Finite Group Databasebeta

← #10817439

Subgroups — #10817439

Order 768.

1030 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82465C211yesmaximal, characteristic, complemented
2C2×C4×#215C211yesmaximal, characteristic
3C2×#4852C211yesmaximal, characteristic
4C2×#4818C211yesmaximal, characteristic
5#76202C211yesmaximal, characteristic
6C3×#1865C211yesmaximal, characteristic, complemented
7#70563C211yesmaximal, characteristic, complemented
8#70276C211yesmaximal, characteristic
9#70253C211yesmaximal, characteristic, complemented
10#70225C211yesmaximal, characteristic, complemented
11#69614C211yesmaximal, characteristic, complemented
12#68907C211yesmaximal, characteristic, complemented
13#68780C211yesmaximal, characteristic
14#68672C211yesmaximal, characteristic
15#68651C211yesmaximal, characteristic, complemented
16#22996—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#944C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C22×#204C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#893C2211yescharacteristic
23C2×D3×C4⋊C4—21no—
24C2×#887C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C3×#389C2211yescharacteristic
30#5229C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37C4×#215—24no—
38#4989—21no—
39#4984C2211yescharacteristic
40#4978—21no—
41#4976—21no—
42#4975C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#808C2211yescharacteristic
48#4933—21no—
49C4×Dic3⋊C4—21no—
50C4×Dic3⋊C4C2211yescharacteristic
51#4929C2211yescharacteristic
52C2×C4×C3⋊C8C2211yescharacteristic
53#4852—24no—
54#4818C418yescomplemented
55#4668—21no—
56#4667—21no—
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59#4551C2211yescharacteristic
60#4550C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×#461—31no—
64C2×C4×SD16—31no—
65C2×C4×SD16—31no—
66#2365—31no—
67D4×C4⋊C4—31no—
68#1956—31no—
69#1948—31no—
70#1933—31no—
71#1933—31no—
72#1877—31no—
73#1877—31no—
74#1865—31no—
75#1865D311yescharacteristic, complemented
76C8×C4⋊C4—31no—
77C4×#102—31no—
78C22×C3×Q8C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×C4×D3—21no—
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4×Q8—22no—
87C3×C4×Q8—41no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C42—21no—
94C2×C3×C42C2311yescharacteristic
95#944—22no—
96#944C2×C414yescomplemented
97C2×#215—22no—
98C2×#215—22no—
99C2×#215C2×C414yes—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4D411yescharacteristic
112C2×Dic3⋊C4D411yescharacteristic
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C2×#204—22no—
120C2×#204—22no—
121C2×#204C2×C414yes—
122#893—22no—
123#893—22no—
124#893—41no—
125#893C2×C414yes—
126D3×C4⋊C4—44no—
127#887—22no—
128#887—41no—
129C4×D12—22no—
130C4×D12—41no—
131C3×#106—22no—
132C3×#103—41no—
133C3×#103C2×C414yescomplemented
134C3×#95—21no—
135#835—22no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95D411yescharacteristic
143C3⋊#95D411yescharacteristic
144#821—22no—
145#816—41no—
146Dic3⋊C8C2×C414yescomplemented
147C8×Dic3—22no—
148C8×Dic3—22no—
149#810—22no—
150#810C2×C414yescomplemented
151#809D414yes—
152#808—22no—
153C4×C3⋊C8—22no—
154C22×SD16—31no—
155C22×SD16—31no—
156C2×C4⋊Q8—31no—
157C2×C4×Q8—31no—
158C2×C4×Q8C2×D311yescharacteristic
159C2×C4×D4—31no—
160C2×C4×D4—31no—
161C2×C4×D4—31no—
162C22×C4⋊C4—61no—
163#461—38no—
164C4×SD16—64no—
165C4×SD16—64no—
166C2×#106—31no—
167C2×#106C2×D311yescharacteristic
168C2×C4⋊C8—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#103—31no—
173C2×#103C2×D311yescharacteristic
174C2×#103C2×D311yescharacteristic
175C2×#102—31no—
176C2×#102—31no—
177C2×C4×C8—31no—
178C2×C4×C8—31no—
179C2×C4×C8—31no—
180#391—31no—
181#391—31no—
182#389—31no—
183#389—61no—
184#389C2×D311yescharacteristic
185#387—61no—
186#387—61no—
187#385—61no—
188#385—61no—
189#384—31no—
190#380—61no—
191C4×C4⋊C4—31no—
192C4×C4⋊C4—31no—
193C4×C4⋊C4—61no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340C2×D311yescharacteristic
198#336—31no—
199#336C2×D311yescharacteristic
200C23×D3—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8C22×C412yes—
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
220C22×Dic3C2×D411yescharacteristic
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12C22×C412yes—
225C2×D12C2×D412yes—
226C2×D12#4112yes—
227C2×C4×D3—24no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×C4×D3—42no—
231C2×C4×D3—42no—
232C2×C4×D3—42no—
233C2×C4×D3—42no—
234C2×C4×D3—81no—
235C2×C4×D3—81no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—21no—
241C2×Dic6C22×C412yes—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8C22×C412yes—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—42no—
253C3×C4⋊C4—42no—
254C3×C4⋊C4C22×C412yes—
255C3×C4⋊C4C2×D412yes—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—42no—
259#215—28no—
260(C2×D3)⋊C4—24no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—42no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—42no—
268Dic3⋊C4—24no—
269Dic3⋊C4—24no—
270Dic3⋊C4—22no—
271Dic3⋊C4—42no—
272Dic3⋊C4—42no—
273Dic3⋊C4—81no—
274Dic3⋊C4SD1614yescomplemented
275C4×Dic3—21no—
276C4×Dic3—21no—
277C4×Dic3—42no—
278C4×Dic3—42no—
279C4×Dic3C22×C412yes—
280C4×Dic3#4112yes—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8C22×C412yes—
286C2×C3⋊C8C2×D412yes—
287#204—28no—
288C22×Q8—31no—
289C22×Q8C22×D311yescharacteristic
290C22×D4—31no—
291C23×C4—61no—
292C23×C4—61no—
293C23×C4—61no—
294C2×SD16—34no—
295C2×SD16—34no—
296C2×SD16—62no—
297C2×SD16—62no—
298C2×SD16—62no—
299C2×SD16—62no—
300C22×C8—31no—
301C22×C8—31no—
302C22×C8C22×D311yescharacteristic
303C22×C8C22×D311yescharacteristic
304C4⋊Q8—34no—
305C4⋊Q8—62no—
306C4×Q8—22no—
307C4×Q8—41no—
308C4×Q8—62no—
309C4×Q8—121no—
310C4×D4—34no—
311C4×D4—62no—
312C4×D4—62no—
313C4×D4—62no—
314C4×D4—121no—
315C4×D4—121no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4—61no—
328C2×C4⋊C4—124no—
329C2×C4⋊C4C22×D311yescharacteristic
330C2×C4⋊C4C22×D311yescharacteristic
331C2×C4⋊C4C22×D311yescharacteristic
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C42—21no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—61no—
339C2×C42—61no—
340C2×C42C22×D311yescharacteristic
341#106—22no—
342#106—62no—
343C4⋊C8—34no—
344#103—34no—
345#103—41no—
346#103—62no—
347#103—62no—
348#103—121no—
349#103C4×D314yescomplemented
350#102—34no—
351#102—62no—
352C4×C8—62no—
353C4×C8—62no—
354C4×C8—62no—
355#95—21no—
356#95—31no—
357#95—31no—
358#95—61no—
359#95—61no—
360#95—61no—
361#95—61no—
362#95—61no—
363#95—61no—
364C23×C3C22×D411yescharacteristic
365C22×D3—22no—
366C22×D3—22no—
367C22×D3—22no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C22×D3—41no—
372C3×Q8—22no—
373C3×Q8—22no—
374C3×Q8—41no—
375C3×Q8—41no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4C2×SD1612yes—
394C2×C3×C4C23×C411yescharacteristic
395C2×C3×C4C22×D411yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4H(5,𝔽2)11yescharacteristic
399C2×C3×C42−1+411yescharacteristic
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—22no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3C4×D412yes—
417C2×Dic3C4×D412yes—
418C2×Dic3C2×SD1612yes—
419C2×Dic3C2×SD1612yes—
420C2×Dic3#13512yes—
421D12—22no—
422D12—22no—
423D12—22no—
424D12C4×D414yes—
425C4×D3—48no—
426C4×D3—84no—
427C4×D3—84no—
428Dic6—22no—
429Dic6—22no—
430Dic6—41no—
431Dic6—41no—
432C3×C8—22no—
433C3×C8—41no—
434C3⋊C8—22no—
435C3⋊C8—22no—
436C24—61no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—32no—
443C2×Q8—61no—
444C2×Q8—61no—
445C2×Q8—61no—
446C2×Q8—61no—
447C2×Q8—61no—
448C2×Q8C2×C4×D312yes—
449C2×D4—32no—
450C2×D4—32no—
451C2×D4—32no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—61no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—31no—
460C22×C4—31no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—64no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—122no—
468C22×C4—122no—
469C22×C4—122no—
470C22×C4—122no—
471C22×C4—122no—
472C22×C4—122no—
473C22×C4—241no—
474C22×C4—241no—
475C22×C4D3×D411yescharacteristic
476C22×C4D3×D411yescharacteristic
477C22×C4C23×D311yescharacteristic
478SD16—68no—
479SD16—68no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—32no—
486C2×C8—32no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8C2×C4×D312yes—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—34no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—62no—
500C4⋊C4—64no—
501C4⋊C4—64no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—122no—
507C4⋊C4—122no—
508C4⋊C4—122no—
509C4⋊C4—241no—
510C4⋊C4C2×C4×D312yes—
511C4⋊C4D3×D412yes—
512C22⋊C4—64no—
513C22⋊C4—122no—
514C22⋊C4—122no—
515C42—21no—
516C42—21no—
517C42—32no—
518C42—32no—
519C42—42no—
520C42—61no—
521C42—61no—
522C42—122no—
523C42—122no—
524C22×C3C2×C4×D411yescharacteristic
525C22×C3D4211yescharacteristic
526C22×C3#54811yescharacteristic
527C22×C3C22×SD1611yescharacteristic
528C22×C3C2×#13511yescharacteristic
529C22×C3#57711yescharacteristic
530C22×C3#57811yescharacteristic
531C2×D3—24no—
532C2×D3—42no—
533C2×D3—42no—
534C2×D3—42no—
535C2×D3—42no—
536C2×D3—42no—
537C2×D3—42no—
538C3×C4—22no—
539C3×C4—42no—
540C3×C4—42no—
541C3×C4—41no—
542C3×C4—42no—
543C3×C4—81no—
544C3×C4C2×C4×D411yescharacteristic
545C3×C4#52011yescharacteristic
546C3×C4#54711yescharacteristic
547C3×C4#55311yescharacteristic
548Dic3—22no—
549Dic3—22no—
550Dic3—42no—
551Dic3—41no—
552Dic3—81no—
553Dic3—81no—
554Dic3C4×SD1614yescomplemented
555C23—62no—
556C23—62no—
557C23—62no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23C2×D3×D411yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8—62no—
568Q8—62no—
569Q8—121no—
570Q8—121no—
571D4—34no—
572D4—62no—
573D4—62no—
574D4—62no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—32no—
584C2×C4—32no—
585C2×C4—32no—
586C2×C4—32no—
587C2×C4—32no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—62no—
604C2×C4—61no—
605C2×C4—128no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—244no—
615C2×C4—244no—
616C2×C4D3×SD1612yes—
617C2×C4C22×C4×D311yescharacteristic
618C2×C4C2×D3×D411yescharacteristic
619C2×C4#100411yescharacteristic
620C2×C4#100711yescharacteristic
621C2×C4D3×#4111yescharacteristic
622C2×C4D3×#4111yescharacteristic
623C8—22no—
624C8—41no—
625C8—62no—
626C8—62no—
627C2×C3#236111yescharacteristic
628C2×C3C2×C4×SD1611yescharacteristic
629C2×C3#294711yescharacteristic
630C2×C3D4×SD1611yescharacteristic
631C2×C3#328311yescharacteristic
632C2×C3#335911yescharacteristic
633C2×C3#336011yescharacteristic
634D3—44no—
635C22—64no—
636C22—122no—
637C22—122no—
638C22—122no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22C4×D3×D411yescharacteristic
643C22#559011yescharacteristic
644C22#563311yescharacteristic
645C22C2×D3×SD1611yescharacteristic
646C22D3×#13511yescharacteristic
647C22#577811yescharacteristic
648C22#577911yescharacteristic
649C4—22no—
650C4—34no—
651C4—42no—
652C4—41no—
653C4—42no—
654C4—42no—
655C4—62no—
656C4—62no—
657C4—81no—
658C4—121no—
659C4—122no—
660C4—241no—
661C4—241no—
662C4#550411yescharacteristic
663C4C4×D3×D411yescharacteristic
664C4#563111yescharacteristic
665C4#566111yescharacteristic
666C3#2299611yescharacteristic, complemented, Sylow
667C2—124no—
668C2#7439711yescharacteristic
669C2C4×D3×SD1611yescharacteristic
670C2#7983011yescharacteristic
671C2#8006311yescharacteristic
672C2#8026911yescharacteristic
673C2#8103711yescharacteristic
674C2#8121511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1865C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#1865D3
hypocenterC3#22996
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
1. #82465C2semidirect
6. C3×#1865C2semidirect
7. #70563C2semidirect
9. #70253C2semidirect
10. #70225C2semidirect
11. #69614C2semidirect
12. #68907C2semidirect
15. #68651C2semidirect
54. #4818C4semidirect
75. #1865D3semidirect
96. #944C2×C4semidirect
133. C3×#103C2×C4semidirect
146. Dic3⋊C8C2×C4semidirect
150. #810C2×C4semidirect
274. Dic3⋊C4SD16semidirect
349. #103C4×D3semidirect
554. Dic3C4×SD16semidirect
666. C3#22996semidirect

Maximal subgroups

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

subgroupindex
#824652
C2×C4×#2152
C2×#48522
C2×#48182
#762022
C3×#18652
#705632
#702762
#702532
#702252
#696142
#689072
#687802
#686722
#686512
#229963