Finite Group Databasebeta

← #10853173

Subgroups — #10853173

Order 768.

732 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83058C211yesmaximal, characteristic
2#76200C211yesmaximal, characteristic
3C3×#2059C211yesmaximal, characteristic, complemented
4#71738C211yesmaximal, characteristic, complemented
5#70633C211yesmaximal, characteristic
6#70609C211yesmaximal, characteristic, complemented
7#70578C211yesmaximal, characteristic
8#70563C211yesmaximal, characteristic, complemented
9#69610C211yesmaximal, characteristic, complemented
10#68872C211yesmaximal, characteristic
11#68867C211yesmaximal, characteristic
12#68843C211yesmaximal, characteristic, complemented
13#68838C211yesmaximal, characteristic, complemented
14#68667C211yesmaximal, characteristic
15#68624C211yesmaximal, characteristic, complemented
16#26014—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C22×ΓL(1,𝔽25)C2211yescharacteristic
19C2×#892—21no—
20C2×#892C2211yescharacteristic
21C2×D3×C4⋊C4C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C2×(C2×D3)⋊C8C2211yescharacteristic
26C2×#818C2211yescharacteristic
27C2×Dic3⋊C8C2211yescharacteristic
28#4994—21no—
29#4993—21no—
30#4987C2211yescharacteristic
31#4987C2211yescharacteristic
32#4984C2211yescharacteristic
33#4984C2211yescharacteristic
34#4978—21no—
35#4978—21no—
36#4975C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38#4972—21no—
39C2×#808C2211yescharacteristic
40C4×(C2×D3)⋊C4C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42#4928C2211yescharacteristic
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44#4664—21no—
45#4655—21no—
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
48C3×#336C2211yescharacteristic
49#4551C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2441—31no—
61#2344—31no—
62#2100—31no—
63#2088—31no—
64#2086—31no—
65#2059D311yescharacteristic, complemented
66#1989—31no—
67#1948—31no—
68#1921—31no—
69#1918—31no—
70#1854—31no—
71#1851—31no—
72#1824—31no—
73#1809—31no—
74#1736—31no—
75C22×C4×D3—21no—
76C22×C4×D3C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×#125—22no—
80C3×#125—41no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4D411yescharacteristic
93C2×(C2×D3)⋊C4D411yescharacteristic
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C2×ΓL(1,𝔽25)—22no—
111C2×ΓL(1,𝔽25)—22no—
112C2×ΓL(1,𝔽25)—41no—
113#892—22no—
114#892—41no—
115#892—44no—
116D3×C4⋊C4—22no—
117D3×C4⋊C4Q814yes—
118C3×#107—22no—
119C3×#106—41no—
120C3×#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95D411yescharacteristic
129C3⋊#95D411yescharacteristic
130(C2×D3)⋊C8—22no—
131#818—22no—
132Dic3⋊C8—22no—
133#808—41no—
134C4⋊(C3⋊C8)—41no—
135C22×M16—31no—
136C2×#125C2×D311yescharacteristic
137C2×#117—31no—
138C2×#117—61no—
139C22×C4⋊C4—31no—
140C2×#107—31no—
141C2×#107C2×D311yescharacteristic
142C2×#106—31no—
143C2×#106C2×D311yescharacteristic
144C2×C4⋊C8—31no—
145C2×C4⋊C8—31no—
146C2×C22⋊C8—31no—
147#399—61no—
148#396—61no—
149#389—31no—
150#389—31no—
151#389—31no—
152#389—61no—
153#389C2×D311yescharacteristic
154#387—61no—
155#387—61no—
156#387—61no—
157#384—31no—
158#381—61no—
159#380—31no—
160#380—31no—
161C4×C4⋊C4—31no—
162C4×C4⋊C4—31no—
163C4×C22⋊C4—31no—
164#340—31no—
165#340—31no—
166#340—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340C2×D311yescharacteristic
175#340C2×D311yescharacteristic
176#336—31no—
177#336C2×D311yescharacteristic
178C23×D3C2×D411yescharacteristic
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2411yescharacteristic
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3C2×D411yescharacteristic
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C2×C4×D3—21no—
194C2×C4×D3—21no—
195C2×C4×D3—21no—
196C2×C4×D3—21no—
197C2×C4×D3—24no—
198C2×C4×D3—42no—
199C2×C4×D3—42no—
200C2×C4×D3C2×Q812yes—
201C2×C4×D3#4112yes—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8#4112yes—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—42no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4—42no—
217C3×C4⋊C4C2×Q812yes—
218C3×C4⋊C4#4112yes—
219C3×C42—21no—
220C3×C42—21no—
221(C2×D3)⋊C4—22no—
222(C2×D3)⋊C4—41no—
223(C2×D3)⋊C4—42no—
224(C2×D3)⋊C4—42no—
225(C2×D3)⋊C4—81no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—42no—
234C4⋊Dic3—42no—
235C4⋊Dic3C2×Q812yes—
236Dic3⋊C4—24no—
237Dic3⋊C4—42no—
238Dic3⋊C4—81no—
239Dic3⋊C4—81no—
240C4×Dic3—21no—
241C4×Dic3—21no—
242C4×Dic3—42no—
243C4×Dic3—42no—
244C4×Dic3—42no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251ΓL(1,𝔽25)—44no—
252C23×C4—31no—
253C23×C4—61no—
254C2×M16—62no—
255C2×M16—62no—
256C2×M16—121no—
257C22×C8—31no—
258C22×C8—31no—
259C22×C8C22×D311yescharacteristic
260C22×C8C22×D311yescharacteristic
261#125—22no—
262#125—41no—
263#117—62no—
264#117—124no—
265#117—121no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—34no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C22⋊C4—31no—
286C2×C22⋊C4—31no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C42—31no—
291C2×C42—61no—
292C2×C42—61no—
293C2×C42—61no—
294C2×C42C22×D311yescharacteristic
295#107—22no—
296#107—62no—
297#106—41no—
298#106—121no—
299C4⋊C8—62no—
300C4⋊C8—121no—
301C22⋊C8—62no—
302#95—21no—
303#95—31no—
304#95—31no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312C23×C3C22×D411yescharacteristic
313C22×D3—21no—
314C22×D3—21no—
315C22×D3—21no—
316C22×D3—21no—
317C22×D3—21no—
318C22×D3—21no—
319C22×D3C22⋊Q812yes—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4C22×Q811yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C4C2×#4111yescharacteristic
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C4H(5,𝔽2)11yescharacteristic
343C2×C3×C42−1+411yescharacteristic
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—22no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3C22⋊Q812yes—
364C2×Dic3AGL(1,ℤ/8)12yes—
365C4×D3—22no—
366C4×D3—22no—
367C4×D3—22no—
368C4×D3—22no—
369C4×D3—48no—
370C3×C8—22no—
371C3×C8—41no—
372C3⋊C8—41no—
373C3⋊C8—41no—
374C24—31no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—31no—
380C22×C4—31no—
381C22×C4—32no—
382C22×C4—32no—
383C22×C4—31no—
384C22×C4—64no—
385C22×C4—61no—
386C22×C4—61no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—122no—
394C22×C4—122no—
395C22×C4D3×D411yescharacteristic
396C22×C4D3×D411yescharacteristic
397C22×C4C23×D311yescharacteristic
398M16—124no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—61no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8—61no—
409C2×C8—61no—
410C2×C8#23912yes—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—32no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—64no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—122no—
430C4⋊C4—122no—
431C4⋊C4—122no—
432C4⋊C4—241no—
433C4⋊C4—241no—
434C4⋊C4#23712yes—
435C4⋊C4D3×Q812yes—
436C22⋊C4—62no—
437C22⋊C4—121no—
438C22⋊C4—122no—
439C22⋊C4—122no—
440C22⋊C4—241no—
441C42—21no—
442C42—21no—
443C42—61no—
444C42—61no—
445C42—122no—
446C42—122no—
447C42—122no—
448C22×C3C2×C22⋊Q811yescharacteristic
449C22×C3#54611yescharacteristic
450C22×C3#54711yescharacteristic
451C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
452C22×C3#57511yescharacteristic
453C22×C3#57711yescharacteristic
454C22×C3#57811yescharacteristic
455C2×D3—22no—
456C2×D3—22no—
457C2×D3—22no—
458C2×D3—41no—
459C2×D3—41no—
460C2×D3—41no—
461C2×D3—41no—
462C3×C4—41no—
463C3×C4—42no—
464C3×C4—42no—
465C3×C4—41no—
466C3×C4—81no—
467C3×C4—81no—
468C3×C4#53311yescharacteristic
469C3×C4#54411yescharacteristic
470C3×C4#55111yescharacteristic
471C3×C4#55211yescharacteristic
472Dic3—22no—
473Dic3—22no—
474Dic3—42no—
475Dic3—41no—
476Dic3—81no—
477Dic3—81no—
478Dic3—81no—
479C23—32no—
480C23—61no—
481C23—61no—
482C23—61no—
483C23—61no—
484C23—61no—
485C23—61no—
486C23C2×D3×D411yescharacteristic
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—32no—
496C2×C4—32no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—62no—
519C2×C4—128no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4C2×#23711yescharacteristic
532C2×C4#100411yescharacteristic
533C2×C4C2×D3×Q811yescharacteristic
534C2×C4C2×#23911yescharacteristic
535C2×C4#100711yescharacteristic
536C2×C4D3×#4111yescharacteristic
537C8—22no—
538C8—41no—
539C8—121no—
540C8—121no—
541C2×C3#243811yescharacteristic
542C2×C3#307711yescharacteristic
543C2×C3#308211yescharacteristic
544C2×C3#331711yescharacteristic
545C2×C3#333011yescharacteristic
546C2×C3#334511yescharacteristic
547C2×C3#335211yescharacteristic
548D3—42no—
549C22—62no—
550C22—62no—
551C22—62no—
552C22—121no—
553C22—121no—
554C22—121no—
555C22—121no—
556C22#559111yescharacteristic
557C22D3×C22⋊Q811yescharacteristic
558C22#572611yescharacteristic
559C22#576811yescharacteristic
560C22#577111yescharacteristic
561C22D3×AGL(1,ℤ/8)11yescharacteristic
562C22#577811yescharacteristic
563C4—42no—
564C4—41no—
565C4—41no—
566C4—42no—
567C4—62no—
568C4—62no—
569C4—81no—
570C4—81no—
571C4—121no—
572C4—122no—
573C4—241no—
574C4—241no—
575C4—241no—
576C4#563711yescharacteristic
577C4#563911yescharacteristic
578C4#572811yescharacteristic
579C4#572911yescharacteristic
580C3#2601411yescharacteristic, complemented, Sylow
581C2—122no—
582C2#7725111yescharacteristic
583C2#8011611yescharacteristic
584C2#8025611yescharacteristic
585C2#8116611yescharacteristic
586C2#8123211yescharacteristic
587C2#8192811yescharacteristic
588C2#8199211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2059C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2059D3
hypocenterC3#26014
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2059

Complements

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

NH ≅ G/Ntype
3. C3×#2059C2semidirect
4. #71738C2semidirect
6. #70609C2semidirect
8. #70563C2semidirect
9. #69610C2semidirect
12. #68843C2semidirect
13. #68838C2semidirect
15. #68624C2semidirect
65. #2059D3semidirect
580. C3#26014semidirect

Maximal subgroups

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

subgroupindex
#830582
#762002
C3×#20592
#717382
#706332
#706092
#705782
#705632
#696102
#688722
#688672
#688432
#688382
#686672
#686242
#260143