Finite Group Databasebeta

← #229848

Subgroups — #229848

Order 512.

688 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23054C211yesmaximal, characteristic
2#23050C211yesmaximal, characteristic
3C4×#467C211yesmaximal, characteristic
4C2×#1587C211yesmaximal, characteristic, complemented
5C2×#1564C211yesmaximal, characteristic
6#9664C211yesmaximal, characteristic, complemented
7#9599C211yesmaximal, characteristic, complemented
8#9548C211yesmaximal, characteristic, complemented
9#9544C211yesmaximal, characteristic, complemented
10#8938C211yesmaximal, characteristic
11#8938C211yesmaximal, characteristic
12#8848C211yesmaximal, characteristic, complemented
13#8844C211yesmaximal, characteristic, complemented
14#8288C211yesmaximal, characteristic, complemented
15#8145C211yesmaximal, characteristic
16C2×#467C411yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C4×C22⋊Q8C2211yescharacteristic
24C4×C22⋊Q8C2211yescharacteristic
25#1984C2211yescharacteristic
26#1984C2211yescharacteristic
27#1969C411yescharacteristic
28#1942C411yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1933C411yescharacteristic
32#1933C411yescharacteristic
33#1924—21no—
34#1924—21no—
35#1923—21no—
36#1923—21no—
37#1916—21no—
38#1916—21no—
39#1892C411yescharacteristic
40#1892C411yescharacteristic
41#1808C411yescharacteristic
42C4×#107C2211yescharacteristic
43#1771—21no—
44#1770C2211yescharacteristic
45C8×C4⋊C4—21no—
46C8×C4⋊C4C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1587—24no—
52#1564—24no—
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×Q16C2×C411yescharacteristic
70C23×C8C2×C411yescharacteristic
71C23×C8C2×C411yescharacteristic
72C2×C22⋊Q8C2×C411yescharacteristic
73C2×C22⋊Q8C2×C411yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#467—24no—
78#446—22no—
79#446—41no—
80C8×Q8—22no—
81C8×Q8—41no—
82C4×Q16—22no—
83C4×Q16—22no—
84#431—22no—
85#431—22no—
86#431—41no—
87#431—41no—
88C2×#107C2×C411yescharacteristic
89C2×#107C2×C411yescharacteristic
90C2×#107C2×C411yescharacteristic
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8D411yescharacteristic
109C2×C22⋊C8D411yescharacteristic
110C2×#97—21no—
111C2×#97—21no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8C2×C411yescharacteristic
119C2×C4×C8C2×C411yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#391C2×C411yescharacteristic
127#391C2×C411yescharacteristic
128#389C2×C411yescharacteristic
129#389C2×C411yescharacteristic
130#387—21no—
131#387—21no—
132#385—21no—
133#385—21no—
134#380C2×C411yescharacteristic
135#380C2×C411yescharacteristic
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2311yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336D411yescharacteristic
158#336D411yescharacteristic
159#335—22no—
160#335—22no—
161#334—22no—
162#334D414yes—
163Q8⋊C8—22no—
164Q8⋊C8—22no—
165Q8⋊C8—41no—
166Q8⋊C8—41no—
167#322—22no—
168#322—22no—
169C22×Q8C22×C411yescharacteristic
170C22×Q8C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4#4111yescharacteristic
174C2×Q16—22no—
175C2×Q16—22no—
176C2×Q16—41no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—41no—
189C22×C8C22×C411yescharacteristic
190C22×C8C22×C411yescharacteristic
191C22×C8C22×C411yescharacteristic
192C22×C8C22×C411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8M1614yes—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—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⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42C22×C411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42#4111yescharacteristic
236C2×C42C2411yescharacteristic
237#107—22no—
238#107—22no—
239#107—41no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8C2×D412yes—
251C4⋊C8C2×D412yes—
252#103—22no—
253#103—22no—
254#103—22no—
255#103—22no—
256#103—22no—
257#103—22no—
258C22⋊C8—24no—
259C22⋊C8—41no—
260C22⋊C8—41no—
261C22⋊C8—42no—
262C22⋊C8—42no—
263#97—42no—
264#97—42no—
265C4×C8—22no—
266C4×C8—22no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—42no—
278C4×C8—42no—
279C4×C8—42no—
280C4×C8—42no—
281C4×C8C2×D412yes—
282C4×C8#4112yes—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293C24C4×D411yescharacteristic
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8C2×M1612yes—
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—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—41no—
328C22×C4C4×D411yescharacteristic
329C22×C4C4×D411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4C2×M1612yes—
332C22×C4#13112yes—
333C22×C4C23×C411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C2×#4111yescharacteristic
336Q16—44no—
337C2×C8—28no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—44no—
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×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8C4×D412yes—
379C2×C8C4×D412yes—
380C2×C8#13112yes—
381C4⋊C4—21no—
382C4⋊C4—24no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4C2×M1612yes—
401C22⋊C4—24no—
402C22⋊C4—42no—
403C22⋊C4—81no—
404C22⋊C4—81no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42C22×D411yescharacteristic
420C42C2×#4111yescharacteristic
421C42H(5,𝔽2)11yescharacteristic
422C422−1+411yescharacteristic
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23#43512yes—
430C23C2×C4×D411yescharacteristic
431Q8—22no—
432Q8—22no—
433Q8—41no—
434Q8—41no—
435Q8—41no—
436Q8—41no—
437Q8—41no—
438Q8—41no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—42no—
455C2×C4—42no—
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—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
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#43512yes—
483C2×C4#44312yes—
484C2×C4C2×C4×D411yescharacteristic
485C2×C4#52011yescharacteristic
486C2×C4D4211yescharacteristic
487C2×C4#54711yescharacteristic
488C2×C4#54811yescharacteristic
489C2×C4#55311yescharacteristic
490C2×C4C22×M1611yescharacteristic
491C2×C4C2×#13111yescharacteristic
492C2×C4C2×#13111yescharacteristic
493C2×C4#56811yescharacteristic
494C2×C4#57511yescharacteristic
495C2×C4#57511yescharacteristic
496C2×C4#57811yescharacteristic
497C2×C4#57811yescharacteristic
498C8—22no—
499C8—22no—
500C8—22no—
501C8—42no—
502C8—42no—
503C8—41no—
504C8—41no—
505C8—41no—
506C8—41no—
507C8—41no—
508C8—42no—
509C8—42no—
510C8—81no—
511C8—81no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22#236111yescharacteristic
520C22C2×#43511yescharacteristic
521C22#293511yescharacteristic
522C22#294311yescharacteristic
523C22#295011yescharacteristic
524C22C2×#44311yescharacteristic
525C22#295711yescharacteristic
526C4—22no—
527C4—21no—
528C4—21no—
529C4—22no—
530C4—42no—
531C4—42no—
532C4—41no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4—81no—
537C4—81no—
538C4—81no—
539C4#292911yescharacteristic
540C4D4×M1611yescharacteristic
541C4#299011yescharacteristic
542C4#299211yescharacteristic
543C4#326911yescharacteristic
544C4#326911yescharacteristic
545C4#334911yescharacteristic
546C4#334911yescharacteristic
547C2—42no—
548C2#1786811yescharacteristic
549C2#1788011yescharacteristic
550C2#1807411yescharacteristic
551C2#1808611yescharacteristic
552C2#2298711yescharacteristic
553C2#2301811yescharacteristic
554C2#2303011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1587C2semidirect
6. #9664C2semidirect
7. #9599C2semidirect
8. #9548C2semidirect
9. #9544C2semidirect
12. #8848C2semidirect
13. #8844C2semidirect
14. #8288C2semidirect

Maximal subgroups

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

subgroupindex
#230542
#230502
C4×#4672
C2×#15872
C2×#15642
#96642
#95992
#95482
#95442
#89382
#88482
#88442
#82882
#81452