Finite Group Databasebeta

← #274471

Subgroups — #274471

Order 512.

674 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29936C211yesmaximal, characteristic
2#25902C211yesmaximal, characteristic
3#22600C211yesmaximal, characteristic
4#21834C211yesmaximal, characteristic
5#10593C211yesmaximal, characteristic, complemented
6#10589C211yesmaximal, characteristic, complemented
7#10541C211yesmaximal, characteristic, complemented
8#10541C211yesmaximal, characteristic, complemented
9#9745C211yesmaximal, characteristic, complemented
10#9737C211yesmaximal, characteristic, complemented
11#9576C211yesmaximal, characteristic, complemented
12#9568C211yesmaximal, characteristic, complemented
13#9130C211yesmaximal, characteristic
14#8831C211yesmaximal, characteristic
15#8342C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×#487C2211yescharacteristic
18C2×#487C2211yescharacteristic
19C2×#433—21no—
20C2×#431C2211yescharacteristic
21#2857C2211yescharacteristic
22#2285C2211yescharacteristic
23#2099—21no—
24#2083—21no—
25#2075—21no—
26#2075—21no—
27#2024—21no—
28#2024—21no—
29#1990—21no—
30#1990—21no—
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1988C2211yescharacteristic
34#1988C2211yescharacteristic
35#1987C2211yescharacteristic
36#1987C2211yescharacteristic
37#1987C2211yescharacteristic
38#1987C2211yescharacteristic
39#1947C2211yescharacteristic
40#1947C2211yescharacteristic
41#1939C2211yescharacteristic
42#1939C2211yescharacteristic
43#1917—21no—
44#1916—21no—
45#1872—21no—
46#1872—21no—
47#1824—21no—
48#1824—21no—
49#1818—21no—
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1770C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×Q8—21no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—41no—
74C2×C4⋊Q8—41no—
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2311yescharacteristic
77C2×C4⋊Q8C2311yescharacteristic
78C2×C22⋊Q8—21no—
79C2×#117—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C42C2311yescharacteristic
83#487—22no—
84#487—22no—
85#487—41no—
86#487—41no—
87#433—44no—
88#431—41no—
89#431—41no—
90C2×#107—21no—
91C2×#107—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#103—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8D411yescharacteristic
108C2×C22⋊C8D411yescharacteristic
109C2×#97—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391—21no—
115#389—21no—
116#389—21no—
117#384D411yescharacteristic
118#384D411yescharacteristic
119#384C2311yescharacteristic
120#382—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380D411yescharacteristic
125#380D411yescharacteristic
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—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#336—21no—
153#336D411yescharacteristic
154#336D411yescharacteristic
155#335—41no—
156#334—41no—
157#333—22no—
158#333—22no—
159#333—41no—
160#333—41no—
161#332—41no—
162#332—41no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—42no—
167C22×Q8—42no—
168C22×Q8—41no—
169C22×Q8—82no—
170C22×Q8—82no—
171C23×C4—41no—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4#4111yescharacteristic
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C4⋊Q8—24no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—42no—
197C4⋊Q8—42no—
198C4⋊Q8—42no—
199C4⋊Q8—42no—
200C4⋊Q8—42no—
201C4⋊Q8#4112yes—
202C4⋊Q8#4112yes—
203C22⋊Q8—42no—
204C22⋊Q8—42no—
205#117—42no—
206#117—42no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—41no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42C2411yescharacteristic
244#107—81no—
245#107—81no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—42no—
259C4⋊C8—42no—
260#103—42no—
261#103—42no—
262#103—42no—
263#103—42no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C22⋊C8—42no—
267C22⋊C8—42no—
268C22⋊C8—81no—
269#97—42no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—42no—
275C4×C8—42no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—41no—
281#95—41no—
282C24#12111yescharacteristic
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—44no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—82no—
295C2×Q8—81no—
296C2×Q8—81no—
297C2×Q8—81no—
298C2×Q8—81no—
299C2×Q8—88no—
300C2×Q8—88no—
301C2×Q8—82no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—44no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4#12111yescharacteristic
328C22×C4#12111yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4#13512yes—
331C22×C4AGL(1,ℤ/8)12yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
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—81no—
395C4⋊C4—81no—
396C4⋊C4—82no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—82no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42C2×#4111yescharacteristic
411C42C2×#4111yescharacteristic
412C42H(5,𝔽2)11yescharacteristic
413C42H(5,𝔽2)11yescharacteristic
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23#46312yes—
421C23C2×#12111yescharacteristic
422Q8—82no—
423Q8—82no—
424Q8—82no—
425Q8—82no—
426Q8—88no—
427Q8—81no—
428Q8—81no—
429Q8—81no—
430Q8—81no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—88no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4#46312yes—
476C2×C4#47112yes—
477C2×C4#52511yescharacteristic
478C2×C4#53511yescharacteristic
479C2×C4#54611yescharacteristic
480C2×C4#54611yescharacteristic
481C2×C4#55111yescharacteristic
482C2×C4#55111yescharacteristic
483C2×C4C2×#13511yescharacteristic
484C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
485C2×C4#57711yescharacteristic
486C2×C4#57811yescharacteristic
487C2×C4#57811yescharacteristic
488C2×C4#57811yescharacteristic
489C2×C4#57811yescharacteristic
490C2×C4#57811yescharacteristic
491C8—41no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—81no—
496C8—81no—
497C8—81no—
498C8—81no—
499C8—81no—
500C8—81no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22#241111yescharacteristic
509C22C2×#46311yescharacteristic
510C22#304411yescharacteristic
511C22#305911yescharacteristic
512C22#306311yescharacteristic
513C22C2×#47111yescharacteristic
514C22#306811yescharacteristic
515C4—22no—
516C4—21no—
517C4—21no—
518C4—22no—
519C4—41no—
520C4—81no—
521C4—81no—
522C4—82no—
523C4—82no—
524C4—81no—
525C4—81no—
526C4—161no—
527C4—161no—
528C4#312111yescharacteristic
529C4#312711yescharacteristic
530C4#322311yescharacteristic
531C4#322511yescharacteristic
532C4#330011yescharacteristic
533C4#330011yescharacteristic
534C4#330811yescharacteristic
535C4#330811yescharacteristic
536C2—42no—
537C2#1919911yescharacteristic
538C2#1923611yescharacteristic
539C2#1953111yescharacteristic
540C2#1957111yescharacteristic
541C2#2448111yescharacteristic
542C2#2451411yescharacteristic
543C2#2452111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
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
5. #10593C2semidirect
6. #10589C2semidirect
7. #10541C2semidirect
8. #10541C2semidirect
9. #9745C2semidirect
10. #9737C2semidirect
11. #9576C2semidirect
12. #9568C2semidirect

Maximal subgroups

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

subgroupindex
#299362
#259022
#226002
#218342
#105932
#105892
#105412
#97452
#97372
#95762
#95682
#91302
#88312
#83422