Finite Group Databasebeta

← #231479

Subgroups — #231479

Order 512.

687 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#496C211yesmaximal, characteristic
2#17895C211yesmaximal, characteristic, complemented
3#17878C211yesmaximal, characteristic, complemented
4#17877C211yesmaximal, characteristic, complemented
5#17855C211yesmaximal, characteristic, complemented
6#9672C211yesmaximal, characteristic, complemented
7#9670C211yesmaximal, characteristic, complemented
8#9668C211yesmaximal, characteristic, complemented
9#9667C211yesmaximal, characteristic, complemented
10#8963C211yesmaximal, characteristic, complemented
11#8961C211yesmaximal, characteristic, complemented
12#8960C211yesmaximal, characteristic, complemented
13#8959C211yesmaximal, characteristic, complemented
14#8269C211yesmaximal, characteristic, complemented
15#8268C211yesmaximal, characteristic, complemented
16C2×#496C411yescharacteristic
17#2982C2211yescharacteristic
18#2975C2211yescharacteristic
19#2975C2211yescharacteristic
20#2972C2211yescharacteristic
21C2×C4×SD16C2211yescharacteristic
22C2×C4×SD16C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24C2×C4×D8C2211yescharacteristic
25C4×#127C2211yescharacteristic
26C4×#124C2211yescharacteristic
27#1973C411yescharacteristic
28#1973C411yescharacteristic
29#1972C411yescharacteristic
30#1972C411yescharacteristic
31#1961C411yescharacteristic
32#1959C411yescharacteristic
33#1847C411yescharacteristic
34C4×#97C2211yescharacteristic
35#1594—22no—
36#1589—22no—
37#1588—22no—
38#1586—22no—
39#1583—22no—
40#1581—22no—
41SD16⋊C8—22no—
42C8×SD16—22no—
43#1564C2211yescharacteristic
44#1564C2211yescharacteristic
45#1564C2211yescharacteristic
46#1564C2211yescharacteristic
47#1479C2211yescharacteristic
48#1479C2211yescharacteristic
49#1478C2211yescharacteristic
50#1478C2211yescharacteristic
51#1478C2211yescharacteristic
52#1478C2211yescharacteristic
53#1478C2211yescharacteristic
54#1478C2211yescharacteristic
55#1461C2211yescharacteristic
56#1460C2211yescharacteristic
57#1460C2211yescharacteristic
58#1460C2211yescharacteristic
59#1452C2211yescharacteristic
60#1452C2211yescharacteristic
61#1450C2211yescharacteristic
62#1450C2211yescharacteristic
63#1276C2211yescharacteristic
64#1276C2211yescharacteristic
65#1276C2211yescharacteristic
66#1276C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C22×SD16C2×C411yescharacteristic
69C22×D8C2×C411yescharacteristic
70C22×D8C2×C411yescharacteristic
71C2×#127C2×C411yescharacteristic
72C2×#124C2×C411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×D4—41no—
75C2×C4×D4C2311yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78#496—24no—
79C8×Q8—22no—
80C8×Q8—22no—
81C4×SD16—21no—
82C4×SD16—21no—
83C4×SD16—21no—
84C4×SD16—21no—
85C4×SD16—21no—
86C4×SD16—21no—
87C4×SD16—21no—
88C4×SD16—21no—
89C4×D8—21no—
90C4×D8—21no—
91C4×D8—21no—
92C4×D8—21no—
93C4×D8—21no—
94C4×D8—21no—
95C4×D8—21no—
96C4×D8—21no—
97#436—22no—
98#436—22no—
99#436—22no—
100#436—22no—
101C8×D4—22no—
102C8×D4—22no—
103#431—41no—
104#431C2311yescharacteristic
105#431C2311yescharacteristic
106#431C2311yescharacteristic
107#431C2311yescharacteristic
108#431C2311yescharacteristic
109#431C2311yescharacteristic
110#431C2311yescharacteristic
111#431C2311yescharacteristic
112C2×#107C2×C411yescharacteristic
113C2×#107C2×C411yescharacteristic
114C2×#106C2×C411yescharacteristic
115C2×#106C2×C411yescharacteristic
116C2×#103C2×C411yescharacteristic
117C2×#103C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×#102C2×C411yescharacteristic
123C2×#102C2×C411yescharacteristic
124#407—41no—
125#407—41no—
126C2×C22⋊C8—41no—
127C2×#97C2×C411yescharacteristic
128C2×#97C2×C411yescharacteristic
129C2×#97C2×C411yescharacteristic
130C2×#97C2×C411yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133#391C2×C411yescharacteristic
134#390—41no—
135#390C2×C411yescharacteristic
136#390C2×C411yescharacteristic
137#390C2×C411yescharacteristic
138#388—41no—
139#384C2×C411yescharacteristic
140#382C2×C411yescharacteristic
141C4×C22⋊C4—41no—
142C43C2311yescharacteristic
143#335—21no—
144#335—21no—
145#335D412yescomplemented
146#335D412yescomplemented
147#334—21no—
148#334—21no—
149#334D412yescomplemented
150#334D412yescomplemented
151Q8⋊C8—22no—
152Q8⋊C8—22no—
153D4⋊C8—22no—
154D4⋊C8—22no—
155D4⋊C8—22no—
156D4⋊C8—22no—
157D4⋊C8—22no—
158D4⋊C8—22no—
159#324—41no—
160#323—41no—
161#323—41no—
162#323—41no—
163#322—21no—
164#322—21no—
165#322—21no—
166#322—21no—
167#322—21no—
168#322—21no—
169C82—21no—
170C82—21no—
171C22×Q8C22×C411yescharacteristic
172C22×D4—41no—
173C22×D4C22×C411yescharacteristic
174C22×D4C22×C411yescharacteristic
175C22×D4C22×C411yescharacteristic
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C2×SD16—22no—
180C2×SD16—21no—
181C2×SD16—21no—
182C2×SD16—21no—
183C2×SD16—21no—
184C2×SD16—22no—
185C2×SD16—41no—
186C2×SD16M1614yescomplemented
187C2×D8—22no—
188C2×D8—21no—
189C2×D8—21no—
190C2×D8—21no—
191C2×D8—21no—
192C2×D8—22no—
193C2×D8—22no—
194C2×D8—41no—
195C2×M16—42no—
196C2×M16—42no—
197C22×C8—42no—
198C22×C8C22×C411yescharacteristic
199C22×C8C22×C411yescharacteristic
200C22×C8C22×C411yescharacteristic
201C22×C8C22×C411yescharacteristic
202#127—22no—
203#127—22no—
204#124—22no—
205#124—22no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—84no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C2411yescharacteristic
242#107—21no—
243#107—21no—
244#107—21no—
245#107—21no—
246#106—21no—
247#106—21no—
248#106—21no—
249#106—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—42no—
255C4⋊C8C2×D412yes—
256C4⋊C8C2×D412yes—
257C4⋊C8C2×D412yes—
258C4⋊C8C2×D412yes—
259#103—21no—
260#103—21no—
261#103—21no—
262#103—21no—
263#102—21no—
264#102—21no—
265#102—21no—
266#102—21no—
267#102—21no—
268#102—21no—
269#102—21no—
270#102—21no—
271#102—21no—
272#102—21no—
273#102—21no—
274#102—21no—
275C22⋊C8—21no—
276C22⋊C8—21no—
277C22⋊C8—21no—
278C22⋊C8—21no—
279C22⋊C8—21no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—42no—
284C22⋊C8—42no—
285C22⋊C8—42no—
286#97—22no—
287#97—22no—
288#97—22no—
289#97—41no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—42no—
299C4×C8C2×D411yescharacteristic
300C4×C8C2×D411yescharacteristic
301C4×C8C2×D411yescharacteristic
302C4×C8C2×D411yescharacteristic
303C4×C8#4111yescharacteristic
304C4×C8#4111yescharacteristic
305C4×C8#4111yescharacteristic
306C4×C8#4111yescharacteristic
307#95—41no—
308#95—41no—
309C24—21no—
310C24—21no—
311C24—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8C2×M1612yes—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—44no—
336C2×D4—82no—
337C2×D4—82no—
338C2×D4C2×M1612yes—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4C23×C411yescharacteristic
356C22×C4H(5,𝔽2)11yescharacteristic
357C22×C4H(5,𝔽2)11yescharacteristic
358SD16—24no—
359SD16—24no—
360SD16—42no—
361SD16—42no—
362D8—42no—
363D8—42no—
364D8—42no—
365D8—42no—
366M16—162no—
367M16—162no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—162no—
383C2×C8C4×D411yescharacteristic
384C2×C8C4×D411yescharacteristic
385C2×C8C4×D411yescharacteristic
386C2×C8C4×D411yescharacteristic
387C2×C8C4×D411yescharacteristic
388C2×C8C4×D411yescharacteristic
389C2×C8C4×D411yescharacteristic
390C2×C8C4×D411yescharacteristic
391C2×C8C2×M1612yes—
392C2×C8#13112yes—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—82no—
404C22⋊C4—44no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—42no—
418C42—41no—
419C42—41no—
420C42—81no—
421C42#13112yes—
422C42C22×D411yescharacteristic
423C42C22×D411yescharacteristic
424C42C2×#4111yescharacteristic
425C42C2×#4111yescharacteristic
426C23—22no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23#51811yescharacteristic
442Q8—22no—
443Q8—22no—
444Q8—41no—
445Q8—41no—
446D4—22no—
447D4—22no—
448D4—41no—
449D4—41no—
450D4—41no—
451D4—41no—
452D4—41no—
453D4—41no—
454D4—41no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—88no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
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—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—161no—
482C2×C4—161no—
483C2×C4—161no—
484C2×C4C2×C4×D411yescharacteristic
485C2×C4C2×C4×D411yescharacteristic
486C2×C4D4211yescharacteristic
487C2×C4#55011yescharacteristic
488C2×C4#55011yescharacteristic
489C2×C4#55911yescharacteristic
490C2×C4C22×M1611yescharacteristic
491C2×C4C2×#13111yescharacteristic
492C2×C4C2×#13111yescharacteristic
493C2×C4#56811yescharacteristic
494C8—21no—
495C8—21no—
496C8—21no—
497C8—21no—
498C8—21no—
499C8—21no—
500C8—42no—
501C8—42no—
502C8—42no—
503C8—42no—
504C8—81no—
505C8—81no—
506C8—81no—
507C8—81no—
508C8#43512yes—
509C8#43512yes—
510C22—21no—
511C22—21no—
512C22—42no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—161no—
525C22—161no—
526C22—161no—
527C22#236611yescharacteristic
528C22#297411yescharacteristic
529C22#298111yescharacteristic
530C4—21no—
531C4—21no—
532C4—21no—
533C4—42no—
534C4—42no—
535C4—81no—
536C4—81no—
537C4—81no—
538C4—81no—
539C4—81no—
540C4C2×#43511yescharacteristic
541C4#293111yescharacteristic
542C4D4×M1611yescharacteristic
543C4#293511yescharacteristic
544C4#328812yes—
545C2—21no—
546C2—21no—
547C2—82no—
548C2—161no—
549C2—161no—
550C2#2317411yescharacteristic
551C2#2319711yescharacteristic
552C2#2320211yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22#2366
prime radicalC2×#496C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17895C2semidirect
3. #17878C2semidirect
4. #17877C2semidirect
5. #17855C2semidirect
6. #9672C2semidirect
7. #9670C2semidirect
8. #9668C2semidirect
9. #9667C2semidirect
10. #8963C2semidirect
11. #8961C2semidirect
12. #8960C2semidirect
13. #8959C2semidirect
14. #8269C2semidirect
15. #8268C2semidirect
145. #335D4semidirect
146. #335D4semidirect
149. #334D4semidirect
150. #334D4semidirect
186. C2×SD16M16semidirect

Maximal subgroups

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

subgroupindex
C4×#4962
#178952
#178782
#178772
#178552
#96722
#96702
#96682
#96672
#89632
#89612
#89602
#89592
#82692
#82682