Finite Group Databasebeta

← #231506

Subgroups — #231506

Order 512.

640 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23800C211yesmaximal, characteristic
2#17878C211yesmaximal, characteristic, complemented
3#17877C211yesmaximal, characteristic
4#17856C211yesmaximal, characteristic, complemented
5#17855C211yesmaximal, characteristic, complemented
6#9672C211yesmaximal, characteristic, complemented
7#9671C211yesmaximal, characteristic, complemented
8#9668C211yesmaximal, characteristic
9#9667C211yesmaximal, characteristic, complemented
10#8961C211yesmaximal, characteristic, complemented
11#8960C211yesmaximal, characteristic, complemented
12#8960C211yesmaximal, characteristic, complemented
13#8957C211yesmaximal, characteristic, complemented
14#8270C211yesmaximal, characteristic, complemented
15#8270C211yesmaximal, characteristic, complemented
16#2982C2211yescharacteristic
17#2982C2211yescharacteristic
18#2975C2211yescharacteristic
19#2972C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×SD16C2211yescharacteristic
22C2×C4×SD16C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24C4×#124C2211yescharacteristic
25C4×#124C2211yescharacteristic
26#2097C411yescharacteristic
27#2097C411yescharacteristic
28#2092C411yescharacteristic
29#2092C411yescharacteristic
30#2015C411yescharacteristic
31#2015C411yescharacteristic
32#2012C411yescharacteristic
33#2012C411yescharacteristic
34#1744C2211yescharacteristic
35#1588—22no—
36#1586—22no—
37#1583—22no—
38#1581—22no—
39#1579—22no—
40SD16⋊C8—22no—
41C8×Q16—22no—
42C8×SD16—22no—
43#1564C2211yescharacteristic
44#1564C2211yescharacteristic
45#1564C2211yescharacteristic
46#1564C2211yescharacteristic
47#1479C2211yescharacteristic
48#1479C2211yescharacteristic
49#1479C2211yescharacteristic
50#1479C2211yescharacteristic
51#1478C2211yescharacteristic
52#1478C2211yescharacteristic
53#1478C2211yescharacteristic
54#1478C2211yescharacteristic
55#1461C2211yescharacteristic
56#1461C2211yescharacteristic
57#1460C2211yescharacteristic
58#1460C2211yescharacteristic
59#1452C2211yescharacteristic
60#1452C2211yescharacteristic
61#1450C2211yescharacteristic
62#1450C2211yescharacteristic
63#1276C2211yescharacteristic
64#1276C2211yescharacteristic
65#1276C2211yescharacteristic
66#1276C2211yescharacteristic
67C22×Q16C2×C411yescharacteristic
68C22×SD16C2×C411yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×D8C2×C411yescharacteristic
71C2×#124C2×C411yescharacteristic
72C2×#124C2×C411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C8×Q8—22no—
78C8×Q8—22no—
79C8×Q8—22no—
80C8×Q8—22no—
81C4×Q16—21no—
82C4×Q16—21no—
83C4×Q16—21no—
84C4×Q16—21no—
85C4×SD16—21no—
86C4×SD16—21no—
87C4×SD16—21no—
88C4×SD16—21no—
89C4×SD16—21no—
90C4×SD16—21no—
91C4×SD16—21no—
92C4×SD16—21no—
93C4×D8—21no—
94C4×D8—21no—
95C4×D8—21no—
96C4×D8—21no—
97#436—22no—
98#436—22no—
99C8×D4—22no—
100C8×D4—22no—
101#431—41no—
102#431—41no—
103#431C2311yescharacteristic
104#431C2311yescharacteristic
105#431C2311yescharacteristic
106#431C2311yescharacteristic
107#431C2311yescharacteristic
108#431C2311yescharacteristic
109#431C2311yescharacteristic
110#431C2311yescharacteristic
111C2×#107C2×C411yescharacteristic
112C2×#107C2×C411yescharacteristic
113C2×#106C2×C411yescharacteristic
114C2×#106C2×C411yescharacteristic
115C2×#103C2×C411yescharacteristic
116C2×#103C2×C411yescharacteristic
117C2×#103C2×C411yescharacteristic
118C2×#103C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×#102C2×C411yescharacteristic
123#407—41no—
124C2×C22⋊C8—41no—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127#391C2×C411yescharacteristic
128#391C2×C411yescharacteristic
129#390C2×C411yescharacteristic
130#390C2×C411yescharacteristic
131#388—41no—
132#388—41no—
133#382C2×C411yescharacteristic
134#382C2×C411yescharacteristic
135C4×C22⋊C4—41no—
136C4×C22⋊C4—41no—
137C43C2311yescharacteristic
138#336C2×C411yescharacteristic
139#336C2×C411yescharacteristic
140#336C2×C411yescharacteristic
141#336C2×C411yescharacteristic
142#335—21no—
143#335—21no—
144#335D412yes—
145#335D412yes—
146#334—21no—
147#334—21no—
148#334D412yes—
149#334D412yes—
150Q8⋊C8—22no—
151Q8⋊C8—22no—
152Q8⋊C8—22no—
153Q8⋊C8—22no—
154D4⋊C8—22no—
155D4⋊C8—22no—
156D4⋊C8—22no—
157D4⋊C8—22no—
158#324—41no—
159#324—41no—
160#323—41no—
161#323—41no—
162#322—21no—
163#322—21no—
164#322—21no—
165#322—21no—
166#322—21no—
167#322—21no—
168C82—21no—
169C82—21no—
170C22×Q8C22×C411yescharacteristic
171C22×Q8C22×C411yescharacteristic
172C22×D4C22×C411yescharacteristic
173C22×D4C22×C411yescharacteristic
174C23×C4—21no—
175C23×C4—21no—
176C2×Q16—22no—
177C2×Q16—22no—
178C2×Q16—21no—
179C2×Q16—21no—
180C2×SD16—22no—
181C2×SD16—22no—
182C2×SD16—22no—
183C2×SD16—21no—
184C2×SD16—21no—
185C2×SD16—21no—
186C2×SD16—21no—
187C2×SD16—41no—
188C2×D8—22no—
189C2×D8—21no—
190C2×D8—21no—
191C2×D8—41no—
192C2×M16—42no—
193C22×C8—42no—
194C22×C8C22×C411yescharacteristic
195C22×C8C22×C411yescharacteristic
196C22×C8C22×C411yescharacteristic
197C22×C8C22×C411yescharacteristic
198#124—22no—
199#124—22no—
200#124—22no—
201#124—41no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—21no—
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—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
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—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C2411yescharacteristic
237#107—21no—
238#107—21no—
239#107—21no—
240#107—21no—
241#106—21no—
242#106—21no—
243#106—21no—
244#106—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C4⋊C8C2×D412yes—
252C4⋊C8C2×D412yes—
253C4⋊C8C2×D412yes—
254C4⋊C8C2×D412yes—
255#103—21no—
256#103—21no—
257#103—21no—
258#103—21no—
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—
271C22⋊C8—21no—
272C22⋊C8—21no—
273C22⋊C8—21no—
274C22⋊C8—21no—
275C22⋊C8—21no—
276C22⋊C8—21no—
277C22⋊C8—21no—
278C22⋊C8—21no—
279C22⋊C8—42no—
280C22⋊C8—42no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—42no—
290C4×C8—42no—
291C4×C8C2×D411yescharacteristic
292C4×C8C2×D411yescharacteristic
293C4×C8C2×D411yescharacteristic
294C4×C8C2×D411yescharacteristic
295C4×C8#4111yescharacteristic
296C4×C8#4111yescharacteristic
297C4×C8#4111yescharacteristic
298C4×C8#4111yescharacteristic
299#95—41no—
300#95—41no—
301#95—41no—
302#95—41no—
303C24—21no—
304C24—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8#13112yes—
316C2×Q8#13112yes—
317C2×D4—21no—
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#13112yes—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4C23×C411yescharacteristic
344C22×C4H(5,𝔽2)11yescharacteristic
345C22×C4H(5,𝔽2)11yescharacteristic
346Q16—24no—
347Q16—24no—
348SD16—24no—
349SD16—24no—
350SD16—42no—
351SD16—42no—
352D8—42no—
353D8—42no—
354M16—162no—
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—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—162no—
372C2×C8C4×D411yescharacteristic
373C2×C8C4×D411yescharacteristic
374C2×C8C4×D411yescharacteristic
375C2×C8C4×D411yescharacteristic
376C2×C8C4×D411yescharacteristic
377C2×C8C4×D411yescharacteristic
378C2×C8C4×D411yescharacteristic
379C2×C8C4×D411yescharacteristic
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—42no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—81no—
410C42C22×D411yescharacteristic
411C42C22×D411yescharacteristic
412C42C2×#4111yescharacteristic
413C42C2×#4111yescharacteristic
414C23—22no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23#51811yescharacteristic
425Q8—22no—
426Q8—22no—
427Q8—22no—
428Q8—22no—
429Q8—41no—
430Q8—41no—
431Q8—41no—
432Q8—41no—
433D4—22no—
434D4—22no—
435D4—41no—
436D4—41no—
437D4—41no—
438D4—41no—
439D4—41no—
440D4—41no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—161no—
466C2×C4—161no—
467C2×C4C2×C4×D411yescharacteristic
468C2×C4C2×C4×D411yescharacteristic
469C2×C4D4211yescharacteristic
470C2×C4#55011yescharacteristic
471C2×C4#55011yescharacteristic
472C2×C4#55911yescharacteristic
473C2×C4C2×#13111yescharacteristic
474C2×C4C2×#13111yescharacteristic
475C2×C4C2×#13111yescharacteristic
476C2×C4#56811yescharacteristic
477C8—21no—
478C8—21no—
479C8—21no—
480C8—21no—
481C8—21no—
482C8—21no—
483C8—21no—
484C8—21no—
485C8—42no—
486C8—42no—
487C8—42no—
488C8—42no—
489C8—81no—
490C8—81no—
491C8—81no—
492C8—81no—
493C22—21no—
494C22—21no—
495C22—42no—
496C22—82no—
497C22—82no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—161no—
504C22—161no—
505C22#236611yescharacteristic
506C22#297611yescharacteristic
507C22#297711yescharacteristic
508C4—21no—
509C4—21no—
510C4—21no—
511C4—42no—
512C4—42no—
513C4—81no—
514C4—81no—
515C4—81no—
516C4—81no—
517C4—81no—
518C4—81no—
519C4#292911yescharacteristic
520C4#292911yescharacteristic
521C4#293311yescharacteristic
522C4#293311yescharacteristic
523C4#329012yes—
524C2—21no—
525C2—21no—
526C2—82no—
527C2—161no—
528C2#2317711yescharacteristic
529C2#2319711yescharacteristic
530C2#2319911yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22#2366
prime radicalC22×D8C2×C4
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. #17878C2semidirect
4. #17856C2semidirect
5. #17855C2semidirect
6. #9672C2semidirect
7. #9671C2semidirect
9. #9667C2semidirect
10. #8961C2semidirect
11. #8960C2semidirect
12. #8960C2semidirect
13. #8957C2semidirect
14. #8270C2semidirect
15. #8270C2semidirect

Maximal subgroups

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

subgroupindex
#238002
#178782
#178772
#178562
#178552
#96722
#96712
#96682
#96672
#89612
#89602
#89572
#82702