Finite Group Databasebeta

← #10831734

Subgroups — #10831734

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5722C211yesmaximal, characteristic, complemented
2C2×#4890C211yesmaximal, characteristic
3C2×#4889C211yesmaximal, characteristic
4#76075C211yesmaximal, characteristic
5C3×#2057C211yesmaximal, characteristic, complemented
6#71792C211yesmaximal, characteristic
7#70642C211yesmaximal, characteristic, complemented
8#70617C211yesmaximal, characteristic, complemented
9#69631C211yesmaximal, characteristic, complemented
10#68909C211yesmaximal, characteristic
11#68905C211yesmaximal, characteristic, complemented
12#68898C211yesmaximal, characteristic
13#68806C211yesmaximal, characteristic, complemented
14#68686C211yesmaximal, characteristic
15#68620C211yesmaximal, characteristic, complemented
16#24341—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×#944—21no—
19C2×#944—21no—
20C22×#204C2211yescharacteristic
21#5722—24no—
22#5722—24no—
23C2×#893C2211yescharacteristic
24C2×#888—21no—
25C2×#888C2211yescharacteristic
26C2×#888C2211yescharacteristic
27C2×#887—21no—
28C2×#887C2211yescharacteristic
29C2×#887C2211yescharacteristic
30C2×C4×Dic6C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×#107C2211yescharacteristic
33C2×C3×#107C2211yescharacteristic
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×#807C2211yescharacteristic
49#4939C2211yescharacteristic
50#4890—24no—
51#4889—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×Q82—31no—
65C2×#474—31no—
66C2×#440—31no—
67#2580—31no—
68#2057—31no—
69#2057D311yescharacteristic, complemented
70#2023—31no—
71#2023—31no—
72#1993—31no—
73#1985—31no—
74#1880—31no—
75#1880—31no—
76#1863—31no—
77#1863—31no—
78#1805—31no—
79C22×C3×Q8—21no—
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×Dic6—21no—
83C22×Dic6—21no—
84C22×Dic6D411yescharacteristic
85C22×Dic6C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C4⋊Q8—22no—
89C3×C4⋊Q8—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97#944—28no—
98#944—28no—
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×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⋊C4D411yescharacteristic
114C2×Dic3⋊C4D411yescharacteristic
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C2×#204—22no—
123C2×#204—22no—
124C2×#204Q814yescomplemented
125#893—22no—
126#893—22no—
127#888—28no—
128#888—22no—
129#888—22no—
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×#107—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#807—22no—
156C22×SD16—31no—
157Q82—64no—
158Q82—64no—
159C2×C4⋊Q8—31no—
160C2×C4⋊Q8—31no—
161C2×C4⋊Q8—61no—
162C2×C4⋊Q8—61no—
163C2×C4⋊Q8—61no—
164C2×C4⋊Q8C2×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#474—64no—
172#440—64no—
173C2×#107—31no—
174C2×#107C2×D311yescharacteristic
175C2×#107C2×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—
204C2×C3×Q8—24no—
205C2×C3×Q8—81no—
206C2×C3×Q8—81no—
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×D12—21no—
225C2×D12—21no—
226C2×D12C2×Q812yes—
227C2×C4×D3—42no—
228C2×C4×D3—81no—
229C2×C4×D3—81no—
230C2×Dic6—22no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—22no—
235C2×Dic6—22no—
236C2×Dic6—24no—
237C2×Dic6—24no—
238C2×Dic6—81no—
239C2×Dic6—81no—
240C2×Dic6—81no—
241C2×Dic6—81no—
242C2×Dic6C2×D412yes—
243C2×Dic6C2×Q812yes—
244C2×Dic6#4112yes—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8C2×Q812yes—
251C3×C4⋊C4—24no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—81no—
259C3×C4⋊C4C2×D412yes—
260C3×C4⋊C4C2×D412yes—
261C3×C42—21no—
262C3×C42—21no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—42no—
266(C2×D3)⋊C4—81no—
267(C2×D3)⋊C4—81no—
268C4⋊Dic3—21no—
269C4⋊Dic3—22no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3C2×D412yes—
273Dic3⋊C4—24no—
274Dic3⋊C4—24no—
275Dic3⋊C4—24no—
276Dic3⋊C4—22no—
277Dic3⋊C4—22no—
278Dic3⋊C4—24no—
279Dic3⋊C4—24no—
280Dic3⋊C4—81no—
281C4×Dic3—24no—
282C4×Dic3—24no—
283C4×Dic3—21no—
284C4×Dic3—21no—
285C4×Dic3—24no—
286C4×Dic3C2×D412yes—
287C4×Dic3C2×D412yes—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294#204—28no—
295C22×Q8—21no—
296C22×Q8—31no—
297C22×Q8—31no—
298C22×Q8—61no—
299C22×Q8—61no—
300C22×D4—31no—
301C23×C4—61no—
302C2×SD16—34no—
303C2×SD16—62no—
304C2×SD16—62no—
305C22×C8—31no—
306C22×C8—31no—
307C22×C8C22×D311yescharacteristic
308C22×C8C22×D311yescharacteristic
309C4⋊Q8—22no—
310C4⋊Q8—22no—
311C4⋊Q8—34no—
312C4⋊Q8—62no—
313C4⋊Q8—62no—
314C4⋊Q8—68no—
315C4⋊Q8—68no—
316C4⋊Q8—68no—
317C4⋊Q8—62no—
318C4×Q8—34no—
319C4×Q8—62no—
320C4×Q8—62no—
321C4×Q8—62no—
322C4×Q8—62no—
323C4×Q8—62no—
324C4×Q8—62no—
325C4×Q8—68no—
326C4×Q8—62no—
327C4×D4—62no—
328C4×D4—62no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—31no—
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⋊C4C22×D311yescharacteristic
343C2×C4⋊C4C22×D311yescharacteristic
344C2×C4⋊C4C22×D311yescharacteristic
345C2×C4⋊C4C22×D311yescharacteristic
346C2×C22⋊C4—61no—
347C2×C22⋊C4—61no—
348C2×C22⋊C4—61no—
349C2×C22⋊C4—61no—
350C2×C22⋊C4—61no—
351C2×C42—31no—
352C2×C42—31no—
353C2×C42—61no—
354C2×C42—61no—
355C2×C42—61no—
356C2×C42C22×D311yescharacteristic
357#107—22no—
358#107—22no—
359#107—62no—
360C4⋊C8—62no—
361#103—62no—
362#103—62no—
363#103—62no—
364#102—62no—
365#102—62no—
366#102—62no—
367#97—62no—
368#95—21no—
369#95—61no—
370#95—61no—
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—84no—
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×C4#13712yes—
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—21no—
406C2×Dic3—22no—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—22no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3#12012yes—
424C2×Dic3#12012yes—
425C2×Dic3#13712yes—
426D12—22no—
427D12—22no—
428D12—41no—
429D12—41no—
430C4×D3—84no—
431Dic6—28no—
432Dic6—22no—
433Dic6—22no—
434Dic6—22no—
435Dic6—84no—
436Dic6—84no—
437Dic6C22⋊Q814yes—
438C3×C8—22no—
439C3×C8—41no—
440C3⋊C8—41no—
441C3⋊C8—41no—
442C24—61no—
443C2×Q8—24no—
444C2×Q8—32no—
445C2×Q8—32no—
446C2×Q8—32no—
447C2×Q8—62no—
448C2×Q8—62no—
449C2×Q8—64no—
450C2×Q8—64no—
451C2×Q8—62no—
452C2×Q8—61no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—81no—
456C2×Q8—81no—
457C2×Q8—241no—
458C2×Q8—241no—
459C2×Q8—241no—
460C2×Q8—241no—
461C2×D4—32no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—61no—
465C2×D4—61no—
466C2×D4—61no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—31no—
474C22×C4—31no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—122no—
479C22×C4—241no—
480C22×C4—241no—
481C22×C4D3×D411yescharacteristic
482C22×C4D3×D411yescharacteristic
483C22×C4C23×D311yescharacteristic
484SD16—68no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8D3×Q812yes—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—32no—
505C4⋊C4—62no—
506C4⋊C4—62no—
507C4⋊C4—62no—
508C4⋊C4—64no—
509C4⋊C4—64no—
510C4⋊C4—64no—
511C4⋊C4—64no—
512C4⋊C4—64no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—61no—
516C4⋊C4—81no—
517C4⋊C4—241no—
518C4⋊C4D3×D412yes—
519C4⋊C4D3×D412yes—
520C22⋊C4—122no—
521C22⋊C4—122no—
522C22⋊C4—122no—
523C22⋊C4—241no—
524C22⋊C4—241no—
525C42—21no—
526C42—21no—
527C42—32no—
528C42—32no—
529C42—64no—
530C42—64no—
531C42—64no—
532C42—61no—
533C42—61no—
534C22×C3C2×#12011yescharacteristic
535C22×C3#54811yescharacteristic
536C22×C3D4×Q811yescharacteristic
537C22×C3C2×#13711yescharacteristic
538C22×C3C2×#13711yescharacteristic
539C22×C3#57611yescharacteristic
540C22×C3#57611yescharacteristic
541C2×D3—42no—
542C2×D3—42no—
543C2×D3—42no—
544C2×D3—81no—
545C2×D3—81no—
546C2×D3—81no—
547C2×D3—81no—
548C3×C4—22no—
549C3×C4—42no—
550C3×C4—42no—
551C3×C4—41no—
552C3×C4—81no—
553C3×C4—81no—
554C3×C4C2×C22⋊Q811yescharacteristic
555C3×C4#53611yescharacteristic
556C3×C4#54811yescharacteristic
557C3×C4D4×Q811yescharacteristic
558Dic3—24no—
559Dic3—22no—
560Dic3—22no—
561Dic3—22no—
562Dic3—41no—
563Dic3—42no—
564Dic3—81no—
565C23—62no—
566C23—121no—
567C23—121no—
568C23—121no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23C2×D3×D411yescharacteristic
573Q8—34no—
574Q8—68no—
575Q8—62no—
576Q8—62no—
577Q8—62no—
578Q8—84no—
579Q8—244no—
580Q8—244no—
581D4—62no—
582D4—62no—
583D4—121no—
584D4—121no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—32no—
593C2×C4—32no—
594C2×C4—32no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—62no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—62no—
617C2×C4—61no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—244no—
624C2×C4#91812yes—
625C2×C4C2×D3×D411yescharacteristic
626C2×C4C2×D3×D411yescharacteristic
627C2×C4C2×D3×Q811yescharacteristic
628C2×C4#100711yescharacteristic
629C2×C4#100711yescharacteristic
630C2×C4D3×#4111yescharacteristic
631C8—22no—
632C8—41no—
633C8—121no—
634C8—121no—
635C2×C3#240911yescharacteristic
636C2×C3#300411yescharacteristic
637C2×C3#300811yescharacteristic
638C2×C3#337511yescharacteristic
639C2×C3#337511yescharacteristic
640C2×C3#338511yescharacteristic
641C2×C3#338511yescharacteristic
642D3—82no—
643C22—122no—
644C22—122no—
645C22—122no—
646C22—241no—
647C22—241no—
648C22—241no—
649C22—241no—
650C22D3×#12011yescharacteristic
651C22#563311yescharacteristic
652C22#572711yescharacteristic
653C22#575711yescharacteristic
654C22C2×#91811yescharacteristic
655C22#577411yescharacteristic
656C22D3×#13711yescharacteristic
657C4—22no—
658C4—41no—
659C4—42no—
660C4—42no—
661C4—62no—
662C4—62no—
663C4—64no—
664C4—62no—
665C4—81no—
666C4—81no—
667C4—121no—
668C4—122no—
669C4—241no—
670C4D3×C22⋊Q811yescharacteristic
671C4#563311yescharacteristic
672C4#572411yescharacteristic
673C4#572711yescharacteristic
674C3#2434111yescharacteristic, complemented, Sylow
675C2—242no—
676C2#7717211yescharacteristic
677C2#8000411yescharacteristic
678C2#8018311yescharacteristic
679C2#8095611yescharacteristic
680C2#8113911yescharacteristic
681C2#8195411yescharacteristic
682C2#8196511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2057C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2057D3
hypocenterC3#24341
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×#5722C2semidirect
5. C3×#2057C2semidirect
7. #70642C2semidirect
8. #70617C2semidirect
9. #69631C2semidirect
11. #68905C2semidirect
13. #68806C2semidirect
15. #68620C2semidirect
69. #2057D3semidirect
124. C2×#204Q8semidirect
674. C3#24341semidirect

Maximal subgroups

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

subgroupindex
C2×#57222
C2×#48902
C2×#48892
#760752
C3×#20572
#717922
#706422
#706172
#696312
#689092
#689052
#688982
#688062
#686862
#686202
#243413