Finite Group Databasebeta

← #285899

Subgroups — #285899

Order 512.

673 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27909C211yesmaximal, characteristic
2#27680C211yesmaximal, characteristic, complemented
3#24653C211yesmaximal, characteristic
4#22953C211yesmaximal, characteristic, complemented
5#10651C211yesmaximal, characteristic, complemented
6#10618C211yesmaximal, characteristic, complemented
7#10599C211yesmaximal, characteristic, complemented
8#10475C211yesmaximal, characteristic, complemented
9#9621C211yesmaximal, characteristic, complemented
10#9572C211yesmaximal, characteristic, complemented
11#9222C211yesmaximal, characteristic, complemented
12#9141C211yesmaximal, characteristic
13#8831C211yesmaximal, characteristic, complemented
14#8645C211yesmaximal, characteristic, complemented
15#8043C211yesmaximal, characteristic, complemented
16C2×#480—21no—
17C2×#480—21no—
18C2×#460C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22#2856C2211yescharacteristic
23C4×C22⋊Q8C2211yescharacteristic
24C2×#384C2211yescharacteristic
25#2099—21no—
26#2096—21no—
27#2093—21no—
28#2093—21no—
29#2034—21no—
30#2000—21no—
31#1984C2211yescharacteristic
32#1982C2211yescharacteristic
33#1968—21no—
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1948C2211yescharacteristic
37#1948C2211yescharacteristic
38#1948C2211yescharacteristic
39#1940—21no—
40#1939C2211yescharacteristic
41#1934—21no—
42#1916—21no—
43#1872—21no—
44#1853—21no—
45#1851—21no—
46#1845C2211yescharacteristic
47#1845C2211yescharacteristic
48#1803C2211yescharacteristic
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1797C2211yescharacteristic
52#1770C2211yescharacteristic
53#1753—21no—
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#331C2211yescharacteristic
58C2×#331C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×SD16—21no—
68C22×M16—21no—
69C2×#127C2311yescharacteristic
70C2×C22⋊Q8D411yescharacteristic
71C2×#121—21no—
72C2×#121—21no—
73C2×#121—21no—
74C2×C4×Q8C2311yescharacteristic
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C42C2311yescharacteristic
79#480—44no—
80#480—44no—
81#460—22no—
82#460—41no—
83C8×Q8—22no—
84C8×Q8—41no—
85#431—22no—
86#431—41no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94#423—22no—
95#423—41no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8D411yescharacteristic
118C2×C22⋊C8D411yescharacteristic
119C2×#97—21no—
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#391D411yescharacteristic
125#390—21no—
126#390—21no—
127#390—21no—
128#389D411yescharacteristic
129#387—21no—
130#385—21no—
131#384—21no—
132#384—21no—
133#384—21no—
134#384C2311yescharacteristic
135#384C2311yescharacteristic
136#384C2311yescharacteristic
137#380D411yescharacteristic
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2311yescharacteristic
140C4×C22⋊C4—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336D411yescharacteristic
153#336D411yescharacteristic
154#335—41no—
155#334—22no—
156#331—22no—
157#331—41no—
158C22×Q8C2×D411yescharacteristic
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—41no—
163C23×C4—41no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4#4111yescharacteristic
167C2×SD16—42no—
168C2×SD16—42no—
169C2×SD16—81no—
170C2×M16—42no—
171C2×M16—42no—
172C2×M16—81no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183#127—21no—
184#127—21no—
185#127—21no—
186#127—21no—
187C22⋊Q8—22no—
188C22⋊Q8—41no—
189#121—42no—
190#121—42no—
191#121—81no—
192#121—81no—
193#121—82no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—81no—
216C2×C4⋊C4—81no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42#4111yescharacteristic
235C2×C42C2411yescharacteristic
236#107—42no—
237#107—42no—
238#107—42no—
239#107—42no—
240#106—81no—
241#106—81no—
242#106—81no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—42no—
254C4⋊C8—42no—
255C4⋊C8—81no—
256C4⋊C8#4112yes—
257C4⋊C8#4112yes—
258#103—81no—
259#103—81no—
260#103—81no—
261#102—42no—
262#102—42no—
263#102—42no—
264#102—42no—
265#102—42no—
266#102—42no—
267#102—81no—
268C22⋊C8—22no—
269C22⋊C8—42no—
270C22⋊C8—42no—
271C22⋊C8SD1614yescomplemented
272#97—81no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—42no—
278C4×C8—42no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284C24—41no—
285C24#12111yescharacteristic
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8#13712yes—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—81no—
307C2×D4—81no—
308C2×D4—82no—
309C2×D4—82no—
310C2×D4—82no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4#12111yescharacteristic
336C22×C4#12111yescharacteristic
337C22×C4#12111yescharacteristic
338C22×C4C2×SD1612yes—
339C22×C4AGL(1,ℤ/8)12yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C2×#4111yescharacteristic
343SD16—84no—
344M16—84no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8C2×SD1612yes—
374C2×C8C2×SD1612yes—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—42no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—81no—
398C4⋊C4—81no—
399C4⋊C4—81no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C22⋊C4—42no—
404C22⋊C4—82no—
405C22⋊C4—81no—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42C2×#4111yescharacteristic
419C42C2×#4111yescharacteristic
420C42H(5,𝔽2)11yescharacteristic
421C422−1+411yescharacteristic
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—81no—
429C23—81no—
430C23—81no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23#46012yes—
436C23C2×#12111yescharacteristic
437Q8—22no—
438Q8—22no—
439Q8—41no—
440Q8—41no—
441D4—81no—
442D4—81no—
443D4—81no—
444D4—81no—
445D4—81no—
446D4—81no—
447D4—81no—
448D4—81no—
449D4—161no—
450D4—161no—
451D4—164no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—164no—
493C2×C4#46012yes—
494C2×C4#47212yes—
495C2×C4#52911yescharacteristic
496C2×C4#53911yescharacteristic
497C2×C4#54611yescharacteristic
498C2×C4#54611yescharacteristic
499C2×C4#54711yescharacteristic
500C2×C4#54711yescharacteristic
501C2×C4C22×SD1611yescharacteristic
502C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
503C2×C4C2×#13711yescharacteristic
504C2×C4#57511yescharacteristic
505C2×C4#57611yescharacteristic
506C2×C4#57611yescharacteristic
507C2×C4#57711yescharacteristic
508C2×C4#57811yescharacteristic
509C8—22no—
510C8—22no—
511C8—41no—
512C8—41no—
513C8—42no—
514C8—42no—
515C8—81no—
516C8—81no—
517C8—81no—
518C8—81no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—161no—
527C22—161no—
528C22—161no—
529C22—161no—
530C22—161no—
531C22—161no—
532C22—161no—
533C22#242511yescharacteristic
534C22C2×#46011yescharacteristic
535C22#304611yescharacteristic
536C22#305611yescharacteristic
537C22#305811yescharacteristic
538C22C2×#47211yescharacteristic
539C22#306911yescharacteristic
540C4—22no—
541C4—21no—
542C4—21no—
543C4—22no—
544C4—41no—
545C4—42no—
546C4—81no—
547C4—81no—
548C4—81no—
549C4—161no—
550C4—161no—
551C4—161no—
552C4#329811yescharacteristic
553C4#330611yescharacteristic
554C4#330911yescharacteristic
555C4#331411yescharacteristic
556C4#333511yescharacteristic
557C4#334311yescharacteristic
558C4#334611yescharacteristic
559C4#334711yescharacteristic
560C2—42no—
561C2—321no—
562C2#1917111yescharacteristic
563C2#1920311yescharacteristic
564C2#1950311yescharacteristic
565C2#1954611yescharacteristic
566C2#2520411yescharacteristic
567C2#2524011yescharacteristic
568C2#2526411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #27680C2semidirect
4. #22953C2semidirect
5. #10651C2semidirect
6. #10618C2semidirect
7. #10599C2semidirect
8. #10475C2semidirect
9. #9621C2semidirect
10. #9572C2semidirect
11. #9222C2semidirect
13. #8831C2semidirect
14. #8645C2semidirect
15. #8043C2semidirect
271. C22⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#279092
#276802
#246532
#229532
#106512
#106182
#105992
#104752
#96212
#95722
#92222
#91412
#88312
#86452
#80432