, characteristic, complemented
12#68825C211yesmaximal, characteristic, complemented
13#68823C211yesmaximal, characteristic
14#68662C211yesmaximal, characteristic, complemented
15#68630C211yesmaximal, characteristic, complemented
16#29693—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18#5691—24no—
19#5691—24no—
20C2×#898—21no—
21C2×#897C2211yescharacteristic
22C2×#895C2211yescharacteristic
23C2×#894—21no—
24C2×#893—21no—
25C2×#893C2211yescharacteristic
26C2×#892—21no—
27C2×#890C2211yescharacteristic
28C2×#887C2211yescharacteristic
29C2×#875C2211yescharacteristic
30C2×C42×D3C2211yescharacteristic
31C2×C3×#107C2211yescharacteristic
32C2×C3×#106C2211yescharacteristic
33C3×#389C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×(C2×D3)⋊C8C2211yescharacteristic
37C2×(C2×D3)⋊C8C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42#4989—21no—
43#4987C2211yescharacteristic
44#4987C2211yescharacteristic
45#4980—21no—
46#4977C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#809C2211yescharacteristic
50C2×#809C2211yescharacteristic
51C2×#808C2211yescharacteristic
52C2×#807C2211yescharacteristic
53#4928C2211yescharacteristic
54#4877—24no—
55#4871—24no—
56#4668—21no—
57#4652—21no—
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60C3×#336C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#529—31no—
65C2×#479—31no—
66C2×#478—31no—
67#2851—31no—
68#2059—31no—
69#2059D311yescharacteristic, complemented
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×D12—21no—
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×C4×D3D411yescharacteristic
83C22×C4×D3D411yescharacteristic
84C22×C4×D3C2311yescharacteristic
85C22×Dic6C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×#125—22no—
89C3×#125—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3D411yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118#898—28no—
119#897—22no—
120#897—22no—
121#895—22no—
122#895—22no—
123#894—28no—
124#893—28no—
125#893—22no—
126#893D414yes—
127#892—28no—
128#890—22no—
129#890—22no—
130#887—22no—
131#887D414yes—
132#875—22no—
133#875—22no—
134C42×D3—22no—
135C42×D3—22no—
136C3×#107—22no—
137C3×#106—22no—
138C3×#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144#821—22no—
145#821—41no—
146(C2×D3)⋊C8—41no—
147(C2×D3)⋊C8—41no—
148#816—22no—
149#816—41no—
150Dic3⋊C8—22no—
151Dic3⋊C8—22no—
152#810—22no—
153#810—41no—
154#809—22no—
155#809—41no—
156#808—22no—
157#807—22no—
158#529—64no—
159#529—64no—
160C2×#126—61no—
161C2×#125—31no—
162C2×#125C2×D311yescharacteristic
163C2×#124—31no—
164C2×#123—61no—
165C2×C22⋊Q8—31no—
166C2×#121—31no—
167C2×C4×Q8—31no—
168C2×C4×D4—31no—
169C2×C4×D4—61no—
170C2×#117—61no—
171C22×C42—31no—
172#479—64no—
173#478—64no—
174C2×#107—31no—
175C2×#107C2×D311yescharacteristic
176C2×#106—31no—
177C2×#106C2×D311yescharacteristic
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#389C2×D311yescharacteristic
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#340C2×D311yescharacteristic
202#340C2×D311yescharacteristic
203#336—31no—
204#336C2×D311yescharacteristic
205C23×D3—21no—
206C23×D3C2×D411yescharacteristic
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—24no—
225C2×D12—21no—
226C2×D12—42no—
227C2×D12—42no—
228C2×D12C2×D412yes—
229C2×C4×D3—22no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—22no—
237C2×C4×D3—22no—
238C2×C4×D3—24no—
239C2×C4×D3—41no—
240C2×C4×D3—41no—
241C2×C4×D3—41no—
242C2×C4×D3—42no—
243C2×C4×D3—42no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6C2×D412yes—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C3×C4⋊C4—24no—
257C3×C4⋊C4—24no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—42no—
265C3×C4⋊C4C2×D412yes—
266C3×C4⋊C4C2×D412yes—
267C3×C42—21no—
268C3×C42—21no—
269(C2×D3)⋊C4—24no—
270(C2×D3)⋊C4—24no—
271(C2×D3)⋊C4—24no—
272(C2×D3)⋊C4—24no—
273(C2×D3)⋊C4—24no—
274(C2×D3)⋊C4—42no—
275(C2×D3)⋊C4—42no—
276C4⋊Dic3—24no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3—42no—
282Dic3⋊C4—24no—
283Dic3⋊C4—42no—
284Dic3⋊C4—42no—
285C4×Dic3—22no—
286C4×Dic3—22no—
287C4×Dic3—24no—
288C4×Dic3C2×D412yes—
289C4×Dic3C2×D412yes—
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×Q8—31no—
297C22×D4—31no—
298C22×D4—61no—
299C23×C4—31no—
300C23×C4—31no—
301C23×C4—31no—
302C23×C4—61no—
303C22×C8—31no—
304C22×C8—31no—
305C22×C8C22×D311yescharacteristic
306C22×C8C22×D311yescharacteristic
307#126—68no—
308#125—22no—
309#125—22no—
310#125—62no—
311#125—62no—
312#124—62no—
313#124—62no—
314#123—68no—
315C22⋊Q8—62no—
316C22⋊Q8—62no—
317#121—62no—
318#121—62no—
319C4×Q8—34no—
320C4×Q8—62no—
321C4×D4—34no—
322C4×D4—62no—
323C4×D4—68no—
324#117—68no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4C22×D311yescharacteristic
336C2×C4⋊C4C22×D311yescharacteristic
337C2×C4⋊C4C22×D311yescharacteristic
338C2×C4⋊C4C22×D311yescharacteristic
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C22⋊C4—61no—
342C2×C22⋊C4—61no—
343C2×C22⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C22⋊C4—61no—
346C2×C42—31no—
347C2×C42—31no—
348C2×C42—31no—
349C2×C42—62no—
350C2×C42—62no—
351C2×C42—61no—
352C2×C42C22×D311yescharacteristic
353#107—22no—
354#107—62no—
355#106—22no—
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—21no—
370#95—61no—
371#95—61no—
372#95—61no—
373#95—61no—
374#95—61no—
375C23×C3C22×D411yescharacteristic
376C22×D3—21no—
377C22×D3—21no—
378C22×D3—21no—
379C22×D3—21no—
380C22×D3—21no—
381C22×D3—21no—
382C22×D3—22no—
383C22×D3—41no—
384C22×D3—41no—
385C22×D3—41no—
386C22×D3—41no—
387C22×D3—41no—
388C22×D3—41no—
389C22×D3#13512yes—
390C2×C3×C4—22no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—22no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—41no—
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×C4C22×D411yescharacteristic
409C2×C3×C4C22×D411yescharacteristic
410C2×C3×C4H(5,𝔽2)11yescharacteristic
411C2×C3×C42−1+411yescharacteristic
412C2×C3×C42−1+411yescharacteristic
413C2×C3×C42−1+411yescharacteristic
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—22no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3#12012yes—
431C2×Dic3#12012yes—
432D12—22no—
433D12—22no—
434D12—48no—
435D12—41no—
436D12—41no—
437C4×D3—44no—
438C4×D3—44no—
439C4×D3—48no—
440C4×D3—41no—
441C4×D3—41no—
442C4×D3—41no—
443C4×D3—41no—
444Dic6—22no—
445Dic6—22no—
446Dic6—41no—
447Dic6—41no—
448C3×C8—41no—
449C3×C8—41no—
450C3⋊C8—41no—
451C3⋊C8—41no—
452C24—31no—
453C24—61no—
454C2×Q8—32no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×Q8—61no—
460C2×D4—32no—
461C2×D4—64no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—61no—
465C2×D4—61no—
466C2×D4—61no—
467C2×D4—122no—
468C2×D4—122no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—31no—
474C22×C4—31no—
475C22×C4—31no—
476C22×C4—62no—
477C22×C4—62no—
478C22×C4—64no—
479C22×C4—62no—
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—61no—
489C22×C4—121no—
490C22×C4—121no—
491C22×C4—121no—
492C22×C4—122no—
493C22×C4—122no—
494C22×C4D3×D411yescharacteristic
495C22×C4D3×D411yescharacteristic
496C22×C4C23×D311yescharacteristic
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
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—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—64no—
519C4⋊C4—64no—
520C4⋊C4—61no—
521C4⋊C4—61no—
522C4⋊C4—61no—
523C4⋊C4—61no—
524C4⋊C4—122no—
525C4⋊C4—122no—
526C4⋊C4—122no—
527C4⋊C4D3×D412yes—
528C4⋊C4D3×D412yes—
529C22⋊C4—64no—
530C22⋊C4—64no—
531C22⋊C4—64no—
532C22⋊C4—64no—
533C22⋊C4—64no—
534C22⋊C4—122no—
535C22⋊C4—122no—
536C42—21no—
537C42—21no—
538C42—32no—
539C42—32no—
540C42—62no—
541C42—62no—
542C42—64no—
543C22×C3C2×#12011yescharacteristic
544C22×C3#53611yescharacteristic
545C22×C3#53711yescharacteristic
546C22×C3C2×#13511yescharacteristic
547C22×C3#57511yescharacteristic
548C22×C3#57511yescharacteristic
549C22×C3#57511yescharacteristic
550C2×D3—22no—
551C2×D3—22no—
552C2×D3—22no—
553C2×D3—41no—
554C2×D3—41no—
555C2×D3—41no—
556C2×D3—41no—
557C2×D3—42no—
558C2×D3—42no—
559C2×D3—42no—
560C2×D3—81no—
561C2×D3—81no—
562C2×D3—81no—
563C2×D3—81no—
564C3×C4—41no—
565C3×C4—42no—
566C3×C4—41no—
567C3×C4—42no—
568C3×C4—81no—
569C3×C4—81no—
570C3×C4#53611yescharacteristic
571C3×C4#53611yescharacteristic
572C3×C4#53711yescharacteristic
573C3×C4#53711yescharacteristic
574Dic3—22no—
575Dic3—22no—
576Dic3—42no—
577Dic3—41no—
578Dic3—81no—
579Dic3—81no—
580C23—32no—
581C23—61no—
582C23—61no—
583C23—61no—
584C23—61no—
585C23—61no—
586C23—61no—
587C23—62no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23C2×D3×D411yescharacteristic
595Q8—62no—
596Q8—62no—
597Q8—121no—
598Q8—121no—
599D4—62no—
600D4—62no—
601D4—128no—
602D4—121no—
603D4—121no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—32no—
613C2×C4—32no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—62no—
631C2×C4—61no—
632C2×C4—124no—
633C2×C4—124no—
634C2×C4—128no—
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—121no—
646C2×C4—121no—
647C2×C4C2×D3×D411yescharacteristic
648C2×C4C2×D3×D411yescharacteristic
649C2×C4#100711yescharacteristic
650C2×C4#100711yescharacteristic
651C2×C4#100711yescharacteristic
652C2×C4#100911yescharacteristic
653C8—41no—
654C8—41no—
655C8—121no—
656C8—121no—
657C2×C3#261611yescharacteristic
658C2×C3#300211yescharacteristic
659C2×C3#300511yescharacteristic
660C2×C3#323211yescharacteristic
661C2×C3#324011yescharacteristic
662C2×C3#325011yescharacteristic
663C2×C3#325611yescharacteristic
664D3—42no—
665D3—82no—
666C22—62no—
667C22—62no—
668C22—62no—
669C22—121no—
670C22—121no—
671C22—121no—
672C22—121no—
673C22—122no—
674C22—122no—
675C22—122no—
676C22—241no—
677C22—241no—
678C22—241no—
679C22—241no—
680C22D3×#12011yescharacteristic
681C22#562911yescharacteristic
682C22#572411yescharacteristic
683C22D3×#13511yescharacteristic
684C22#576811yescharacteristic
685C22#577311yescharacteristic
686C22#577711yescharacteristic
687C4—41no—
688C4—42no—
689C4—41no—
690C4—42no—
691C4—62no—
692C4—62no—
693C4—81no—
694C4—81no—
695C4—121no—
696C4—122no—
697C4—241no—
698C4—241no—
699C4#562911yescharacteristic
700C4#562911yescharacteristic
701C4#572411yescharacteristic
702C4#572411yescharacteristic
703C3#2969311yescharacteristic, complemented, Sylow
704C2—122no—
705C2—242no—
706C2#7720111yescharacteristic
707C2#8008211yescharacteristic
708C2#8022711yescharacteristic
709C2#8118111yescharacteristic
710C2#8126111yescharacteristic
711C2#8192011yescharacteristic
712C2#8198511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2059C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#2059D3
hypocenterC3#29693
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2059

Complements

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

NH ≅ G/Ntype
2. C2×#4877C2semidirect
3. C2×#4871C2semidirect
4. #76285C2semidirect
5. C3×#2059C2semidirect
6. #71739C2semidirect
8. #70582C2semidirect
9. #69628C2semidirect
10. #68905C2semidirect
11. #68893C2semidirect
12. #68825C2semidirect
14. #68662C2semidirect
15. #68630C2semidirect
69. #2059D3semidirect
703. C3#29693semidirect

Maximal subgroups

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

subgroupindex
C2×#56912
C2×#48772
C2×#48712
#762852
C3×#20592
#717392
#706342
#705822
#696282
#689052
#688932
#688252
#688232
#686622
#686302
#296933