Finite Group Databasebeta

← #10831723

Subgroups — #10831723

Order 768.

1036 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5685C211yesmaximal, characteristic, complemented
2C2×#4889C211yesmaximal, characteristic
3C2×#4870C211yesmaximal, characteristic
4#76075C211yesmaximal, characteristic
5C3×#2059C211yesmaximal, characteristic, complemented
6#71743C211yesmaximal, characteristic
7#70618C211yesmaximal, characteristic, complemented
8#70588C211yesmaximal, characteristic, complemented
9#69632C211yesmaximal, characteristic, complemented
10#68909C211yesmaximal, characteristic
11#68905C211yesmaximal, characteristic, complemented
12#68899C211yesmaximal, characteristic
13#68806C211yesmaximal, characteristic, complemented
14#68686C211yesmaximal, characteristic
15#68620C211yesmaximal, characteristic, complemented
16#24340—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C22×#204C2211yescharacteristic
19#5685—24no—
20#5685—24no—
21C2×#893C2211yescharacteristic
22C2×#890C2211yescharacteristic
23C2×#889—21no—
24C2×#889—21no—
25C2×#888—21no—
26C2×#888C2211yescharacteristic
27C2×#887—21no—
28C2×#887C2211yescharacteristic
29C2×#887C2211yescharacteristic
30C2×C4×Dic6C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×#107C2211yescharacteristic
33C2×C3×#106C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#815C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39#4993—21no—
40#4989—21no—
41#4988C2211yescharacteristic
42#4977C2211yescharacteristic
43#4972C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#809C2211yescharacteristic
48C2×#808C2211yescharacteristic
49#4939C2211yescharacteristic
50#4889—24no—
51#4870—24no—
52#4673—21no—
53#4668—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×#476—31no—
66C2×#440—31no—
67#2580—31no—
68#2059—31no—
69#2059D311yescharacteristic, complemented
70#2025—31no—
71#2025—31no—
72#1993—31no—
73#1985—31no—
74#1880—31no—
75#1880—31no—
76#1865—31no—
77#1865—31no—
78#1805—31no—
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×Dic6—21no—
82C22×Dic6D411yescharacteristic
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#125—22no—
87C3×#125—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3D411yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
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×Dic3—21no—
116C2×C4×Dic3C2311yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C2×#204—22no—
121C2×#204—22no—
122C2×#204Q814yescomplemented
123#893—22no—
124#893—22no—
125#890—22no—
126#890—22no—
127#889—28no—
128#889—28no—
129#888—28no—
130#888—22no—
131#888D414yes—
132#887—28no—
133#887—22no—
134#887—22no—
135#887—22no—
136#887D414yes—
137C4×Dic6—22no—
138C4×Dic6—22no—
139C4×Dic6—22no—
140C4×Dic6—22no—
141C3×#107—22no—
142C3×#106—22no—
143C3×#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147#821—22no—
148#816—22no—
149#815—22no—
150Dic3⋊C8—22no—
151#810—22no—
152#810—22no—
153#809—22no—
154#809—22no—
155#808—22no—
156C22×SD16—31no—
157#557—64no—
158#557—64no—
159C2×C4⋊Q8—31no—
160C2×C4⋊Q8—61no—
161C2×#125—31no—
162C2×#125—61no—
163C2×#125—61no—
164C2×#125C2×D311yescharacteristic
165C2×C4×Q8—31no—
166C2×C4×Q8—31no—
167C2×C4×Q8—31no—
168C2×C4×Q8—31no—
169C2×C4×Q8—61no—
170C2×C4×D4—31no—
171#476—64no—
172#440—64no—
173C2×#107C2×D311yescharacteristic
174C2×#106—31no—
175C2×#106C2×D311yescharacteristic
176C2×C4⋊C8—31no—
177C2×#103—31no—
178C2×#103—31no—
179C2×#103—31no—
180C2×#102—31no—
181C2×#102—31no—
182C2×#102—31no—
183C2×#97—31no—
184#399—61no—
185#393—61no—
186#390—31no—
187#390—31no—
188#389—31no—
189#389C2×D311yescharacteristic
190#385—61no—
191#385—61no—
192#382—31no—
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340C2×D311yescharacteristic
198#340C2×D311yescharacteristic
199#336—31no—
200#336—31no—
201#336—31no—
202#336C2×D311yescharacteristic
203C23×D3—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2411yescharacteristic
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12C2×Q812yes—
224C2×C4×D3—42no—
225C2×C4×D3—81no—
226C2×C4×D3—81no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—22no—
231C2×Dic6—24no—
232C2×Dic6—41no—
233C2×Dic6—41no—
234C2×Dic6—42no—
235C2×Dic6—42no—
236C2×Dic6C2×D412yes—
237C2×Dic6C2×Q812yes—
238C2×Dic6#4112yes—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8C2×Q812yes—
245C3×C4⋊C4—24no—
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—81no—
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—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3C2×D412yes—
269Dic3⋊C4—24no—
270Dic3⋊C4—24no—
271Dic3⋊C4—24no—
272Dic3⋊C4—22no—
273Dic3⋊C4—22no—
274Dic3⋊C4—24no—
275Dic3⋊C4—24no—
276Dic3⋊C4—81no—
277C4×Dic3—24no—
278C4×Dic3—24no—
279C4×Dic3—21no—
280C4×Dic3—21no—
281C4×Dic3—24no—
282C4×Dic3C2×D412yes—
283C4×Dic3C2×D412yes—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290#204—28no—
291C22×Q8—31no—
292C22×Q8—31no—
293C22×Q8—61no—
294C22×D4—31no—
295C23×C4—61no—
296C2×SD16—34no—
297C2×SD16—62no—
298C2×SD16—62no—
299C22×C8—31no—
300C22×C8—31no—
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303C4⋊Q8—34no—
304C4⋊Q8—62no—
305C4⋊Q8—68no—
306#125—22no—
307#125—22no—
308#125—62no—
309#125—68no—
310#125—68no—
311#125—62no—
312C4×Q8—34no—
313C4×Q8—62no—
314C4×Q8—62no—
315C4×Q8—62no—
316C4×Q8—62no—
317C4×Q8—62no—
318C4×Q8—62no—
319C4×Q8—68no—
320C4×Q8—62no—
321C4×D4—62no—
322C4×D4—62no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4C22×D311yescharacteristic
339C2×C4⋊C4C22×D311yescharacteristic
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C22⋊C4—61no—
343C2×C22⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C22⋊C4—61no—
346C2×C22⋊C4—61no—
347C2×C42—31no—
348C2×C42—31no—
349C2×C42—61no—
350C2×C42—61no—
351C2×C42—61no—
352C2×C42C22×D311yescharacteristic
353#107—22no—
354#106—22no—
355#106—62no—
356C4⋊C8—62no—
357#103—62no—
358#103—62no—
359#103—62no—
360#102—62no—
361#102—62no—
362#102—62no—
363#97—62no—
364#95—21no—
365#95—61no—
366#95—61no—
367#95—61no—
368C23×C3C22×D411yescharacteristic
369C22×D3—22no—
370C22×D3—41no—
371C22×D3—41no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
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×C4—41no—
394C2×C3×C4C22×D411yescharacteristic
395C2×C3×C4C22×D411yescharacteristic
396C2×C3×C4C22×Q811yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C42−1+411yescharacteristic
399C2×C3×C42−1+411yescharacteristic
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—22no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—22no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3#12012yes—
420C2×Dic3#12012yes—
421D12—22no—
422D12—22no—
423D12—41no—
424D12—41no—
425C4×D3—84no—
426Dic6—22no—
427Dic6—22no—
428Dic6—22no—
429Dic6—44no—
430Dic6—48no—
431Dic6C22⋊Q814yes—
432C3×C8—22no—
433C3×C8—41no—
434C3⋊C8—41no—
435C3⋊C8—41no—
436C24—61no—
437C2×Q8—32no—
438C2×Q8—32no—
439C2×Q8—32no—
440C2×Q8—64no—
441C2×Q8—62no—
442C2×Q8—61no—
443C2×Q8—61no—
444C2×Q8—61no—
445C2×Q8—121no—
446C2×Q8—121no—
447C2×Q8—122no—
448C2×Q8—122no—
449C2×D4—32no—
450C2×D4—61no—
451C2×D4—61no—
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—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—122no—
467C22×C4—241no—
468C22×C4—241no—
469C22×C4D3×D411yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4C23×D311yescharacteristic
472SD16—68no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8D3×Q812yes—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—32no—
494C4⋊C4—62no—
495C4⋊C4—62no—
496C4⋊C4—62no—
497C4⋊C4—64no—
498C4⋊C4—64no—
499C4⋊C4—64no—
500C4⋊C4—64no—
501C4⋊C4—64no—
502C4⋊C4—64no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—81no—
507C4⋊C4—241no—
508C4⋊C4D3×D412yes—
509C4⋊C4D3×D412yes—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—241no—
514C22⋊C4—241no—
515C42—21no—
516C42—21no—
517C42—32no—
518C42—32no—
519C42—64no—
520C42—64no—
521C42—64no—
522C42—61no—
523C42—61no—
524C22×C3C2×#12011yescharacteristic
525C22×C3#54811yescharacteristic
526C22×C3D4×Q811yescharacteristic
527C22×C3#57511yescharacteristic
528C22×C3#57511yescharacteristic
529C22×C3#57711yescharacteristic
530C22×C3#57711yescharacteristic
531C2×D3—42no—
532C2×D3—42no—
533C2×D3—42no—
534C2×D3—81no—
535C2×D3—81no—
536C2×D3—81no—
537C2×D3—81no—
538C3×C4—41no—
539C3×C4—42no—
540C3×C4—41no—
541C3×C4—42no—
542C3×C4—81no—
543C3×C4—81no—
544C3×C4C2×C22⋊Q811yescharacteristic
545C3×C4#53611yescharacteristic
546C3×C4#54811yescharacteristic
547C3×C4D4×Q811yescharacteristic
548Dic3—24no—
549Dic3—22no—
550Dic3—22no—
551Dic3—41no—
552Dic3—42no—
553Dic3—41no—
554Dic3—81no—
555C23—62no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23C2×D3×D411yescharacteristic
563Q8—34no—
564Q8—62no—
565Q8—62no—
566Q8—62no—
567Q8—124no—
568Q8—128no—
569D4—62no—
570D4—62no—
571D4—121no—
572D4—121no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—32no—
582C2×C4—32no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—62no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—62no—
606C2×C4—61no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—244no—
613C2×C4C2×D3×D411yescharacteristic
614C2×C4C2×D3×D411yescharacteristic
615C2×C4C2×D3×Q811yescharacteristic
616C2×C4#100711yescharacteristic
617C2×C4#100711yescharacteristic
618C2×C4D3×#4111yescharacteristic
619C8—22no—
620C8—41no—
621C8—121no—
622C8—121no—
623C2×C3#240911yescharacteristic
624C2×C3#300511yescharacteristic
625C2×C3#301011yescharacteristic
626C2×C3#337111yescharacteristic
627C2×C3#337111yescharacteristic
628C2×C3#338811yescharacteristic
629C2×C3#338811yescharacteristic
630D3—82no—
631C22—122no—
632C22—122no—
633C22—122no—
634C22—241no—
635C22—241no—
636C22—241no—
637C22—241no—
638C22D3×#12011yescharacteristic
639C22#563311yescharacteristic
640C22#572711yescharacteristic
641C22#576811yescharacteristic
642C22#577011yescharacteristic
643C22#577511yescharacteristic
644C22#577711yescharacteristic
645C4—41no—
646C4—42no—
647C4—41no—
648C4—42no—
649C4—62no—
650C4—64no—
651C4—62no—
652C4—81no—
653C4—81no—
654C4—121no—
655C4—122no—
656C4—121no—
657C4—241no—
658C4D3×C22⋊Q811yescharacteristic
659C4#563311yescharacteristic
660C4#572411yescharacteristic
661C4#572711yescharacteristic
662C3#2434011yescharacteristic, complemented, Sylow
663C2—242no—
664C2#7717211yescharacteristic
665C2#8008611yescharacteristic
666C2#8022711yescharacteristic
667C2#8120311yescharacteristic
668C2#8121811yescharacteristic
669C2#8193011yescharacteristic
670C2#8196911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2059C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2059D3
hypocenterC3#24340
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
1. C2×#5685C2semidirect
5. C3×#2059C2semidirect
7. #70618C2semidirect
8. #70588C2semidirect
9. #69632C2semidirect
11. #68905C2semidirect
13. #68806C2semidirect
15. #68620C2semidirect
69. #2059D3semidirect
122. C2×#204Q8semidirect
662. C3#24340semidirect

Maximal subgroups

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

subgroupindex
C2×#56852
C2×#48892
C2×#48702
#760752
C3×#20592
#717432
#706182
#705882
#696322
#689092
#689052
#688992
#688062
#686862
#686202
#243403