Finite Group Databasebeta

← #10858944

Subgroups — #10858944

Order 768.

858 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2442C211yesmaximal, characteristic, complemented
2#73467C211yesmaximal, characteristic
3#71743C211yesmaximal, characteristic, complemented
4#71738C211yesmaximal, characteristic, complemented
5#71560C211yesmaximal, characteristic
6C2×#4529C211yesmaximal, characteristic, complemented
7#69629C211yesmaximal, characteristic, complemented
8#69626C211yesmaximal, characteristic, complemented
9#69624C211yesmaximal, characteristic, complemented
10#69601C211yesmaximal, characteristic, complemented
11#69556C211yesmaximal, characteristic, complemented
12#69542C211yesmaximal, characteristic, complemented
13#69515C211yesmaximal, characteristic
14#69505C211yesmaximal, characteristic, complemented
15#69449C211yesmaximal, characteristic, complemented
16#26422—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C2×#930C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C23×C3⋊C8C2211yescharacteristic
22C2×#67⋊C4—21no—
23C2×C4×D12C2211yescharacteristic
24C3×#400—21no—
25C3×#398—21no—
26C3×#389C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#385—21no—
29C3×#382C2211yescharacteristic
30C3×#380—21no—
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#2442D311yescharacteristic, complemented
61#2361—31no—
62#2101—31no—
63#2088—31no—
64#2065—31no—
65#2025—31no—
66#1989—31no—
67#1919—31no—
68#1893—31no—
69#1874—31no—
70#1864—31no—
71#1853—31no—
72#1850—31no—
73#1804—31no—
74C2×#340—31no—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C22×#67—21no—
78C23×Dic3—21no—
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C3×#125—22no—
82C3×#125—41no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—41no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4D411yescharacteristic
96C2×C3×C22⋊C4D411yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C2311yescharacteristic
99#930—22no—
100#930Q814yes—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3—22no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—41no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—21no—
113C22×C3⋊C8—22no—
114C22×C3⋊C8—41no—
115C22×C3⋊C8—41no—
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120#67⋊C4—44no—
121C4×D12—22no—
122C4×D12—41no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95D411yescharacteristic
129C3×#95D411yescharacteristic
130C3⋊#95—21no—
131C3⋊#95—21no—
132#830—22no—
133#809—22no—
134#809—41no—
135#809—41no—
136#809—41no—
137#808—22no—
138#808—41no—
139C4⋊(C3⋊C8)—41no—
140C4⋊(C3⋊C8)—41no—
141C23×C8—31no—
142C2×#125C2×D311yescharacteristic
143C2×#121—31no—
144C2×C4×D4—31no—
145C2×C4×D4—61no—
146C22×C4⋊C4—31no—
147C22×C4⋊C4C2×D311yescharacteristic
148C2×#106—31no—
149C2×#106—31no—
150C2×C4⋊C8—31no—
151C2×C4⋊C8—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×C22⋊C8—31no—
157#400—21no—
158#398—21no—
159#390—31no—
160#389—31no—
161#389C2×D311yescharacteristic
162#389C2×D311yescharacteristic
163#388—61no—
164#386—61no—
165#385—21no—
166#385—61no—
167#382C2×D311yescharacteristic
168#380—21no—
169#380C2×D311yescharacteristic
170C4×C4⋊C4—31no—
171C4×C22⋊C4C2×D311yescharacteristic
172#340—31no—
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#336—31no—
183#336—31no—
184C24×C3C2×D411yescharacteristic
185C23×D3—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—42no—
195C22×C3×C4—42no—
196C22×C3×C4—81no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×Q812yes—
203C22×C3×C4#4112yes—
204C22×C3×C4C2411yescharacteristic
205C2×#67—24no—
206C2×#67—42no—
207C2×#67—42no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—42no—
212C22×Dic3—42no—
213C22×Dic3—81no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12C2×Q812yes—
220C2×C4×D3—42no—
221C2×C4×D3—42no—
222C2×C4×D3—81no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4—42no—
232C3×C4⋊C4—42no—
233C3×C4⋊C4—81no—
234C3×C4⋊C4—81no—
235C3×C4⋊C4#4112yes—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—42no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—21no—
243(C2×D3)⋊C4—24no—
244(C2×D3)⋊C4—42no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—81no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—42no—
252C4⋊Dic3C2×Q812yes—
253C4⋊Dic3#4112yes—
254Dic3⋊C4—42no—
255C4×Dic3—42no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
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—44no—
269C22×D4—31no—
270C22×D4—61no—
271C23×C4—21no—
272C23×C4—61no—
273C23×C4—61no—
274C23×C4C22×D311yescharacteristic
275C22×C8—31no—
276C22×C8—31no—
277C22×C8—31no—
278C22×C8—31no—
279C22×C8—62no—
280C22×C8—121no—
281C22×C8—121no—
282#125—22no—
283#125—41no—
284#121—34no—
285#121—62no—
286C4×D4—62no—
287C4×D4—121no—
288C4×D4—124no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—41no—
299C2×C4⋊C4—62no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—121no—
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C22⋊C4—21no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4#6711yescharacteristic
313C2×C22⋊C4#6711yescharacteristic
314C2×C42—61no—
315C2×C42C22×D311yescharacteristic
316C2×C42C22×D311yescharacteristic
317#106—62no—
318#106—121no—
319C4⋊C8—121no—
320C4⋊C8—121no—
321#102—62no—
322#102—121no—
323#102—121no—
324#102—121no—
325C22⋊C8—62no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—61no—
332#95—61no—
333#95#6711yescharacteristic
334#95#6711yescharacteristic
335C23×C3—21no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3C22⋊Q812yes—
340C23×C3#13512yes—
341C23×C3AGL(1,ℤ/8)12yes—
342C23×C3C22×D411yescharacteristic
343C22×D3—22no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
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—84no—
381C2×C3×C4C22⋊Q812yes—
382C2×C3×C4C22×Q811yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387C2×C3×C4H(5,𝔽2)11yescharacteristic
388#67—48no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—84no—
399D12—22no—
400D12—22no—
401D12—41no—
402D12—41no—
403C4×D3—84no—
404C3⋊C8—41no—
405C3⋊C8—41no—
406C3⋊C8—41no—
407C3⋊C8—41no—
408C24—61no—
409C24C2×#6711yescharacteristic
410C2×D4—32no—
411C2×D4—64no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—122no—
418C2×D4—122no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—81no—
433C22×C4—122no—
434C22×C4—122no—
435C22×C4—122no—
436C22×C4—122no—
437C22×C4—241no—
438C22×C4—241no—
439C22×C4D3×Q812yes—
440C22×C4#23912yes—
441C22×C4C2×#6711yescharacteristic
442C22×C4C2×#6711yescharacteristic
443C22×C4C2×#6711yescharacteristic
444C22×C4C2×#6711yescharacteristic
445C22×C4C2×#6711yescharacteristic
446C22×C4C23×D311yescharacteristic
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—124no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—32no—
468C4⋊C4—32no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—81no—
477C4⋊C4—81no—
478C4⋊C4—122no—
479C4⋊C4—122no—
480C4⋊C4#23712yes—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—42no—
484C22⋊C4—64no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—241no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—122no—
493C22×C3—22no—
494C22×C3—22no—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—22no—
498C22×C3—22no—
499C22×C3#47814yes—
500C22×C3C2×C22⋊Q811yescharacteristic
501C22×C3#54611yescharacteristic
502C22×C3#54611yescharacteristic
503C22×C3C2×#13511yescharacteristic
504C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
505C22×C3#57711yescharacteristic
506C22×C3#57711yescharacteristic
507C2×D3—42no—
508C2×D3—42no—
509C2×D3—42no—
510C2×D3—81no—
511C2×D3—81no—
512C2×D3—81no—
513C2×D3—81no—
514C3×C4—22no—
515C3×C4—41no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—42no—
519C3×C4—41no—
520C3×C4—81no—
521C3×C4—81no—
522C3×C4—81no—
523C3×C4#52911yescharacteristic
524C3×C4#53811yescharacteristic
525C3×C4#55411yescharacteristic
526C3×C4#55411yescharacteristic
527Dic3—42no—
528Dic3—42no—
529Dic3—81no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—62no—
535C23—121no—
536C23—121no—
537C23—121no—
538C23—121no—
539C23—121no—
540C23—121no—
541C23#94612yes—
542C23#94912yes—
543C23#95012yes—
544C23C22×#6711yescharacteristic
545D4—62no—
546D4—62no—
547D4—128no—
548D4—121no—
549D4—121no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—84no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—244no—
591C2×C4—244no—
592C2×C4#94612yes—
593C2×C4C2×#23711yescharacteristic
594C2×C4#100411yescharacteristic
595C2×C4C2×D3×Q811yescharacteristic
596C2×C4C2×#23911yescharacteristic
597C2×C4D3×#4111yescharacteristic
598C2×C4#100911yescharacteristic
599C8—121no—
600C8—121no—
601C8—121no—
602C8—121no—
603C2×C3—24no—
604C2×C3#245011yescharacteristic
605C2×C3C2×#47811yescharacteristic
606C2×C3#308411yescharacteristic
607C2×C3#330111yescharacteristic
608C2×C3#330111yescharacteristic
609C2×C3#330611yescharacteristic
610C2×C3#331011yescharacteristic
611D3—82no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—122no—
619C22—122no—
620C22—122no—
621C22—241no—
622C22—241no—
623C22—241no—
624C22—241no—
625C22#506614yes—
626C22#580611yescharacteristic
627C22C2×#94611yescharacteristic
628C22C2×#94911yescharacteristic
629C22C2×#95011yescharacteristic
630C22#582911yescharacteristic
631C22#583611yescharacteristic
632C22#583711yescharacteristic
633C4—22no—
634C4—41no—
635C4—42no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—122no—
643C4—122no—
644C4—241no—
645C4#562411yescharacteristic
646C4#563511yescharacteristic
647C4#563611yescharacteristic
648C4#564411yescharacteristic
649C3#2642211yescharacteristic, complemented, Sylow
650C2—24no—
651C2—242no—
652C2#8276511yescharacteristic
653C2C2×#506611yescharacteristic
654C2#8347611yescharacteristic
655C2#8349611yescharacteristic
656C2#8353511yescharacteristic
657C2#8359911yescharacteristic
658C2#8361811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2442C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#2442D3
hypocenterC3#26422
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
1. C3×#2442C2semidirect
3. #71743C2semidirect
4. #71738C2semidirect
6. C2×#4529C2semidirect
7. #69629C2semidirect
8. #69626C2semidirect
9. #69624C2semidirect
10. #69601C2semidirect
11. #69556C2semidirect
12. #69542C2semidirect
14. #69505C2semidirect
15. #69449C2semidirect
60. #2442D3semidirect
649. C3#26422semidirect

Maximal subgroups

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

subgroupindex
C3×#24422
#734672
#717432
#717382
#715602
C2×#45292
#696292
#696262
#696242
#696012
#695562
#695422
#695152
#695052
#694492
#264223