Finite Group Databasebeta

← #10845255

Subgroups — #10845255

Order 768.

872 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83260C211yesmaximal, characteristic, complemented
2C2×#5089C211yesmaximal, characteristic, complemented
3C2×#4844C211yesmaximal, characteristic
4#73654C211yesmaximal, characteristic, complemented
5C3×#1988C211yesmaximal, characteristic, complemented
6#70469C211yesmaximal, characteristic, complemented
7#69474C211yesmaximal, characteristic, complemented
8#68915C211yesmaximal, characteristic, complemented
9#68909C211yesmaximal, characteristic, complemented
10#68905C211yesmaximal, characteristic, complemented
11#68903C211yesmaximal, characteristic, complemented
12#68897C211yesmaximal, characteristic
13#68893C211yesmaximal, characteristic, complemented
14#68863C211yesmaximal, characteristic, complemented
15#68862C211yesmaximal, characteristic
16#25335—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C22×D24C2211yescharacteristic
19C22×#204C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×#875C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26#5231—21no—
27#5231—21no—
28C2×#821C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32#5089—24no—
33#4988C2211yescharacteristic
34#4988C2211yescharacteristic
35#4988C2211yescharacteristic
36#4987—21no—
37#4987—21no—
38#4977C2211yescharacteristic
39#4977C2211yescharacteristic
40#4973C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#809C2211yescharacteristic
47#4940—21no—
48C4×(C2×D3)⋊C4—21no—
49C4×C4⋊Dic3C2211yescharacteristic
50C2×#803C2211yescharacteristic
51#4844—24no—
52#4670—21no—
53#4668—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#489—31no—
63C2×#463—31no—
64#2563—31no—
65#2367—31no—
66#2074—31no—
67#2073—31no—
68#2055—31no—
69#2054—31no—
70#1988D311yescharacteristic, complemented
71#1940—31no—
72#1880—31no—
73#1879—31no—
74#1805—31no—
75#1804—31no—
76#1762—31no—
77C22×C3×Q8—21no—
78C22×D12C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4⋊Q8—22no—
86C3×C4⋊Q8—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
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×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3D411yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C2×D24—22no—
114C2×D24—22no—
115C2×D24—41no—
116C2×#204—22no—
117C2×#204—22no—
118C2×#204—41no—
119#893—22no—
120#893—41no—
121#887—22no—
122#887—41no—
123#875—22no—
124#875D414yes—
125C4×D12—22no—
126C4×D12—22no—
127C3×C4⋊C8—22no—
128C3×#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137#821—22no—
138#816—22no—
139Dic3⋊C8—41no—
140Dic3⋊C8—41no—
141#810—22no—
142#810—22no—
143#809—22no—
144#809—22no—
145#809—22no—
146#809—22no—
147#803—22no—
148C22×SD16—31no—
149C22×D8—31no—
150C2×C4⋊Q8C2×D311yescharacteristic
151C2×#124—31no—
152C2×C4×Q8—31no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155#489—64no—
156#463—64no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97—31no—
169#393—61no—
170#393—61no—
171#391—31no—
172#390—31no—
173#390—31no—
174#390—31no—
175#388—61no—
176#388—61no—
177#385—61no—
178#382—31no—
179#382—31no—
180#380—61no—
181#380—61no—
182C4×C4⋊C4—31no—
183C4×C4⋊C4C2×D311yescharacteristic
184C4×C22⋊C4—61no—
185#340—31no—
186#340—31no—
187#340C2×D311yescharacteristic
188#340C2×D311yescharacteristic
189#340C2×D311yescharacteristic
190#340C2×D311yescharacteristic
191#336—31no—
192#336—31no—
193C23×D3—21no—
194C23×D3—21no—
195C2×C3×Q8—42no—
196C2×C3×Q8—81no—
197C2×C3×Q8—81no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12C2×D412yes—
222C2×D12#4112yes—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
225C2×C4×D3—42no—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×C4×D3—81no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6C2×D412yes—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—42no—
249C3×C4⋊C4#4112yes—
250C3×C42—81no—
251C3×C42C2×D412yes—
252C3×C42#4112yes—
253(C2×D3)⋊C4—24no—
254(C2×D3)⋊C4—24no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—81no—
259(C2×D3)⋊C4—81no—
260(C2×D3)⋊C4—81no—
261(C2×D3)⋊C4—81no—
262(C2×D3)⋊C4—81no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—41no—
266C4⋊Dic3—41no—
267Dic3⋊C4—42no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272C4×Dic3—42no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279D24—44no—
280#204—44no—
281C22×Q8—21no—
282C22×Q8—31no—
283C22×D4—31no—
284C22×D4—31no—
285C23×C4—61no—
286C23×C4—61no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—121no—
290C2×D8—62no—
291C2×D8—62no—
292C2×D8—121no—
293C22×C8—31no—
294C22×C8—31no—
295C22×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299#124—34no—
300#124—62no—
301C4×Q8—62no—
302C4×Q8—121no—
303C4×D4—62no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—121no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C42—21no—
327C2×C42—31no—
328C2×C42—31no—
329C2×C42—61no—
330C2×C42C22×D311yescharacteristic
331C4⋊C8—22no—
332C4⋊C8—121no—
333C4⋊C8—121no—
334#103—62no—
335#103—62no—
336#103—62no—
337#102—62no—
338#102—62no—
339#102—62no—
340#102—62no—
341#102—62no—
342#97—62no—
343#95—21no—
344#95—31no—
345#95—31no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352C23×C3C22×D411yescharacteristic
353C22×D3—22no—
354C22×D3—22no—
355C22×D3—41no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C3×Q8—84no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—22no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4#12112yes—
385C2×C3×C4#12112yes—
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4H(5,𝔽2)11yescharacteristic
391C2×C3×C42−1+411yescharacteristic
392C2×Dic3—22no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410D12—22no—
411D12—22no—
412D12—22no—
413D12—22no—
414D12—41no—
415D12—41no—
416D12—41no—
417D12—41no—
418C4×D3—84no—
419C4×D3—84no—
420Dic6—22no—
421Dic6—22no—
422Dic6—41no—
423Dic6—41no—
424C3×C8—41no—
425C3×C8—41no—
426C3⋊C8—41no—
427C3⋊C8—41no—
428C24—61no—
429C24—61no—
430C2×Q8—32no—
431C2×Q8—42no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8—81no—
438C2×Q8—81no—
439C2×D4—32no—
440C2×D4—32no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—122no—
462C22×C4—122no—
463C22×C4—122no—
464C22×C4—122no—
465C22×C4—122no—
466C22×C4—241no—
467C22×C4D3×D411yescharacteristic
468C22×C4D3×D411yescharacteristic
469C22×C4C23×D311yescharacteristic
470SD16—124no—
471D8—124no—
472C2×C8—21no—
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—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—42no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—121no—
495C4⋊C4—121no—
496C4⋊C4—122no—
497C4⋊C4#23912yes—
498C22⋊C4—64no—
499C22⋊C4—64no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—241no—
504C22⋊C4—241no—
505C22⋊C4—241no—
506C22⋊C4—241no—
507C22⋊C4—241no—
508C42—61no—
509C42—61no—
510C42—61no—
511C42—61no—
512C42—81no—
513C42—122no—
514C42#23512yes—
515C42D3×D412yes—
516C22×C3C2×#12111yescharacteristic
517C22×C3#54611yescharacteristic
518C22×C3#54811yescharacteristic
519C22×C3#57511yescharacteristic
520C22×C3#57511yescharacteristic
521C22×C3#57611yescharacteristic
522C22×C3#57711yescharacteristic
523C2×D3—42no—
524C2×D3—42no—
525C2×D3—42no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—81no—
532C2×D3—81no—
533C2×D3—81no—
534C2×D3—81no—
535C2×D3—81no—
536C2×D3—81no—
537C3×C4—22no—
538C3×C4—22no—
539C3×C4—42no—
540C3×C4—81no—
541C3×C4—81no—
542C3×C4—81no—
543C3×C4#52911yescharacteristic
544C3×C4#54211yescharacteristic
545C3×C4#54611yescharacteristic
546C3×C4#54711yescharacteristic
547Dic3—42no—
548Dic3—41no—
549Dic3—41no—
550Dic3—41no—
551Dic3—41no—
552Dic3—81no—
553C23—62no—
554C23—62no—
555C23—121no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23C2×D3×D411yescharacteristic
568Q8—62no—
569Q8—62no—
570Q8—84no—
571Q8—121no—
572Q8—121no—
573D4—62no—
574D4—62no—
575D4—62no—
576D4—62no—
577D4—121no—
578D4—121no—
579D4—121no—
580D4—121no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—41no—
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—62no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—244no—
616C2×C4—244no—
617C2×C4#88412yes—
618C2×C4#88412yes—
619C2×C4C2×#23511yescharacteristic
620C2×C4C2×D3×D411yescharacteristic
621C2×C4C2×#23911yescharacteristic
622C2×C4#100711yescharacteristic
623C2×C4D3×#4111yescharacteristic
624C2×C4#100911yescharacteristic
625C8—41no—
626C8—41no—
627C8—121no—
628C8—121no—
629C2×C3#242711yescharacteristic
630C2×C3#304011yescharacteristic
631C2×C3#304511yescharacteristic
632C2×C3#331911yescharacteristic
633C2×C3#332111yescharacteristic
634C2×C3#337011yescharacteristic
635C2×C3#337611yescharacteristic
636D3—82no—
637D3—82no—
638C22—122no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22—122no—
643C22—122no—
644C22—241no—
645C22—241no—
646C22—241no—
647C22—241no—
648C22—241no—
649C22—241no—
650C22—241no—
651C22—241no—
652C22C2×#88411yescharacteristic
653C22#563011yescharacteristic
654C22#563311yescharacteristic
655C22#575711yescharacteristic
656C22#576211yescharacteristic
657C22#577711yescharacteristic
658C22#577711yescharacteristic
659C4—22no—
660C4—22no—
661C4—42no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—122no—
666C4—121no—
667C4—121no—
668C4—121no—
669C4—121no—
670C4—241no—
671C4#558011yescharacteristic
672C4#558211yescharacteristic
673C4#567611yescharacteristic
674C4#572611yescharacteristic
675C3#2533511yescharacteristic, complemented, Sylow
676C2—242no—
677C2—242no—
678C2#7662411yescharacteristic
679C2#7876711yescharacteristic
680C2#7897511yescharacteristic
681C2#8095511yescharacteristic
682C2#8098311yescharacteristic
683C2#8100511yescharacteristic
684C2#8102511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1988C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1988D3
hypocenterC3#25335
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
1. #83260C2semidirect
2. C2×#5089C2semidirect
4. #73654C2semidirect
5. C3×#1988C2semidirect
6. #70469C2semidirect
7. #69474C2semidirect
8. #68915C2semidirect
9. #68909C2semidirect
10. #68905C2semidirect
11. #68903C2semidirect
13. #68893C2semidirect
14. #68863C2semidirect
70. #1988D3semidirect
675. C3#25335semidirect

Maximal subgroups

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

subgroupindex
#832602
C2×#50892
C2×#48442
#736542
C3×#19882
#704692
#694742
#689152
#689092
#689052
#689032
#688972
#688932
#688632
#688622
#253353