Finite Group Databasebeta

← #10845251

Subgroups — #10845251

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
1C2×#4844C211yesmaximal, characteristic
2C2×#4714C211yesmaximal, characteristic, complemented
3#73807C211yesmaximal, characteristic, complemented
4#73655C211yesmaximal, characteristic, complemented
5C3×#1940C211yesmaximal, characteristic, complemented
6#70486C211yesmaximal, characteristic, complemented
7#70482C211yesmaximal, characteristic, complemented
8#70481C211yesmaximal, characteristic
9#70480C211yesmaximal, characteristic, complemented
10#70460C211yesmaximal, characteristic, complemented
11#69812C211yesmaximal, characteristic, complemented
12#68686C211yesmaximal, characteristic, complemented
13#68673C211yesmaximal, characteristic
14#68638C211yesmaximal, characteristic, complemented
15#68637C211yesmaximal, characteristic, complemented
16#25335—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C22×#204C2211yescharacteristic
19C2×#875C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×#869C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C3×C4×C4⋊C4C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37#4990—21no—
38#4989—21no—
39#4987—21no—
40#4987—21no—
41#4939C2211yescharacteristic
42#4939C2211yescharacteristic
43#4939C2211yescharacteristic
44#4936C2211yescharacteristic
45#4936C2211yescharacteristic
46C4×C4⋊Dic3C2211yescharacteristic
47#4929C2211yescharacteristic
48#4844—24no—
49#4714—24no—
50#4673—21no—
51#4673—21no—
52#4669—21no—
53#4666—21no—
54C3×#336C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
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#1988—31no—
71#1940D311yescharacteristic, complemented
72#1880—31no—
73#1879—31no—
74#1805—31no—
75#1804—31no—
76#1762—31no—
77C22×D12C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×C4×D3—21no—
81C22×Dic6—21no—
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
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×C4⋊Dic3—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×D24—22no—
112C2×D24—22no—
113C2×D24—41no—
114C2×#204—22no—
115C2×#204—22no—
116C2×#204—41no—
117#875—22no—
118#875D414yes—
119C4×D12—22no—
120C4×D12—22no—
121C4×D12—22no—
122C4×D12—41no—
123#869—22no—
124#869—22no—
125C4×Dic6—22no—
126C4×Dic6—41no—
127C3×C4⋊C8—22no—
128C3×C4⋊C8—41no—
129C3×C4⋊C8—41no—
130C3×#97—22no—
131C3×#95D411yescharacteristic
132C3×#95D411yescharacteristic
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#821—22no—
141#821—22no—
142#821—22no—
143#821—22no—
144#821—22no—
145#816—22no—
146#816—22no—
147#816—22no—
148C22×SD16—31no—
149C22×D8—31no—
150C2×C4⋊Q8—31no—
151C2×#124—31no—
152C2×C4×Q8—31no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155#489—64no—
156#463—64no—
157C2×C4⋊C8C2×D311yescharacteristic
158C2×C4⋊C8C2×D311yescharacteristic
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×#97C2×D311yescharacteristic
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#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336C2×D311yescharacteristic
192#336C2×D311yescharacteristic
193C23×D3—21no—
194C23×D3—21no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2411yescharacteristic
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12C2×D412yes—
219C2×D12#4112yes—
220C2×C4×D3—42no—
221C2×C4×D3—42no—
222C2×C4×D3—42no—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
225C2×C4×D3—81no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—42no—
232C2×Dic6—81no—
233C2×Dic6—81no—
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—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C3×C4⋊C4—41no—
248C3×C4⋊C4—41no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42C2×D412yes—
254C3×C42#4112yes—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—81no—
261(C2×D3)⋊C4—81no—
262(C2×D3)⋊C4—81no—
263(C2×D3)⋊C4—81no—
264(C2×D3)⋊C4—81no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—42no—
275C4⋊Dic3#4112yes—
276Dic3⋊C4—42no—
277C4×Dic3—42no—
278C4×Dic3—81no—
279D24—44no—
280#204—44no—
281C22×Q8—31no—
282C22×Q8—61no—
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×C8C22×D311yescharacteristic
294C22×C8C22×D311yescharacteristic
295C22×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297C4⋊Q8—62no—
298C4⋊Q8—62no—
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—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4D1211yescharacteristic
315C2×C4⋊C4D1211yescharacteristic
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—61no—
327C2×C42—61no—
328C2×C42C22×D311yescharacteristic
329C2×C42C22×D311yescharacteristic
330C2×C42C22×D311yescharacteristic
331C4⋊C8—22no—
332C4⋊C8—41no—
333C4⋊C8—41no—
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—22no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95D1211yescharacteristic
351#95D1211yescharacteristic
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—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4#12112yes—
384C2×C3×C4#12112yes—
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4H(5,𝔽2)11yescharacteristic
390C2×C3×C42−1+411yescharacteristic
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409D12—22no—
410D12—22no—
411D12—22no—
412D12—22no—
413D12—41no—
414D12—41no—
415D12—41no—
416D12—41no—
417C4×D3—84no—
418C4×D3—84no—
419Dic6—22no—
420Dic6—22no—
421Dic6—41no—
422Dic6—41no—
423Dic6—84no—
424C3×C8—41no—
425C3×C8—41no—
426C3×C8—41no—
427C3×C8—41no—
428C24—61no—
429C24—61no—
430C2×Q8—32no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—122no—
437C2×Q8—241no—
438C2×Q8—241no—
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—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—122no—
458C22×C4—122no—
459C22×C4—122no—
460C22×C4—122no—
461C22×C4—122no—
462C22×C4—241no—
463C22×C4C2×D1211yescharacteristic
464C22×C4C2×D1211yescharacteristic
465C22×C4C2×D1211yescharacteristic
466C22×C4C2×D1211yescharacteristic
467C22×C4C2×D1211yescharacteristic
468C22×C4C2×D1211yescharacteristic
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—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C4⋊C4—32no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—122no—
497C4⋊C4—122no—
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—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—122no—
513C42—241no—
514C42D3×D412yes—
515C42#23912yes—
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—41no—
540C3×C4—41no—
541C3×C4—41no—
542C3×C4—41no—
543C3×C4#52911yescharacteristic
544C3×C4#54211yescharacteristic
545C3×C4#54611yescharacteristic
546C3×C4#54711yescharacteristic
547Dic3—42no—
548Dic3—42no—
549Dic3—81no—
550Dic3—81no—
551Dic3—81no—
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—
567C23C22×D1211yescharacteristic
568Q8—62no—
569Q8—62no—
570Q8—121no—
571Q8—121no—
572Q8—244no—
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—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
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—121no—
615C2×C4—244no—
616C2×C4—244no—
617C2×C4#89512yes—
618C2×C4#89512yes—
619C2×C4C2×D3×D411yescharacteristic
620C2×C4#100411yescharacteristic
621C2×C4C2×#23911yescharacteristic
622C2×C4#100711yescharacteristic
623C2×C4D3×#4111yescharacteristic
624C2×C4D3×#4111yescharacteristic
625C8—41no—
626C8—41no—
627C8—41no—
628C8—41no—
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×#89511yescharacteristic
653C22#555411yescharacteristic
654C22#557611yescharacteristic
655C22#574811yescharacteristic
656C22#574811yescharacteristic
657C22#575111yescharacteristic
658C22#575211yescharacteristic
659C4—22no—
660C4—22no—
661C4—41no—
662C4—41no—
663C4—41no—
664C4—41no—
665C4—122no—
666C4—122no—
667C4—241no—
668C4—241no—
669C4—241no—
670C4—241no—
671C4#561211yescharacteristic
672C4#563111yescharacteristic
673C4#569111yescharacteristic
674C4#573311yescharacteristic
675C3#2533511yescharacteristic, complemented, Sylow
676C2—242no—
677C2—242no—
678C2#7621911yescharacteristic
679C2#7905811yescharacteristic
680C2#7908811yescharacteristic
681C2#7950611yescharacteristic
682C2#7951611yescharacteristic
683C2#7997111yescharacteristic
684C2#7997411yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1940C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1940D3
hypocenterC3#25335
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1940

Complements

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

NH ≅ G/Ntype
2. C2×#4714C2semidirect
3. #73807C2semidirect
4. #73655C2semidirect
5. C3×#1940C2semidirect
6. #70486C2semidirect
7. #70482C2semidirect
9. #70480C2semidirect
10. #70460C2semidirect
11. #69812C2semidirect
12. #68686C2semidirect
14. #68638C2semidirect
15. #68637C2semidirect
71. #1940D3semidirect
675. C3#25335semidirect

Maximal subgroups

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

subgroupindex
C2×#48442
C2×#47142
#738072
#736552
C3×#19402
#704862
#704822
#704812
#704802
#704602
#698122
#686862
#686732
#686382
#686372
#253353