Finite Group Databasebeta

← #10855201

Subgroups — #10855201

Order 768.

908 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4819C211yesmaximal, characteristic
2C2×C4×D24C211yesmaximal, characteristic
3#76505C211yesmaximal, characteristic, complemented
4#76203C211yesmaximal, characteristic, complemented
5C3×#1933C211yesmaximal, characteristic, complemented
6#70276C211yesmaximal, characteristic, complemented
7#70251C211yesmaximal, characteristic, complemented
8#70245C211yesmaximal, characteristic, complemented
9#69784C211yesmaximal, characteristic, complemented
10#69687C211yesmaximal, characteristic, complemented
11#69626C211yesmaximal, characteristic, complemented
12#68903C211yesmaximal, characteristic, complemented
13#68897C211yesmaximal, characteristic
14#68895C211yesmaximal, characteristic, complemented
15#68838C211yesmaximal, characteristic, complemented
16#26160—31nomaximal, Sylow
17C2×#947C2211yescharacteristic
18C22×#215C2211yescharacteristic
19C22×D24C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#389C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35#4989—21no—
36#4988C2211yescharacteristic
37#4987—21no—
38#4984C2211yescharacteristic
39#4984C2211yescharacteristic
40#4977C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C4×Dic3⋊C4C2211yescharacteristic
46#4819—24no—
47C4×D24—24no—
48#4675—21no—
49#4674—21no—
50#4668—21no—
51#4668—21no—
52#4664—21no—
53D3×#95—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#463—31no—
63C2×C4×D8—31no—
64#2447—31no—
65#2349—31no—
66#2097—31no—
67#2086—31no—
68#2073—31no—
69#2042—31no—
70#1933D311yescharacteristic, complemented
71#1916—31no—
72#1879—31no—
73#1877—31no—
74#1877—31no—
75#1864—31no—
76#1864—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×C4×D3—21no—
83C22×C4×D3—21no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91#947—22no—
92#947—41no—
93C2×#215—22no—
94C2×#215—22no—
95C2×#215—22no—
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⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4D411yescharacteristic
109C2×Dic3⋊C4D411yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C2×D24—22no—
115C2×D24—22no—
116C2×D24D414yes—
117#893—22no—
118#893—22no—
119#893—22no—
120#893—41no—
121C4×D12—22no—
122C4×D12—22no—
123C4×D12—41no—
124C4×D12—41no—
125C3×#103—22no—
126C3×#103—22no—
127C3×C4×C8—22no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131#835—41no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140#821—22no—
141#821—22no—
142#818—22no—
143Dic3⋊C8—22no—
144#809—22no—
145#809—22no—
146#809—22no—
147#808—41no—
148C22×SD16—31no—
149C22×D8—31no—
150C2×#124—31no—
151C2×C4×D4—31no—
152C2×C4×D4—31no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155#463—64no—
156C4×D8—64no—
157C2×#107—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103C2×D311yescharacteristic
162C2×#103C2×D311yescharacteristic
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×C4×C8C2×D311yescharacteristic
169#397—61no—
170#397—61no—
171#391—31no—
172#391C2×D311yescharacteristic
173#390—31no—
174#389—31no—
175#389—31no—
176#389C2×D311yescharacteristic
177#385—61no—
178#385—61no—
179#385—61no—
180#382—31no—
181#381—61no—
182#380—61no—
183C4×C4⋊C4—31no—
184C2×#95—61no—
185#340—31no—
186#340—31no—
187#340C2×D311yescharacteristic
188#340C2×D311yescharacteristic
189#336—31no—
190#336—31no—
191#336—31no—
192#336—31no—
193C23×D3—21no—
194C23×D3—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12C2×D412yes—
223C2×D12C2×D412yes—
224C2×D12#4112yes—
225C2×D12#4112yes—
226C2×C4×D3—24no—
227C2×C4×D3—42no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×C4×D3—42no—
231C2×C4×D3—42no—
232C2×C4×D3—42no—
233C2×C4×D3—81no—
234C2×C4×D3—81no—
235C2×C4×D3—81no—
236C2×C4×D3—81no—
237C2×C4×D3—81no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8C2×D412yes—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—81no—
250C3×C4⋊C4#4112yes—
251C3×C42—21no—
252C3×C42—21no—
253#215—44no—
254(C2×D3)⋊C4—24no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—81no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—81no—
267C4⋊Dic3—81no—
268Dic3⋊C4—22no—
269Dic3⋊C4—41no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272C4×Dic3—21no—
273C4×Dic3—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280D24—28no—
281C22×Q8C22×D311yescharacteristic
282C22×D4—31no—
283C22×D4—31no—
284C23×C4—61no—
285C23×C4—61no—
286C23×C4—61no—
287C23×C4—61no—
288C2×SD16—62no—
289C2×SD16—62no—
290C2×SD16—62no—
291C2×D8—34no—
292C2×D8—62no—
293C2×D8—62no—
294C22×C8—31no—
295C22×C8—31no—
296C22×C8C22×D311yescharacteristic
297C22×C8C22×D311yescharacteristic
298#124—62no—
299#124—121no—
300C4×D4—62no—
301C4×D4—62no—
302C4×D4—62no—
303C4×D4—62no—
304C4×D4—62no—
305C4×D4—121no—
306C4×D4—121no—
307C4×D4—121no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
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—31no—
327C2×C42—31no—
328C2×C42C22×D311yescharacteristic
329#107—62no—
330#106—121no—
331C4⋊C8—62no—
332#103—22no—
333#103—22no—
334#103—121no—
335#102—62no—
336#102—62no—
337#102—62no—
338#102—62no—
339#102—62no—
340C4×C8—22no—
341#95—21no—
342#95—21no—
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—22no—
356C22×D3—22no—
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—41no—
368C3×Q8—41no—
369C3×Q8—41no—
370C3×Q8—41no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—22no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
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—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4#13512yes—
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4H(5,𝔽2)11yescharacteristic
394C2×C3×C42−1+411yescharacteristic
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—21no—
406C2×Dic3—21no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413D12—22no—
414D12—22no—
415D12—22no—
416D12—22no—
417D12—22no—
418D12—41no—
419D12—41no—
420D12#12314yes—
421C4×D3—84no—
422C4×D3—84no—
423C4×D3—84no—
424C4×D3—84no—
425C3×C8—22no—
426C3×C8—41no—
427C3⋊C8—41no—
428C3⋊C8—41no—
429C24—61no—
430C24—61no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×D4—32no—
438C2×D4—32no—
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—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—64no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—122no—
461C22×C4—122no—
462C22×C4—122no—
463C22×C4—122no—
464C22×C4—122no—
465C22×C4—122no—
466C22×C4—241no—
467C22×C4—241no—
468C22×C4—241no—
469C22×C4—241no—
470C22×C4—241no—
471C22×C4D3×D411yescharacteristic
472C22×C4D3×D411yescharacteristic
473C22×C4C23×D311yescharacteristic
474SD16—124no—
475D8—68no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8D3×D412yes—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—62no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—81no—
499C4⋊C4—121no—
500C4⋊C4—241no—
501C4⋊C4—241no—
502C4⋊C4#23512yes—
503C22⋊C4—64no—
504C22⋊C4—122no—
505C22⋊C4—122no—
506C22⋊C4—122no—
507C22⋊C4—122no—
508C22⋊C4—122no—
509C22⋊C4—122no—
510C22⋊C4—241no—
511C42—21no—
512C42—21no—
513C42—61no—
514C42—61no—
515C42—61no—
516C42—61no—
517C22×C3#52511yescharacteristic
518C22×C3D4211yescharacteristic
519C22×C3#54811yescharacteristic
520C22×C3C2×#13511yescharacteristic
521C22×C3#57511yescharacteristic
522C22×C3#57611yescharacteristic
523C22×C3#57711yescharacteristic
524C2×D3—24no—
525C2×D3—42no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3—42no—
530C2×D3—42no—
531C2×D3—42no—
532C2×D3—42no—
533C2×D3—42no—
534C2×D3—81no—
535C2×D3—81no—
536C2×D3—81no—
537C2×D3—81no—
538C3×C4—22no—
539C3×C4—42no—
540C3×C4—41no—
541C3×C4—81no—
542C3×C4—81no—
543C3×C4—81no—
544C3×C4C2×#12311yescharacteristic
545C3×C4#53911yescharacteristic
546C3×C4#54711yescharacteristic
547C3×C4#55311yescharacteristic
548Dic3—41no—
549Dic3—41no—
550Dic3—41no—
551Dic3—41no—
552Dic3—81no—
553Dic3—81no—
554C23—62no—
555C23—62no—
556C23—62no—
557C23—62no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23C2×D3×D411yescharacteristic
569Q8—41no—
570Q8—41no—
571Q8—41no—
572Q8—41no—
573D4—34no—
574D4—62no—
575D4—62no—
576D4—62no—
577D4—62no—
578D4—62no—
579D4—121no—
580D4—121no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
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—41no—
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—121no—
616C2×C4—244no—
617C2×C4—244no—
618C2×C4—244no—
619C2×C4—244no—
620C2×C4#91912yes—
621C2×C4C2×#23511yescharacteristic
622C2×C4C2×D3×D411yescharacteristic
623C2×C4D3×#4111yescharacteristic
624C2×C4D3×#4111yescharacteristic
625C2×C4#100911yescharacteristic
626C2×C4#101011yescharacteristic
627C8—22no—
628C8—41no—
629C8—121no—
630C8—121no—
631C2×C3#244311yescharacteristic
632C2×C3#310211yescharacteristic
633C2×C3#311811yescharacteristic
634C2×C3#326711yescharacteristic
635C2×C3#328011yescharacteristic
636C2×C3#336011yescharacteristic
637C2×C3#337011yescharacteristic
638D3—44no—
639D3—82no—
640C22—64no—
641C22—122no—
642C22—122no—
643C22—122no—
644C22—122no—
645C22—122no—
646C22—122no—
647C22—122no—
648C22—122no—
649C22—122no—
650C22—241no—
651C22—241no—
652C22—241no—
653C22—241no—
654C22#554711yescharacteristic
655C22#560911yescharacteristic
656C22#567711yescharacteristic
657C22#576211yescharacteristic
658C22C2×#91911yescharacteristic
659C22#576811yescharacteristic
660C22#576911yescharacteristic
661C4—22no—
662C4—41no—
663C4—42no—
664C4—81no—
665C4—81no—
666C4—81no—
667C4—121no—
668C4—121no—
669C4—121no—
670C4—121no—
671C4—241no—
672C4—241no—
673C4#553011yescharacteristic
674C4#555111yescharacteristic
675C4D3×#12311yescharacteristic
676C4#570711yescharacteristic
677C3#2616011yescharacteristic, complemented, Sylow
678C2—124no—
679C2—242no—
680C2#7584811yescharacteristic
681C2#7969611yescharacteristic
682C2#7974611yescharacteristic
683C2#8067511yescharacteristic
684C2#8077311yescharacteristic
685C2#8162111yescharacteristic
686C2#8166011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1933C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1933D3
hypocenterC3#26160
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1933

Complements

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

NH ≅ G/Ntype
3. #76505C2semidirect
4. #76203C2semidirect
5. C3×#1933C2semidirect
6. #70276C2semidirect
7. #70251C2semidirect
8. #70245C2semidirect
9. #69784C2semidirect
10. #69687C2semidirect
11. #69626C2semidirect
12. #68903C2semidirect
14. #68895C2semidirect
15. #68838C2semidirect
70. #1933D3semidirect
677. C3#26160semidirect

Maximal subgroups

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

subgroupindex
C2×#48192
C2×C4×D242
#765052
#762032
C3×#19332
#702762
#702512
#702452
#697842
#696872
#696262
#689032
#688972
#688952
#688382
#261603