Finite Group Databasebeta

← #210793

Subgroups — #210793

Order 512.

718 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22102C211yesmaximal, characteristic
2#22100C211yesmaximal, characteristic
3C4×#467C211yesmaximal, characteristic
4C2×#1587C211yesmaximal, characteristic, complemented
5C2×C8×M16C211yesmaximal, characteristic
6#9664C211yesmaximal, characteristic, complemented
7#9599C211yesmaximal, characteristic, complemented
8#9553C211yesmaximal, characteristic, complemented
9#9553C211yesmaximal, characteristic, complemented
10#9077C211yesmaximal, characteristic
11#9077C211yesmaximal, characteristic
12#8595C211yesmaximal, characteristic
13#8288C211yesmaximal, characteristic, complemented
14#8207C211yesmaximal, characteristic, complemented
15C8×#103C211yesmaximal, characteristic, complemented
16C2×#467C411yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C4×C22⋊Q8C2211yescharacteristic
24C4×C22⋊Q8C2211yescharacteristic
25#1984C2211yescharacteristic
26#1984C2211yescharacteristic
27#1969C411yescharacteristic
28#1942C411yescharacteristic
29#1940—21no—
30#1939—21no—
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1933C411yescharacteristic
34#1933C411yescharacteristic
35#1924—21no—
36#1924—21no—
37#1923—21no—
38#1923—21no—
39#1917—21no—
40#1916—21no—
41#1892C411yescharacteristic
42#1892C411yescharacteristic
43#1808C411yescharacteristic
44#1797C2211yescharacteristic
45#1797C2211yescharacteristic
46C4×#107C2211yescharacteristic
47#1770C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49C4×#103C2211yescharacteristic
50C4×#103C2211yescharacteristic
51#1587—24no—
52C2×#334C2211yescharacteristic
53C2×#334C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C8×M16—24no—
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61C2×C82C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69C22×Q16C2×C411yescharacteristic
70C22×M16—21no—
71C23×C8C2×C411yescharacteristic
72C23×C8C2×C411yescharacteristic
73C2×C22⋊Q8C2×C411yescharacteristic
74C2×C22⋊Q8C2×C411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77C22×C42C2311yescharacteristic
78#467—24no—
79#446—22no—
80#446—22no—
81C8×Q8—22no—
82C8×Q8—22no—
83C4×Q16—22no—
84C4×Q16—22no—
85#431—22no—
86#431—22no—
87C2×#107C2×C411yescharacteristic
88C2×#107C2×C411yescharacteristic
89C2×#107C2×C411yescharacteristic
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8D411yescharacteristic
95C2×C4⋊C8D411yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#103C2×C411yescharacteristic
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8D411yescharacteristic
109C2×C22⋊C8D411yescharacteristic
110C4×M16—22no—
111C4×M16—22no—
112C2×#97—21no—
113C2×#97—21no—
114C2×#97—21no—
115C2×C4×C8—22no—
116C2×C4×C8—22no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2×C411yescharacteristic
120C2×C4×C8C2×C411yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127#391C2×C411yescharacteristic
128#391C2×C411yescharacteristic
129#389C2×C411yescharacteristic
130#389C2×C411yescharacteristic
131#387—21no—
132#387—21no—
133#385—21no—
134#385—21no—
135#380C2×C411yescharacteristic
136#380C2×C411yescharacteristic
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2311yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
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#336—21no—
158#336—21no—
159#334—22no—
160#334—22no—
161Q8⋊C8—22no—
162Q8⋊C8—22no—
163Q8⋊C8—22no—
164Q8⋊C8—22no—
165#322—22no—
166#322D414yes—
167C82—22no—
168C82—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—
177C2×M16—24no—
178C2×M16—24no—
179C2×M16—24no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—41no—
191C22×C8—41no—
192C22×C8C22×C411yescharacteristic
193C22×C8C22×C411yescharacteristic
194C22×C8C22×C411yescharacteristic
195C22×C8C22×C411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C22⋊Q8—41no—
203C22⋊Q8—41no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×Q8—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⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42#4111yescharacteristic
239C2×C42C2411yescharacteristic
240#107—22no—
241#107—22no—
242#107—41no—
243C4⋊C8—22no—
244C4⋊C8—22no—
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—81no—
257#103—22no—
258#103—22no—
259#103—22no—
260#103—22no—
261#103—22no—
262#103M1614yescomplemented
263C22⋊C8—22no—
264C22⋊C8—22no—
265C22⋊C8—81no—
266C22⋊C8—81no—
267#97—24no—
268#97—42no—
269#97—81no—
270C4×C8—22no—
271C4×C8—22no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—42no—
279C4×C8—81no—
280C4×C8C2×D412yes—
281C4×C8C2×D412yes—
282C4×C8C2×D412yes—
283C4×C8#4112yes—
284C4×C8#4112yes—
285C4×C8#4112yes—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296C24C4×D411yescharacteristic
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×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8C2×M1612yes—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
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—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4C4×D411yescharacteristic
334C22×C4C4×D411yescharacteristic
335C22×C4C4×D411yescharacteristic
336C22×C4C23×C411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C2×#4111yescharacteristic
339Q16—44no—
340M16—216no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—21no—
350C2×C8—21no—
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—21no—
364C2×C8—21no—
365C2×C8—44no—
366C2×C8—44no—
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×C8—41no—
379C2×C8C4×D412yes—
380C2×C8C4×D412yes—
381C2×C8#12112yes—
382C2×C8#12112yes—
383C2×C8C2×M1612yes—
384C2×C8#13112yes—
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—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4C2×M1612yes—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42C22×D411yescharacteristic
424C42C2×#4111yescharacteristic
425C42C2×#4111yescharacteristic
426C42C2×#4111yescharacteristic
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23#44312yes—
434C23C2×C4×D411yescharacteristic
435Q8—22no—
436Q8—22no—
437Q8—41no—
438Q8—41no—
439Q8—41no—
440Q8—41no—
441Q8—41no—
442Q8—41no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
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—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4#44312yes—
488C2×C4C2×C4×D411yescharacteristic
489C2×C4C4×#4111yescharacteristic
490C2×C4C2×#12111yescharacteristic
491C2×C4#52511yescharacteristic
492C2×C4#52911yescharacteristic
493C2×C4#53211yescharacteristic
494C2×C4C22×M1611yescharacteristic
495C2×C4C2×#13111yescharacteristic
496C2×C4#56811yescharacteristic
497C2×C4#56811yescharacteristic
498C2×C4#57711yescharacteristic
499C2×C4#57711yescharacteristic
500C2×C4#57811yescharacteristic
501C2×C4#57811yescharacteristic
502C8—24no—
503C8—24no—
504C8—22no—
505C8—22no—
506C8—22no—
507C8—22no—
508C8—22no—
509C8—22no—
510C8—41no—
511C8—41no—
512C8—81no—
513C8—81no—
514C8—81no—
515C8—81no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22C4×#12111yescharacteristic
524C22D4×M1611yescharacteristic
525C22#293511yescharacteristic
526C22#295011yescharacteristic
527C22#295211yescharacteristic
528C22C2×#44311yescharacteristic
529C22C2×#44311yescharacteristic
530C4—21no—
531C4—21no—
532C4—22no—
533C4—41no—
534C4—42no—
535C4—41no—
536C4—42no—
537C4—81no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—81no—
542C4—81no—
543C4#293311yescharacteristic
544C4D4×M1611yescharacteristic
545C4#296511yescharacteristic
546C4#296711yescharacteristic
547C4#306311yescharacteristic
548C4#306311yescharacteristic
549C4#317811yescharacteristic
550C4#317811yescharacteristic
551C2—42no—
552C2#1805711yescharacteristic
553C2#1805811yescharacteristic
554C2#1808611yescharacteristic
555C2#1808811yescharacteristic
556C2#2149011yescharacteristic
557C2#2153011yescharacteristic
558C2#2154411yescharacteristic
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 ◁ C2×#431 ◁ 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. #9553C2semidirect
9. #9553C2semidirect
13. #8288C2semidirect
14. #8207C2semidirect
15. C8×#103C2semidirect
262. #103M16semidirect

Maximal subgroups

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

subgroupindex
#221022
#221002
C4×#4672
C2×#15872
C2×C8×M162
#96642
#95992
#95532
#90772
#85952
#82882
#82072
C8×#1032