Finite Group Databasebeta

← #10868101

Subgroups — #10868101

Order 768.

806 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82399C211yesmaximal, characteristic
2#76237C211yesmaximal, characteristic, complemented
3C3×#1864C211yesmaximal, characteristic, complemented
4#71609C211yesmaximal, characteristic, complemented
5#70579C211yesmaximal, characteristic, complemented
6#70558C211yesmaximal, characteristic, complemented
7#70206C211yesmaximal, characteristic
8#70193C211yesmaximal, characteristic, complemented
9#70175C211yesmaximal, characteristic
10#70165C211yesmaximal, characteristic, complemented
11#69620C211yesmaximal, characteristic, complemented
12#68872C211yesmaximal, characteristic, complemented
13#68795C211yesmaximal, characteristic, complemented
14#68658C211yesmaximal, characteristic, complemented
15#68624C211yesmaximal, characteristic, complemented
16#27047—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#938C2211yescharacteristic
19C2×C4×#67—21no—
20C2×#898—21no—
21C2×D3×C4⋊C4C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C3×#389C2211yescharacteristic
26#5221C2211yescharacteristic
27C2×#832C2211yescharacteristic
28C2×#832C2211yescharacteristic
29#5210—21no—
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×#818C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34#4987C2211yescharacteristic
35#4987C2211yescharacteristic
36#4984C2211yescharacteristic
37#4978—21no—
38#4975C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4956—21no—
43#4945—21no—
44C4×(C2×D3)⋊C4C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46#4928C2211yescharacteristic
47C2×#803C2211yescharacteristic
48#4675—21no—
49#4660—21no—
50C3×#340C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2496—31no—
61#2361—31no—
62#2071—31no—
63#2065—31no—
64#2033—31no—
65#2033—31no—
66#1947—31no—
67#1943—31no—
68#1924—31no—
69#1921—31no—
70#1920—31no—
71#1864D311yescharacteristic, complemented
72#1853—31no—
73#1824—31no—
74#1762—31no—
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C22×#67—21no—
78C22×C4×D3—21no—
79C22×C4×D3C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4×D4—22no—
83C3×C4×D4—41no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—21no—
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93#938—22no—
94#938—22no—
95C4×#67—44no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4D411yescharacteristic
99C2×(C2×D3)⋊C4D411yescharacteristic
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114#898—44no—
115D3×C4⋊C4—22no—
116D3×C4⋊C4—41no—
117C3×#106—22no—
118C3×#102—22no—
119C3×#102—41no—
120C3×#95—21no—
121#832—22no—
122#832—22no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95D411yescharacteristic
130C3⋊#95D411yescharacteristic
131(C2×D3)⋊C8—22no—
132(C2×D3)⋊C8—41no—
133#818—22no—
134Dic3⋊C8—22no—
135#808—41no—
136#807—22no—
137#803—41no—
138C2×#126—61no—
139C2×#121—31no—
140C2×C4×D4—61no—
141C2×C4×D4C2×D311yescharacteristic
142C22×C4⋊C4—31no—
143C2×#107—31no—
144C2×#107—31no—
145C2×#106—31no—
146C2×#106C2×D311yescharacteristic
147C2×C4⋊C8—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×#102C2×D311yescharacteristic
151C2×#102C2×D311yescharacteristic
152C2×C22⋊C8—31no—
153C2×C22⋊C8—31no—
154C2×#97—31no—
155#400—61no—
156#397—61no—
157#390—31no—
158#389—31no—
159#389—31no—
160#389C2×D311yescharacteristic
161#388—61no—
162#387—61no—
163#386—61no—
164#385—61no—
165#384—31no—
166#380—31no—
167#380—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—31no—
170C4×C22⋊C4—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340C2×D311yescharacteristic
179#336—31no—
180#336C2×D311yescharacteristic
181C24×C3—21no—
182C23×D3C2×D411yescharacteristic
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4#4112yes—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—42no—
193C22×C3×C4—42no—
194C22×C3×C4—81no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2411yescharacteristic
198C2×#67—24no—
199C2×#67—42no—
200C2×#67—42no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—21no—
214C2×C4×D3—42no—
215C2×C4×D3—42no—
216C2×C4×D3—81no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—42no—
229C3×C4⋊C4#4112yes—
230C3×C22⋊C4—42no—
231C3×C42—21no—
232C3×C42—21no—
233#217—24no—
234#217—42no—
235#217—81no—
236(C2×D3)⋊C4—22no—
237(C2×D3)⋊C4—22no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4—42no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—42no—
248C4⋊Dic3#4112yes—
249Dic3⋊C4—42no—
250Dic3⋊C4—42no—
251Dic3⋊C4—42no—
252Dic3⋊C4—42no—
253C4×Dic3—21no—
254C4×Dic3—21no—
255C4×Dic3—42no—
256C4×Dic3—42no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8#4112yes—
263C22×D4—61no—
264C22×D4C22×D311yescharacteristic
265C23×C4—21no—
266C23×C4—31no—
267C23×C4—61no—
268C22×C8—31no—
269C22×C8—31no—
270C22×C8C22×D311yescharacteristic
271C22×C8C22×D311yescharacteristic
272#126—124no—
273#121—62no—
274#121—62no—
275C4×D4—22no—
276C4×D4—41no—
277C4×D4—124no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—121no—
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C22⋊C4—21no—
294C2×C22⋊C4—31no—
295C2×C22⋊C4—31no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C42—31no—
302C2×C42—61no—
303C2×C42—61no—
304C2×C42C22×D311yescharacteristic
305#107—62no—
306#107—62no—
307#106—22no—
308#106—121no—
309C4⋊C8—62no—
310#102—22no—
311#102—41no—
312#102—62no—
313#102—62no—
314C22⋊C8—62no—
315C22⋊C8—121no—
316#97—121no—
317#95—21no—
318#95—31no—
319#95—31no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326C23×C3—22no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3C22×D411yescharacteristic
334C22×D3—21no—
335C22×D3—21no—
336C22×D3—21no—
337C22×D3—21no—
338C22×D3—21no—
339C22×D3—21no—
340C22×D3AGL(1,ℤ/8)12yes—
341C3×D4—22no—
342C3×D4—22no—
343C3×D4—41no—
344C3×D4—41no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—84no—
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4H(5,𝔽2)11yescharacteristic
366C2×C3×C42−1+411yescharacteristic
367#67—48no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—22no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3AGL(1,ℤ/8)12yes—
389C4×D3—41no—
390C4×D3—41no—
391C4×D3—41no—
392C4×D3—41no—
393C4×D3—84no—
394C3×C8—41no—
395C3×C8—41no—
396C3⋊C8—22no—
397C3⋊C8—41no—
398C24—21no—
399C24—31no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—64no—
406C2×D4—122no—
407C2×D4—122no—
408C2×D4#23512yes—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—81no—
428C22×C4—122no—
429C22×C4—122no—
430C22×C4—241no—
431C22×C4D3×D411yescharacteristic
432C22×C4D3×D411yescharacteristic
433C22×C4C23×D311yescharacteristic
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—32no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—32no—
452C4⋊C4—42no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—122no—
461C4⋊C4—122no—
462C4⋊C4—122no—
463C4⋊C4—122no—
464C4⋊C4—122no—
465C4⋊C4#23712yes—
466C22⋊C4—42no—
467C22⋊C4—62no—
468C22⋊C4—62no—
469C22⋊C4—64no—
470C22⋊C4—122no—
471C22⋊C4—122no—
472C22⋊C4—122no—
473C22⋊C4—241no—
474C42—21no—
475C42—21no—
476C42—61no—
477C42—61no—
478C42—122no—
479C42—122no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3#52511yescharacteristic
488C22×C3#54611yescharacteristic
489C22×C3#54711yescharacteristic
490C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
491C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
492C22×C3#57711yescharacteristic
493C22×C3#57811yescharacteristic
494C2×D3—22no—
495C2×D3—22no—
496C2×D3—22no—
497C2×D3—41no—
498C2×D3—41no—
499C2×D3—41no—
500C2×D3—41no—
501C3×C4—41no—
502C3×C4—42no—
503C3×C4—41no—
504C3×C4—81no—
505C3×C4—81no—
506C3×C4#52911yescharacteristic
507C3×C4#53911yescharacteristic
508C3×C4#55311yescharacteristic
509C3×C4#55611yescharacteristic
510Dic3—22no—
511Dic3—41no—
512Dic3—42no—
513Dic3—41no—
514Dic3—81no—
515Dic3—81no—
516Dic3—81no—
517C23—22no—
518C23—32no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—61no—
526C23—61no—
527C23—61no—
528C23—61no—
529C23—61no—
530C23—61no—
531C23C2×D3×D411yescharacteristic
532D4—22no—
533D4—22no—
534D4—41no—
535D4—41no—
536D4—128no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—32no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—62no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—84no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—244no—
579C2×C4C2×#23511yescharacteristic
580C2×C4C2×#23711yescharacteristic
581C2×C4#100411yescharacteristic
582C2×C4#100711yescharacteristic
583C2×C4D3×#4111yescharacteristic
584C2×C4D3×#4111yescharacteristic
585C8—41no—
586C8—41no—
587C8—62no—
588C8—121no—
589C2×C3—82no—
590C2×C3#246611yescharacteristic
591C2×C3#310911yescharacteristic
592C2×C3#312011yescharacteristic
593C2×C3#330811yescharacteristic
594C2×C3#332411yescharacteristic
595C2×C3#334311yescharacteristic
596C2×C3#334511yescharacteristic
597D3—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—62no—
602C22—62no—
603C22—62no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—121no—
609C22—121no—
610C22—121no—
611C22—121no—
612C22#554711yescharacteristic
613C22#559111yescharacteristic
614C22#563111yescharacteristic
615C22#576211yescharacteristic
616C22#577111yescharacteristic
617C22D3×AGL(1,ℤ/8)11yescharacteristic
618C22D3×AGL(1,ℤ/8)11yescharacteristic
619C4—41no—
620C4—42no—
621C4—41no—
622C4—62no—
623C4—81no—
624C4—81no—
625C4—121no—
626C4—121no—
627C4—122no—
628C4—241no—
629C4—241no—
630C4—241no—
631C4#551511yescharacteristic
632C4#553811yescharacteristic
633C4#564111yescharacteristic
634C4#566111yescharacteristic
635C3#2704711yescharacteristic, complemented, Sylow
636C2—82no—
637C2—122no—
638C2#7443411yescharacteristic
639C2#7965711yescharacteristic
640C2#7986911yescharacteristic
641C2#8010911yescharacteristic
642C2#8016511yescharacteristic
643C2#8108311yescharacteristic
644C2#8119611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1864C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#1864D3
hypocenterC3#27047
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #76237C2semidirect
3. C3×#1864C2semidirect
4. #71609C2semidirect
5. #70579C2semidirect
6. #70558C2semidirect
8. #70193C2semidirect
10. #70165C2semidirect
11. #69620C2semidirect
12. #68872C2semidirect
13. #68795C2semidirect
14. #68658C2semidirect
15. #68624C2semidirect
71. #1864D3semidirect
635. C3#27047semidirect

Maximal subgroups

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

subgroupindex
#823992
#762372
C3×#18642
#716092
#705792
#705582
#702062
#701932
#701752
#701652
#696202
#688722
#687952
#686582
#686242
#270473