Finite Group Databasebeta

← #10855165

Subgroups — #10855165

Order 768.

1022 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4872C211yesmaximal, characteristic, complemented
2C2×#4843C211yesmaximal, characteristic
3#76202C211yesmaximal, characteristic, complemented
4#76004C211yesmaximal, characteristic, complemented
5C3×#2086C211yesmaximal, characteristic, complemented
6#70641C211yesmaximal, characteristic, complemented
7#70447C211yesmaximal, characteristic, complemented
8#69627C211yesmaximal, characteristic, complemented
9#69625C211yesmaximal, characteristic, complemented
10#68902C211yesmaximal, characteristic, complemented
11#68897C211yesmaximal, characteristic, complemented
12#68897C211yesmaximal, characteristic, complemented
13#68897C211yesmaximal, characteristic, complemented
14#68894C211yesmaximal, characteristic, complemented
15#68853C211yesmaximal, characteristic, complemented
16#26158—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C22×D24C2211yescharacteristic
19C2×#894—21no—
20C2×#893C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#893C2211yescharacteristic
23C2×D3×C4⋊C4—21no—
24C2×#874C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C2×C3×#107C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C3×#389C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×C8×Dic3C2211yescharacteristic
33#4991—21no—
34#4989—21no—
35#4988C2211yescharacteristic
36#4988C2211yescharacteristic
37#4988C2211yescharacteristic
38#4984C2211yescharacteristic
39#4975C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C4×Dic3⋊C4C2211yescharacteristic
48#4872—24no—
49#4843—24no—
50#4672—21no—
51#4668—21no—
52#4667—21no—
53#4665—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#459—31no—
63C2×C4×D8—31no—
64#2445—31no—
65D4×C4⋊C4—31no—
66#2093—31no—
67#2086D311yescharacteristic, complemented
68#2071—31no—
69#2041—31no—
70#1934—31no—
71#1916—31no—
72#1879—31no—
73#1879—31no—
74#1879—31no—
75#1862—31no—
76#1862—31no—
77C22×D12—21no—
78C22×D12—21no—
79C22×D12C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C22×C4×D3—21no—
84C22×C4×D3—21no—
85C22×C4×D3—21no—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
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×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4D411yescharacteristic
110C2×Dic3⋊C4D411yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C2×D24—22no—
116C2×D24—22no—
117C2×D24—22no—
118C2×D24—22no—
119C2×D24—22no—
120C2×D24D414yes—
121#894—44no—
122#893—22no—
123#893—22no—
124#893—22no—
125#893—22no—
126#893—41no—
127#893—41no—
128D3×C4⋊C4—44no—
129#874—22no—
130#874—41no—
131C4×D12—22no—
132C4×D12—41no—
133C3×#107—22no—
134C3×C4⋊C8—22no—
135C3×#95—21no—
136C3×#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140#821—22no—
141#821—22no—
142C8×Dic3—22no—
143#809—22no—
144#809—22no—
145#809—22no—
146#809—22no—
147#809—22no—
148#809—41no—
149#807—41no—
150C22×D8—31no—
151C22×D8—31no—
152C2×#127—31no—
153C2×#123—61no—
154C2×C4×D4—31no—
155C2×C4×D4—31no—
156C2×C4×D4—31no—
157C2×C4×D4—31no—
158C22×C4⋊C4—61no—
159#459—64no—
160C4×D8—64no—
161C2×#107—31no—
162C2×#107C2×D311yescharacteristic
163C2×C4⋊C8C2×D311yescharacteristic
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102—31no—
171C2×#102—31no—
172C2×C4×C8—31no—
173#394—61no—
174#394—61no—
175#390—31no—
176#390—31no—
177#390—31no—
178#389—31no—
179#389C2×D311yescharacteristic
180#389C2×D311yescharacteristic
181#386—61no—
182#385—61no—
183#385—61no—
184#384—31no—
185#380—61no—
186C4×C4⋊C4—31no—
187#340—31no—
188#340C2×D311yescharacteristic
189#340C2×D311yescharacteristic
190#340C2×D311yescharacteristic
191#336—31no—
192#336—31no—
193#336—31no—
194#336—31no—
195C23×D3—21no—
196C23×D3—21no—
197C23×D3—21no—
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×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
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×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—42no—
225C2×D12—42no—
226C2×D12—42no—
227C2×D12—42no—
228C2×D12—81no—
229C2×D12—81no—
230C2×D12C2×D412yes—
231C2×D12C2×D412yes—
232C2×D12#4112yes—
233C2×D12#4112yes—
234C2×C4×D3—24no—
235C2×C4×D3—42no—
236C2×C4×D3—42no—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C2×C4×D3—42no—
240C2×C4×D3—42no—
241C2×C4×D3—42no—
242C2×C4×D3—42no—
243C2×C4×D3—81no—
244C2×C4×D3—81no—
245C2×C4×D3—81no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8C2×D412yes—
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—21no—
259C3×C4⋊C4—42no—
260C3×C4⋊C4—42no—
261C3×C4⋊C4#4112yes—
262C3×C42—21no—
263C3×C42—21no—
264(C2×D3)⋊C4—24no—
265(C2×D3)⋊C4—42no—
266(C2×D3)⋊C4—42no—
267(C2×D3)⋊C4—42no—
268(C2×D3)⋊C4—42no—
269(C2×D3)⋊C4—42no—
270(C2×D3)⋊C4—42no—
271(C2×D3)⋊C4—81no—
272(C2×D3)⋊C4—81no—
273(C2×D3)⋊C4—81no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—42no—
277Dic3⋊C4—41no—
278Dic3⋊C4—42no—
279Dic3⋊C4—81no—
280Dic3⋊C4D814yescomplemented
281C4×Dic3—21no—
282C4×Dic3—21no—
283C4×Dic3—21no—
284C4×Dic3—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291D24—28no—
292D24—44no—
293C22×D4—31no—
294C22×D4—31no—
295C22×D4—31no—
296C22×D4—61no—
297C22×D4—61no—
298C23×C4—61no—
299C23×C4—61no—
300C23×C4—61no—
301C23×C4—61no—
302C2×D8—34no—
303C2×D8—62no—
304C2×D8—62no—
305C2×D8—62no—
306C2×D8—62no—
307C2×D8—62no—
308C22×C8—31no—
309C22×C8—31no—
310C22×C8C22×D311yescharacteristic
311C22×C8C22×D311yescharacteristic
312#127—62no—
313#127—121no—
314#123—124no—
315C4×D4—62no—
316C4×D4—62no—
317C4×D4—62no—
318C4×D4—62no—
319C4×D4—62no—
320C4×D4—121no—
321C4×D4—121no—
322C4×D4—121no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—124no—
332C2×C4⋊C4C22×D311yescharacteristic
333C2×C4⋊C4C22×D311yescharacteristic
334C2×C4⋊C4C22×D311yescharacteristic
335C2×C4⋊C4C22×D311yescharacteristic
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C22⋊C4—61no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C22⋊C4—61no—
342C2×C22⋊C4—61no—
343C2×C22⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C22⋊C4—61no—
346C2×C42—31no—
347C2×C42—31no—
348C2×C42C22×D311yescharacteristic
349#107—22no—
350#107—121no—
351C4⋊C8—22no—
352#102—62no—
353#102—62no—
354#102—62no—
355#102—62no—
356#102—62no—
357#102—62no—
358#102—62no—
359#102—121no—
360C4×C8—62no—
361#95—21no—
362#95—21no—
363#95—31no—
364#95—31no—
365#95—61no—
366C23×C3C22×D411yescharacteristic
367C22×D3—22no—
368C22×D3—22no—
369C22×D3—22no—
370C22×D3—22no—
371C22×D3—41no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—41no—
381C22×D3—41no—
382C22×D3—41no—
383C22×D3—41no—
384C22×D3—41no—
385C22×D3—41no—
386C22×D3—41no—
387C22×D3—41no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—22no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4C2×D812yes—
406C2×C3×C4C22×D411yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4H(5,𝔽2)11yescharacteristic
411C2×C3×C42−1+411yescharacteristic
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3C2×D812yes—
426C2×Dic3C2×D812yes—
427D12—22no—
428D12—22no—
429D12—22no—
430D12—22no—
431D12—22no—
432D12—41no—
433D12—41no—
434D12—41no—
435D12—41no—
436D12—41no—
437D12—41no—
438D12—84no—
439D12—84no—
440D12#12314yescomplemented
441C4×D3—48no—
442C4×D3—84no—
443C4×D3—84no—
444C4×D3—84no—
445C3×C8—22no—
446C3×C8—41no—
447C3⋊C8—41no—
448C3⋊C8—41no—
449C24—61no—
450C24—61no—
451C24—61no—
452C2×D4—32no—
453C2×D4—32no—
454C2×D4—32no—
455C2×D4—32no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—61no—
459C2×D4—61no—
460C2×D4—61no—
461C2×D4—61no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—61no—
465C2×D4—61no—
466C2×D4—61no—
467C2×D4—61no—
468C2×D4—61no—
469C2×D4—61no—
470C2×D4—122no—
471C2×D4—122no—
472C2×D4—122no—
473C2×D4—122no—
474C2×D4—241no—
475C2×D4—241no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—31no—
481C22×C4—31no—
482C22×C4—31no—
483C22×C4—31no—
484C22×C4—64no—
485C22×C4—61no—
486C22×C4—122no—
487C22×C4—122no—
488C22×C4—122no—
489C22×C4—122no—
490C22×C4—122no—
491C22×C4—122no—
492C22×C4—122no—
493C22×C4—122no—
494C22×C4—241no—
495C22×C4—241no—
496C22×C4—241no—
497C22×C4D3×D411yescharacteristic
498C22×C4D3×D411yescharacteristic
499C22×C4C23×D311yescharacteristic
500D8—68no—
501D8—124no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—61no—
508C2×C8—61no—
509C2×C8—61no—
510C2×C8—61no—
511C2×C8—61no—
512C2×C8—61no—
513C2×C8D3×D412yes—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—34no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—121no—
527C4⋊C4—122no—
528C4⋊C4—122no—
529C4⋊C4—241no—
530C4⋊C4#23912yes—
531C22⋊C4—64no—
532C22⋊C4—122no—
533C22⋊C4—122no—
534C22⋊C4—122no—
535C22⋊C4—122no—
536C22⋊C4—122no—
537C22⋊C4—122no—
538C22⋊C4—241no—
539C22⋊C4—241no—
540C22⋊C4—241no—
541C42—21no—
542C42—21no—
543C42—61no—
544C42—61no—
545C42—61no—
546C42—61no—
547C22×C3#52511yescharacteristic
548C22×C3D4211yescharacteristic
549C22×C3#54811yescharacteristic
550C22×C3C22×D811yescharacteristic
551C22×C3#57511yescharacteristic
552C22×C3#57611yescharacteristic
553C22×C3#57711yescharacteristic
554C2×D3—24no—
555C2×D3—42no—
556C2×D3—42no—
557C2×D3—42no—
558C2×D3—42no—
559C2×D3—42no—
560C2×D3—42no—
561C2×D3—42no—
562C2×D3—42no—
563C2×D3—42no—
564C2×D3—81no—
565C2×D3—81no—
566C2×D3—81no—
567C2×D3—81no—
568C2×D3—81no—
569C2×D3—81no—
570C2×D3—81no—
571C2×D3—81no—
572C2×D3—81no—
573C2×D3—81no—
574C2×D3—81no—
575C3×C4—22no—
576C3×C4—42no—
577C3×C4—41no—
578C3×C4—81no—
579C3×C4—81no—
580C3×C4—81no—
581C3×C4C2×#12311yescharacteristic
582C3×C4#53911yescharacteristic
583C3×C4#54711yescharacteristic
584C3×C4#55311yescharacteristic
585Dic3—22no—
586Dic3—22no—
587Dic3—41no—
588Dic3—41no—
589Dic3—81no—
590C23—62no—
591C23—62no—
592C23—62no—
593C23—62no—
594C23—121no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—121no—
599C23—121no—
600C23—121no—
601C23—121no—
602C23—121no—
603C23—121no—
604C23—121no—
605C23—121no—
606C23—121no—
607C23—121no—
608C23—121no—
609C23—121no—
610C23—121no—
611C23C2×D3×D411yescharacteristic
612D4—34no—
613D4—62no—
614D4—62no—
615D4—62no—
616D4—62no—
617D4—62no—
618D4—121no—
619D4—121no—
620D4—121no—
621D4—121no—
622D4—121no—
623D4—121no—
624D4—244no—
625D4—244no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—22no—
632C2×C4—32no—
633C2×C4—32no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—61no—
646C2×C4—61no—
647C2×C4—61no—
648C2×C4—61no—
649C2×C4—61no—
650C2×C4—61no—
651C2×C4—61no—
652C2×C4—61no—
653C2×C4—61no—
654C2×C4—61no—
655C2×C4—128no—
656C2×C4—121no—
657C2×C4—121no—
658C2×C4—121no—
659C2×C4—244no—
660C2×C4—244no—
661C2×C4—244no—
662C2×C4D3×D812yes—
663C2×C4C2×D3×D411yescharacteristic
664C2×C4C2×#23911yescharacteristic
665C2×C4#100711yescharacteristic
666C2×C4D3×#4111yescharacteristic
667C2×C4D3×#4111yescharacteristic
668C2×C4#100911yescharacteristic
669C8—22no—
670C8—41no—
671C8—121no—
672C8—121no—
673C2×C3#244311yescharacteristic
674C2×C3#310011yescharacteristic
675C2×C3#311911yescharacteristic
676C2×C3#326711yescharacteristic
677C2×C3D4×D811yescharacteristic
678C2×C3#335811yescharacteristic
679C2×C3#337011yescharacteristic
680D3—44no—
681D3—82no—
682D3—161no—
683C22—64no—
684C22—122no—
685C22—122no—
686C22—122no—
687C22—122no—
688C22—122no—
689C22—122no—
690C22—122no—
691C22—122no—
692C22—122no—
693C22—241no—
694C22—241no—
695C22—241no—
696C22—241no—
697C22—241no—
698C22—241no—
699C22—241no—
700C22—241no—
701C22—241no—
702C22—241no—
703C22—241no—
704C22#560911yescharacteristic
705C22#562711yescharacteristic
706C22#563311yescharacteristic
707C22C2×D3×D811yescharacteristic
708C22#576911yescharacteristic
709C22#577711yescharacteristic
710C22#577811yescharacteristic
711C4—22no—
712C4—41no—
713C4—42no—
714C4—62no—
715C4—62no—
716C4—81no—
717C4—81no—
718C4—81no—
719C4—121no—
720C4—121no—
721C4—241no—
722C4D3×#12311yescharacteristic
723C4#569211yescharacteristic
724C4#569511yescharacteristic
725C4#572611yescharacteristic
726C3#2615811yescharacteristic, complemented, Sylow
727C2—124no—
728C2—242no—
729C2—481no—
730C2#7795111yescharacteristic
731C2#8072911yescharacteristic
732C2#8092111yescharacteristic
733C2#8094811yescharacteristic
734C2#8099211yescharacteristic
735C2#8120811yescharacteristic
736C2#8121711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2086C2
socleC23×C3C22×D4
prime radicalC2×#4843C2
hypercenter#2086D3
hypocenterC3#26158
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2086

Complements

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

NH ≅ G/Ntype
1. C2×#4872C2semidirect
3. #76202C2semidirect
4. #76004C2semidirect
5. C3×#2086C2semidirect
6. #70641C2semidirect
7. #70447C2semidirect
8. #69627C2semidirect
9. #69625C2semidirect
10. #68902C2semidirect
11. #68897C2semidirect
12. #68897C2semidirect
13. #68897C2semidirect
14. #68894C2semidirect
15. #68853C2semidirect
67. #2086D3semidirect
280. Dic3⋊C4D8semidirect
440. D12#123semidirect
726. C3#26158semidirect

Maximal subgroups

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

subgroupindex
C2×#48722
C2×#48432
#762022
#760042
C3×#20862
#706412
#704472
#696272
#696252
#689022
#688972
#688942
#688532
#261583