Finite Group Databasebeta

← #229642

Subgroups — #229642

Order 512.

746 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23150C211yesmaximal, characteristic, complemented
2D4×C4⋊C8C211yesmaximal, characteristic
3C4×#461C211yesmaximal, characteristic
4C2×#1582C211yesmaximal, characteristic
5C2×C8×SD16C211yesmaximal, characteristic
6#9643C211yesmaximal, characteristic, complemented
7#9605C211yesmaximal, characteristic, complemented
8#9557C211yesmaximal, characteristic, complemented
9#9553C211yesmaximal, characteristic, complemented
10#8945C211yesmaximal, characteristic
11#8945C211yesmaximal, characteristic
12#8850C211yesmaximal, characteristic, complemented
13#8846C211yesmaximal, characteristic, complemented
14#8264C211yesmaximal, characteristic, complemented
15#8174C211yesmaximal, characteristic
16C2×#461C411yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23C22×C4⋊C8—21no—
24C4×C4⋊Q8C2211yescharacteristic
25C42×D4C2211yescharacteristic
26#1984C2211yescharacteristic
27#1984C2211yescharacteristic
28#1981C411yescharacteristic
29#1948C411yescharacteristic
30#1940—21no—
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1927C411yescharacteristic
34#1927C411yescharacteristic
35#1917—21no—
36#1916—21no—
37#1877C411yescharacteristic
38#1877C411yescharacteristic
39#1850—21no—
40#1850—21no—
41#1842C2211yescharacteristic
42#1802C411yescharacteristic
43#1797—21no—
44#1771—21no—
45#1767C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×#103C2211yescharacteristic
50#1744C2211yescharacteristic
51#1582—24no—
52C8×SD16—24no—
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×SD16C2×C411yescharacteristic
71C23×C8—21no—
72C23×C8—21no—
73C2×C4⋊Q8C2×C411yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4C2311yescharacteristic
79C22×C42—21no—
80#461—24no—
81#446—22no—
82#446—41no—
83C8×Q8—22no—
84C8×Q8—41no—
85C4×SD16—22no—
86C4×SD16—22no—
87C4×SD16—22no—
88C4×SD16—22no—
89C8×D4—22no—
90C8×D4—22no—
91C8×D4—41no—
92C8×D4—41no—
93C2×#106C2×C411yescharacteristic
94C2×#106C2×C411yescharacteristic
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—44no—
102C2×C4⋊C8C2×C411yescharacteristic
103C2×C4⋊C8C2×C411yescharacteristic
104C2×C4⋊C8D411yescharacteristic
105C2×C4⋊C8D411yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#103C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×#97—21no—
121C2×#97—21no—
122C2×C4×C8—21no—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129#391C2×C411yescharacteristic
130#391C2×C411yescharacteristic
131#389—21no—
132#389—21no—
133#384C2×C411yescharacteristic
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142C43C2311yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#336—21no—
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#336C2×C411yescharacteristic
157#336C2×C411yescharacteristic
158#336D411yescharacteristic
159#336D411yescharacteristic
160#334—22no—
161#334—22no—
162Q8⋊C8—22no—
163Q8⋊C8—22no—
164Q8⋊C8—41no—
165Q8⋊C8—41no—
166D4⋊C8—22no—
167D4⋊C8D414yes—
168#322—22no—
169C82—22no—
170C22×Q8C22×C411yescharacteristic
171C22×Q8C22×C411yescharacteristic
172C22×D4C22×C411yescharacteristic
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C2×SD16—22no—
177C2×SD16—22no—
178C2×SD16—22no—
179C2×SD16—22no—
180C2×SD16—22no—
181C2×SD16M1614yes—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—42no—
191C22×C8—42no—
192C22×C8—81no—
193C22×C8—81no—
194C22×C8C22×C411yescharacteristic
195C22×C8C22×C411yescharacteristic
196C22×C8C22×C411yescharacteristic
197C22×C8C22×C411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202C4⋊Q8—22no—
203C4⋊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—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—41no—
218C4×D4C2×D412yes—
219C4×D4#4112yes—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42#4111yescharacteristic
247C2×C42C2411yescharacteristic
248#106—22no—
249#106—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—41no—
260C4⋊C8—42no—
261C4⋊C8—42no—
262C4⋊C8—42no—
263C4⋊C8—42no—
264C4⋊C8—42no—
265C4⋊C8—81no—
266C4⋊C8SD1614yescomplemented
267C4⋊C8C2×D412yes—
268#103—22no—
269#103—22no—
270#103—22no—
271#103—22no—
272#103—22no—
273#103—22no—
274#102—22no—
275#102—22no—
276C22⋊C8—42no—
277C22⋊C8—42no—
278#97—42no—
279#97—42no—
280C4×C8—21no—
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—21no—
290C4×C8—21no—
291C4×C8—42no—
292C4×C8C2×D412yes—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300C24—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8C2×M1612yes—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4C4×D412yes—
317C2×D4C2×M1612yes—
318C2×D4#13112yes—
319C22×C4—24no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—81no—
338C22×C4C4×D411yescharacteristic
339C22×C4C4×D411yescharacteristic
340C22×C4C4×D411yescharacteristic
341C22×C4C4×D411yescharacteristic
342C22×C4C23×C411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C2×#4111yescharacteristic
345SD16—28no—
346SD16—44no—
347C2×C8—21no—
348C2×C8—21no—
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—22no—
359C2×C8—22no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
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×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—84no—
387C2×C8—84no—
388C2×C8C2×M1612yes—
389C2×C8C2×SD1612yes—
390C2×C8C2×SD1612yes—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—81no—
407C4⋊C4C4×D412yes—
408C4⋊C4#13112yes—
409C22⋊C4—24no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—42no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—42no—
428C42—42no—
429C42C2×SD1612yes—
430C42#13512yes—
431C42C22×D411yescharacteristic
432C42C2×#4111yescharacteristic
433C42H(5,𝔽2)11yescharacteristic
434C422−1+411yescharacteristic
435C23—22no—
436C23—22no—
437C23—22no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23C2×C4×D411yescharacteristic
443Q8—22no—
444Q8—22no—
445Q8—41no—
446Q8—41no—
447Q8—41no—
448Q8—41no—
449Q8—41no—
450Q8—41no—
451D4—22no—
452D4—22no—
453D4—22no—
454D4#43514yes—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—48no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—84no—
494C2×C4C4×SD1612yes—
495C2×C4C4×SD1612yes—
496C2×C4C2×C4×D411yescharacteristic
497C2×C4#52011yescharacteristic
498C2×C4D4211yescharacteristic
499C2×C4#54711yescharacteristic
500C2×C4#54811yescharacteristic
501C2×C4#55311yescharacteristic
502C2×C4C22×M1611yescharacteristic
503C2×C4C2×#13111yescharacteristic
504C2×C4C2×#13111yescharacteristic
505C2×C4#56811yescharacteristic
506C2×C4C22×SD1611yescharacteristic
507C2×C4C2×#13511yescharacteristic
508C2×C4#57711yescharacteristic
509C2×C4#57811yescharacteristic
510C8—22no—
511C8—22no—
512C8—22no—
513C8—42no—
514C8—41no—
515C8—41no—
516C8—41no—
517C8—41no—
518C8—41no—
519C8—42no—
520C8—81no—
521C8—81no—
522C8—81no—
523C8—81no—
524C22—24no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22#236111yescharacteristic
532C22#292911yescharacteristic
533C22D4×M1611yescharacteristic
534C22C2×C4×SD1611yescharacteristic
535C22#294711yescharacteristic
536C22#295511yescharacteristic
537C22#295711yescharacteristic
538C4—22no—
539C4—21no—
540C4—21no—
541C4—22no—
542C4—41no—
543C4—41no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4—81no—
551C4C2×#43511yescharacteristic
552C4#293511yescharacteristic
553C4#299011yescharacteristic
554C4#299211yescharacteristic
555C4D4×SD1611yescharacteristic
556C4#328311yescharacteristic
557C4#335911yescharacteristic
558C4#336011yescharacteristic
559C2—44no—
560C2#1790211yescharacteristic
561C2#1792111yescharacteristic
562C2M16×SD1611yescharacteristic
563C2#1805111yescharacteristic
564C2#2298711yescharacteristic
565C2#2299611yescharacteristic
566C2#2301711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23150C2semidirect
6. #9643C2semidirect
7. #9605C2semidirect
8. #9557C2semidirect
9. #9553C2semidirect
12. #8850C2semidirect
13. #8846C2semidirect
14. #8264C2semidirect
266. C4⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#231502
D4×C4⋊C82
C4×#4612
C2×#15822
C2×C8×SD162
#96432
#96052
#95572
#95532
#89452
#88502
#88462
#82642
#81742