Finite Group Databasebeta

← #10858764

Subgroups — #10858764

Order 768.

902 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2440C211yesmaximal, characteristic, complemented
2#73467C211yesmaximal, characteristic
3#71792C211yesmaximal, characteristic, complemented
4#71779C211yesmaximal, characteristic, complemented
5#71560C211yesmaximal, characteristic
6C2×#4529C211yesmaximal, characteristic, complemented
7#69629C211yesmaximal, characteristic, complemented
8#69626C211yesmaximal, characteristic, complemented
9#69610C211yesmaximal, characteristic, complemented
10#69602C211yesmaximal, characteristic, complemented
11#69584C211yesmaximal, characteristic, complemented
12#69572C211yesmaximal, characteristic, complemented
13#69515C211yesmaximal, characteristic
14#69505C211yesmaximal, characteristic, complemented
15#69449C211yesmaximal, characteristic, complemented
16#26410—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C22⋊Q8—21no—
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×#930C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C23×C3⋊C8C2211yescharacteristic
23C2×#67⋊C4—21no—
24C2×C4×D12C2211yescharacteristic
25C3×#398—21no—
26C3×#395—21no—
27C3×#389C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C3×#385—21no—
30C3×#384C2211yescharacteristic
31C3×#380C2211yescharacteristic
32C3×C4×C22⋊C4C2211yescharacteristic
33C2×#830C2211yescharacteristic
34C2×#809C2211yescharacteristic
35C2×#809C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#809C2211yescharacteristic
38C2×#808C2211yescharacteristic
39C2×#808C2211yescharacteristic
40#4958—21no—
41#4939C2211yescharacteristic
42C4×C4⋊Dic3C2211yescharacteristic
43#4928C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4670—21no—
47#4668—21no—
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2440D311yescharacteristic, complemented
61#2361—31no—
62#2100—31no—
63#2088—31no—
64#2065—31no—
65#2023—31no—
66#1987—31no—
67#1920—31no—
68#1893—31no—
69#1874—31no—
70#1864—31no—
71#1853—31no—
72#1850—31no—
73#1802—31no—
74C2×#340—31no—
75C22×C3×Q8—21no—
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C22×#67—21no—
79C23×Dic3—21no—
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C3×C4⋊Q8—22no—
83C3×C4⋊Q8—41no—
84C3×C22⋊Q8—44no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—41no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C22⋊C4D411yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102#930—22no—
103#930Q814yescomplemented
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3—22no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—41no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C22×C3⋊C8—22no—
117C22×C3⋊C8—41no—
118C22×C3⋊C8—41no—
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#67⋊C4—44no—
124C4×D12—22no—
125C4×D12—41no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95D411yescharacteristic
130C3×#95D411yescharacteristic
131C3⋊#95—21no—
132C3⋊#95—21no—
133#830—22no—
134#809—22no—
135#809—41no—
136#809—41no—
137#809—41no—
138#808—22no—
139#808—41no—
140C4⋊(C3⋊C8)—41no—
141C4⋊(C3⋊C8)—41no—
142C23×C8—31no—
143C2×C4⋊Q8C2×D311yescharacteristic
144C2×C22⋊Q8—21no—
145C2×#121—31no—
146C2×C4×D4—31no—
147C2×C4×D4—61no—
148C22×C4⋊C4—31no—
149C22×C4⋊C4C2×D311yescharacteristic
150C2×#106—31no—
151C2×#106—31no—
152C2×C4⋊C8—31no—
153C2×C4⋊C8—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×C22⋊C8—31no—
159#398—21no—
160#395—21no—
161#390—31no—
162#389—31no—
163#389C2×D311yescharacteristic
164#389C2×D311yescharacteristic
165#388—61no—
166#386—61no—
167#385—21no—
168#385—61no—
169#384C2×D311yescharacteristic
170#380C2×D311yescharacteristic
171C4×C4⋊C4—31no—
172C4×C22⋊C4C2×D311yescharacteristic
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#340—31no—
183#336—31no—
184#336—31no—
185C24×C3C2×D411yescharacteristic
186C23×D3—21no—
187C2×C3×Q8—42no—
188C2×C3×Q8—42no—
189C2×C3×Q8—81no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—42no—
199C22×C3×C4—42no—
200C22×C3×C4—81no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×Q812yes—
207C22×C3×C4#4112yes—
208C22×C3×C4C2411yescharacteristic
209C2×#67—24no—
210C2×#67—42no—
211C2×#67—42no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—42no—
216C22×Dic3—42no—
217C22×Dic3—81no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12C2×Q812yes—
224C2×C4×D3—42no—
225C2×C4×D3—42no—
226C2×C4×D3—81no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—81no—
240C3×C4⋊C4#4112yes—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—42no—
243C3×C22⋊C4SD1614yes—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—21no—
248(C2×D3)⋊C4—24no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—81no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—42no—
257C4⋊Dic3C2×Q812yes—
258C4⋊Dic3#4112yes—
259Dic3⋊C4—42no—
260C4×Dic3—42no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—44no—
274C22×Q8—21no—
275C22×D4—31no—
276C22×D4—61no—
277C23×C4—21no—
278C23×C4—61no—
279C23×C4—61no—
280C23×C4C22×D311yescharacteristic
281C22×C8—31no—
282C22×C8—31no—
283C22×C8—31no—
284C22×C8—31no—
285C22×C8—62no—
286C22×C8—121no—
287C22×C8—121no—
288C4⋊Q8—22no—
289C4⋊Q8—41no—
290C22⋊Q8—44no—
291#121—34no—
292#121—62no—
293C4×D4—62no—
294C4×D4—121no—
295C4×D4—124no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—41no—
307C2×C4⋊C4—62no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—121no—
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4#6711yescharacteristic
321C2×C22⋊C4#6711yescharacteristic
322C2×C42—61no—
323C2×C42C22×D311yescharacteristic
324C2×C42C22×D311yescharacteristic
325#106—62no—
326#106—121no—
327C4⋊C8—121no—
328C4⋊C8—121no—
329#102—62no—
330#102—121no—
331#102—121no—
332#102—121no—
333C22⋊C8—62no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—61no—
338#95—61no—
339#95#6711yescharacteristic
340#95#6711yescharacteristic
341C23×C3—21no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—21no—
345C23×C3C22⋊Q812yes—
346C23×C3C2×SD1612yes—
347C23×C3AGL(1,ℤ/8)12yes—
348C23×C3C22×D411yescharacteristic
349C22×D3—22no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C3×Q8—84no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
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—41no—
385C2×C3×C4—84no—
386C2×C3×C4C22⋊Q812yes—
387C2×C3×C4C2×SD1612yes—
388C2×C3×C4C2×SD1612yes—
389C2×C3×C4C22×Q811yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4H(5,𝔽2)11yescharacteristic
394C2×C3×C4H(5,𝔽2)11yescharacteristic
395#67—48no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—84no—
406D12—22no—
407D12—22no—
408D12—41no—
409D12—41no—
410C4×D3—84no—
411C3⋊C8—41no—
412C3⋊C8—41no—
413C3⋊C8—41no—
414C3⋊C8—41no—
415C24—61no—
416C24C2×#6711yescharacteristic
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—81no—
420C2×D4—32no—
421C2×D4—64no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—61no—
427C2×D4—122no—
428C2×D4—122no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—81no—
443C22×C4—122no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4—122no—
447C22×C4—241no—
448C22×C4—241no—
449C22×C4D3×Q812yes—
450C22×C4#23912yes—
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6711yescharacteristic
453C22×C4C2×#6711yescharacteristic
454C22×C4C2×#6711yescharacteristic
455C22×C4C2×#6711yescharacteristic
456C22×C4C23×D311yescharacteristic
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—124no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—32no—
478C4⋊C4—32no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—81no—
489C4⋊C4—122no—
490C4⋊C4—122no—
491C4⋊C4#23712yes—
492C22⋊C4—22no—
493C22⋊C4—42no—
494C22⋊C4—64no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—241no—
498C22⋊C4#21514yes—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—122no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3#47614yes—
511C22×C3C2×C22⋊Q811yescharacteristic
512C22×C3#54611yescharacteristic
513C22×C3#54611yescharacteristic
514C22×C3C22×SD1611yescharacteristic
515C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
516C22×C3#57611yescharacteristic
517C22×C3#57811yescharacteristic
518C2×D3—42no—
519C2×D3—42no—
520C2×D3—42no—
521C2×D3—81no—
522C2×D3—81no—
523C2×D3—81no—
524C2×D3—81no—
525C3×C4—22no—
526C3×C4—22no—
527C3×C4—22no—
528C3×C4—41no—
529C3×C4—41no—
530C3×C4—42no—
531C3×C4—81no—
532C3×C4—81no—
533C3×C4—81no—
534C3×C4#52911yescharacteristic
535C3×C4#53811yescharacteristic
536C3×C4#55411yescharacteristic
537C3×C4#55411yescharacteristic
538Dic3—42no—
539Dic3—42no—
540Dic3—81no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—62no—
546C23—121no—
547C23—121no—
548C23—121no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23#93212yes—
553C23C2×#21512yes—
554C23#94612yes—
555C23C22×#6711yescharacteristic
556Q8—84no—
557D4—62no—
558D4—62no—
559D4—128no—
560D4—121no—
561D4—121no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—84no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—244no—
601C2×C4—244no—
602C2×C4C2×#21512yes—
603C2×C4C2×#21512yes—
604C2×C4#94612yes—
605C2×C4C2×#23711yescharacteristic
606C2×C4#100411yescharacteristic
607C2×C4C2×D3×Q811yescharacteristic
608C2×C4C2×#23911yescharacteristic
609C2×C4D3×#4111yescharacteristic
610C2×C4#100911yescharacteristic
611C8—121no—
612C8—121no—
613C8—121no—
614C8—121no—
615C2×C3—24no—
616C2×C3#245011yescharacteristic
617C2×C3C2×#47611yescharacteristic
618C2×C3#308311yescharacteristic
619C2×C3#329511yescharacteristic
620C2×C3#329811yescharacteristic
621C2×C3#330811yescharacteristic
622C2×C3#331811yescharacteristic
623D3—82no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—122no—
631C22—122no—
632C22—122no—
633C22—241no—
634C22—241no—
635C22—241no—
636C22—241no—
637C22#508414yes—
638C22C2×#93211yescharacteristic
639C22#580611yescharacteristic
640C22C22×#21511yescharacteristic
641C22C2×#94611yescharacteristic
642C22#582911yescharacteristic
643C22#583511yescharacteristic
644C22#583811yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—41no—
649C4—42no—
650C4—41no—
651C4—81no—
652C4—81no—
653C4—81no—
654C4—122no—
655C4—122no—
656C4—241no—
657C4#562411yescharacteristic
658C4#563511yescharacteristic
659C4#563611yescharacteristic
660C4#564411yescharacteristic
661C3#2641011yescharacteristic, complemented, Sylow
662C2—24no—
663C2—242no—
664C2#8276511yescharacteristic
665C2C2×#508411yescharacteristic
666C2#8347511yescharacteristic
667C2#8348511yescharacteristic
668C2#8353211yescharacteristic
669C2#8355011yescharacteristic
670C2#8363411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2440C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#2440D3
hypocenterC3#26410
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
1. C3×#2440C2semidirect
3. #71792C2semidirect
4. #71779C2semidirect
6. C2×#4529C2semidirect
7. #69629C2semidirect
8. #69626C2semidirect
9. #69610C2semidirect
10. #69602C2semidirect
11. #69584C2semidirect
12. #69572C2semidirect
14. #69505C2semidirect
15. #69449C2semidirect
60. #2440D3semidirect
103. #930Q8semidirect
661. C3#26410semidirect

Maximal subgroups

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

subgroupindex
C3×#24402
#734672
#717922
#717792
#715602
C2×#45292
#696292
#696262
#696102
#696022
#695842
#695722
#695152
#695052
#694492
#264103