Finite Group Databasebeta

← #10831789

Subgroups — #10831789

Order 768.

1064 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5721C211yesmaximal, characteristic, complemented
2C2×#4891C211yesmaximal, characteristic
3C2×#4889C211yesmaximal, characteristic
4#76059C211yesmaximal, characteristic
5C3×#2057C211yesmaximal, characteristic, complemented
6#71792C211yesmaximal, characteristic
7#70642C211yesmaximal, characteristic, complemented
8#70618C211yesmaximal, characteristic, complemented
9#69632C211yesmaximal, characteristic, complemented
10#68909C211yesmaximal, characteristic
11#68904C211yesmaximal, characteristic, complemented
12#68899C211yesmaximal, characteristic
13#68805C211yesmaximal, characteristic, complemented
14#68686C211yesmaximal, characteristic
15#68620C211yesmaximal, characteristic, complemented
16#24346—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×Q8×Dic3—21no—
19C2×#944—21no—
20C22×#204C2211yescharacteristic
21#5721—24no—
22#5721—24no—
23C2×#896—21no—
24C2×#893C2211yescharacteristic
25C2×#890C2211yescharacteristic
26C2×#890C2211yescharacteristic
27C2×#889—21no—
28C2×#889—21no—
29C2×#887C2211yescharacteristic
30C2×#887C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C2×C4×Dic6C2211yescharacteristic
33C2×C3×#107C2211yescharacteristic
34C2×C3×#107C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C2×#821C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#815C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40#4994—21no—
41#4988C2211yescharacteristic
42#4975C2211yescharacteristic
43#4972C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#809C2211yescharacteristic
48C2×#808C2211yescharacteristic
49#4939C2211yescharacteristic
50#4891—24no—
51#4889—24no—
52#4676—21no—
53#4667—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#557—31no—
65C2×#478—31no—
66C2×#440—31no—
67#2538—31no—
68#2059—31no—
69#2057D311yescharacteristic, complemented
70#2025—31no—
71#2023—31no—
72#1992—31no—
73#1985—31no—
74#1880—31no—
75#1880—31no—
76#1865—31no—
77#1863—31no—
78#1803—31no—
79C22×C3×Q8—21no—
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×Dic6D411yescharacteristic
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4⋊Q8—22no—
87C3×C4⋊Q8—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95Q8×Dic3—28no—
96#944—28no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4D411yescharacteristic
116C2×Dic3⋊C4D411yescharacteristic
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3C2311yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C2×#204—22no—
125C2×#204—22no—
126C2×#204Q814yescomplemented
127#896—44no—
128#893—22no—
129#893—22no—
130#890—22no—
131#890—22no—
132#890—22no—
133#890D414yes—
134#889—28no—
135#889—28no—
136#887—22no—
137#887—22no—
138#887—22no—
139#887D414yes—
140C4×Dic6—22no—
141C4×Dic6—22no—
142C4×Dic6—22no—
143C4×Dic6—22no—
144C3×#107—22no—
145C3×#107—22no—
146C3×#95—21no—
147C3⋊#95—21no—
148#821—22no—
149#816—22no—
150#815—22no—
151Dic3⋊C8—22no—
152#810—22no—
153#810—22no—
154#809—22no—
155#809—22no—
156#808—22no—
157C22×SD16—31no—
158#557—64no—
159#557—64no—
160C2×C4⋊Q8—61no—
161C2×C4⋊Q8C2×D311yescharacteristic
162C2×#125—31no—
163C2×#125—31no—
164C2×#125—61no—
165C2×#125—61no—
166C2×C22⋊Q8—61no—
167C2×C4×Q8—31no—
168C2×C4×Q8—31no—
169C2×C4×Q8—31no—
170C2×C4×Q8—31no—
171C2×C4×Q8—61no—
172C2×C4×D4—31no—
173#478—64no—
174#440—64no—
175C2×#107C2×D311yescharacteristic
176C2×#107C2×D311yescharacteristic
177C2×#106—31no—
178C2×C4⋊C8—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#103—31no—
182C2×#102—31no—
183C2×#102—31no—
184C2×#102—31no—
185C2×#97—31no—
186#396—61no—
187#396—61no—
188#390—31no—
189#390—31no—
190#389—31no—
191#389C2×D311yescharacteristic
192#384—31no—
193#380—61no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340C2×D311yescharacteristic
199#340C2×D311yescharacteristic
200#336—31no—
201#336—31no—
202#336—31no—
203#336C2×D311yescharacteristic
204C23×D3—21no—
205C2×C3×Q8—24no—
206C2×C3×Q8—42no—
207C2×C3×Q8—42no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2411yescharacteristic
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12C2×Q812yes—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×C4×D3—42no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—22no—
235C2×Dic6—22no—
236C2×Dic6—41no—
237C2×Dic6C2×D412yes—
238C2×Dic6C2×Q812yes—
239C2×Dic6#4112yes—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8C2×Q812yes—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—42no—
254C3×C4⋊C4C2×D412yes—
255C3×C4⋊C4C2×D412yes—
256C3×C42—21no—
257C3×C42—21no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—81no—
262(C2×D3)⋊C4—81no—
263C4⋊Dic3—24no—
264C4⋊Dic3—21no—
265C4⋊Dic3—22no—
266C4⋊Dic3—24no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3C2×D412yes—
270Dic3⋊C4—24no—
271Dic3⋊C4—24no—
272Dic3⋊C4—24no—
273Dic3⋊C4—22no—
274Dic3⋊C4—22no—
275Dic3⋊C4—24no—
276Dic3⋊C4—24no—
277Dic3⋊C4—42no—
278Dic3⋊C4—42no—
279Dic3⋊C4—81no—
280C4×Dic3—24no—
281C4×Dic3—24no—
282C4×Dic3—21no—
283C4×Dic3—21no—
284C4×Dic3—24no—
285C4×Dic3C2×D412yes—
286C4×Dic3C2×D412yes—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293#204—28no—
294C22×Q8—21no—
295C22×Q8—31no—
296C22×Q8—31no—
297C22×D4—31no—
298C23×C4—61no—
299C2×SD16—34no—
300C2×SD16—62no—
301C2×SD16—62no—
302C22×C8—31no—
303C22×C8—31no—
304C22×C8C22×D311yescharacteristic
305C22×C8C22×D311yescharacteristic
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308C4⋊Q8—68no—
309#125—34no—
310#125—62no—
311#125—62no—
312#125—68no—
313#125—68no—
314#125—62no—
315C22⋊Q8—124no—
316C4×Q8—34no—
317C4×Q8—62no—
318C4×Q8—62no—
319C4×Q8—62no—
320C4×Q8—62no—
321C4×Q8—62no—
322C4×Q8—62no—
323C4×Q8—68no—
324C4×Q8—62no—
325C4×D4—62no—
326C4×D4—62no—
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—31no—
334C2×C4⋊C4—61no—
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⋊C4—61no—
344C2×C4⋊C4C22×D311yescharacteristic
345C2×C4⋊C4C22×D311yescharacteristic
346C2×C4⋊C4C22×D311yescharacteristic
347C2×C4⋊C4C22×D311yescharacteristic
348C2×C22⋊C4—61no—
349C2×C22⋊C4—61no—
350C2×C22⋊C4—61no—
351C2×C22⋊C4—61no—
352C2×C22⋊C4—61no—
353C2×C42—31no—
354C2×C42—31no—
355C2×C42—61no—
356C2×C42—61no—
357C2×C42—61no—
358C2×C42C22×D311yescharacteristic
359#107—22no—
360#107—22no—
361#106—62no—
362C4⋊C8—62no—
363#103—62no—
364#103—62no—
365#103—62no—
366#102—62no—
367#102—62no—
368#102—62no—
369#97—62no—
370#95—21no—
371#95—61no—
372C23×C3C22×D411yescharacteristic
373C22×D3—22no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C3×Q8—48no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4AGL(1,ℤ/8)12yes—
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C22×Q811yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C42−1+411yescharacteristic
404C2×C3×C42−1+411yescharacteristic
405C2×Dic3—22no—
406C2×Dic3—22no—
407C2×Dic3—22no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—22no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3#12012yes—
425C2×Dic3#12012yes—
426D12—22no—
427D12—22no—
428D12—41no—
429D12—41no—
430C4×D3—84no—
431Dic6—22no—
432Dic6—22no—
433Dic6—22no—
434Dic6—44no—
435Dic6C22⋊Q814yescomplemented
436C3×C8—22no—
437C3×C8—41no—
438C3⋊C8—41no—
439C3⋊C8—41no—
440C24—61no—
441C2×Q8—24no—
442C2×Q8—32no—
443C2×Q8—32no—
444C2×Q8—32no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—62no—
448C2×Q8—62no—
449C2×Q8—61no—
450C2×Q8—61no—
451C2×Q8—61no—
452C2×Q8—121no—
453C2×D4—32no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—61no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—31no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—122no—
471C22×C4—122no—
472C22×C4—122no—
473C22×C4D3×D411yescharacteristic
474C22×C4D3×D411yescharacteristic
475C22×C4C23×D311yescharacteristic
476SD16—68no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8D3×Q812yes—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—32no—
497C4⋊C4—42no—
498C4⋊C4—62no—
499C4⋊C4—62no—
500C4⋊C4—62no—
501C4⋊C4—64no—
502C4⋊C4—64no—
503C4⋊C4—64no—
504C4⋊C4—64no—
505C4⋊C4—64no—
506C4⋊C4—64no—
507C4⋊C4—64no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—122no—
512C4⋊C4—122no—
513C4⋊C4—241no—
514C4⋊C4D3×D412yes—
515C4⋊C4D3×D412yes—
516C22⋊C4—122no—
517C22⋊C4—122no—
518C22⋊C4—122no—
519C22⋊C4—241no—
520C22⋊C4—241no—
521C42—21no—
522C42—21no—
523C42—32no—
524C42—32no—
525C42—64no—
526C42—64no—
527C42—64no—
528C42—61no—
529C42—61no—
530C22×C3C2×#12011yescharacteristic
531C22×C3#54811yescharacteristic
532C22×C3D4×Q811yescharacteristic
533C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
534C22×C3#57511yescharacteristic
535C22×C3#57711yescharacteristic
536C22×C3#57811yescharacteristic
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—42no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C3×C4—22no—
545C3×C4—42no—
546C3×C4—42no—
547C3×C4—41no—
548C3×C4—81no—
549C3×C4—81no—
550C3×C4C2×C22⋊Q811yescharacteristic
551C3×C4#53611yescharacteristic
552C3×C4#54811yescharacteristic
553C3×C4D4×Q811yescharacteristic
554Dic3—24no—
555Dic3—22no—
556Dic3—22no—
557Dic3—41no—
558Dic3—41no—
559Dic3—42no—
560Dic3—81no—
561C23—62no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23C2×D3×D411yescharacteristic
569Q8—34no—
570Q8—48no—
571Q8—62no—
572Q8—62no—
573Q8—62no—
574Q8—124no—
575D4—62no—
576D4—62no—
577D4—121no—
578D4—121no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
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—41no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—62no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—62no—
611C2×C4—61no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—244no—
618C2×C4#91112yes—
619C2×C4C2×D3×D411yescharacteristic
620C2×C4C2×D3×D411yescharacteristic
621C2×C4C2×D3×Q811yescharacteristic
622C2×C4#100711yescharacteristic
623C2×C4#100711yescharacteristic
624C2×C4D3×#4111yescharacteristic
625C8—22no—
626C8—41no—
627C8—121no—
628C8—121no—
629C2×C3#240911yescharacteristic
630C2×C3#300611yescharacteristic
631C2×C3#301311yescharacteristic
632C2×C3#336811yescharacteristic
633C2×C3#337711yescharacteristic
634C2×C3#338611yescharacteristic
635C2×C3#338811yescharacteristic
636D3—82no—
637C22—122no—
638C22—122no—
639C22—122no—
640C22—241no—
641C22—241no—
642C22—241no—
643C22—241no—
644C22D3×#12011yescharacteristic
645C22#563311yescharacteristic
646C22#572711yescharacteristic
647C22C2×#91111yescharacteristic
648C22#576611yescharacteristic
649C22#577311yescharacteristic
650C22#577811yescharacteristic
651C4—22no—
652C4—41no—
653C4—42no—
654C4—42no—
655C4—62no—
656C4—64no—
657C4—62no—
658C4—81no—
659C4—81no—
660C4—121no—
661C4—122no—
662C4—121no—
663C4—241no—
664C4D3×C22⋊Q811yescharacteristic
665C4#563311yescharacteristic
666C4#572411yescharacteristic
667C4#572711yescharacteristic
668C3#2434611yescharacteristic, complemented, Sylow
669C2—242no—
670C2#7717211yescharacteristic
671C2#8000311yescharacteristic
672C2#8018611yescharacteristic
673C2#8099511yescharacteristic
674C2#8110111yescharacteristic
675C2#8195511yescharacteristic
676C2#8196811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2057C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2057D3
hypocenterC3#24346
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2057

Complements

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

NH ≅ G/Ntype
1. C2×#5721C2semidirect
5. C3×#2057C2semidirect
7. #70642C2semidirect
8. #70618C2semidirect
9. #69632C2semidirect
11. #68904C2semidirect
13. #68805C2semidirect
15. #68620C2semidirect
69. #2057D3semidirect
126. C2×#204Q8semidirect
435. Dic6C22⋊Q8semidirect
668. C3#24346semidirect

Maximal subgroups

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

subgroupindex
C2×#57212
C2×#48912
C2×#48892
#760592
C3×#20572
#717922
#706422
#706182
#696322
#689092
#689042
#688992
#688052
#686862
#686202
#243463