Finite Group Databasebeta

← #227018

Subgroups — #227018

Order 512.

769 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23095C211yesmaximal, characteristic
2#22997C211yesmaximal, characteristic
3#22542C211yesmaximal, characteristic, complemented
4#22521C211yesmaximal, characteristic, complemented
5C2×#1626C211yesmaximal, characteristic
6#9822C211yesmaximal, characteristic
7#9565C211yesmaximal, characteristic, complemented
8#9562C211yesmaximal, characteristic, complemented
9#9222C211yesmaximal, characteristic
10#9222C211yesmaximal, characteristic
11#9057C211yesmaximal, characteristic
12#8896C211yesmaximal, characteristic, complemented
13#8891C211yesmaximal, characteristic, complemented
14C8×#103C211yesmaximal, characteristic, complemented
15C8×#103C211yesmaximal, characteristic, complemented
16C2×#557C2211yescharacteristic
17C2×#486C2211yescharacteristic
18C2×#460C2211yescharacteristic
19C2×#460C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×C8×Q8C2211yescharacteristic
22C2×C4×SD16—21no—
23#2366C2211yescharacteristic
24#2355C2211yescharacteristic
25Q8×C4⋊C4C2211yescharacteristic
26#1991—21no—
27#1990C2211yescharacteristic
28#1989—21no—
29#1988C2211yescharacteristic
30#1981—21no—
31#1981C2211yescharacteristic
32#1968—21no—
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1956—21no—
36#1956C2211yescharacteristic
37#1948C2211yescharacteristic
38#1948C2211yescharacteristic
39#1934—21no—
40#1926—21no—
41#1880—21no—
42#1877—21no—
43#1864—21no—
44#1834C2211yescharacteristic
45#1805—21no—
46#1803—21no—
47#1803C2211yescharacteristic
48#1803C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C4×#103—21no—
52C4×#103—21no—
53C4×#103C2211yescharacteristic
54C4×#103C2211yescharacteristic
55C4×#102—21no—
56#1626C418yescomplemented
57C2×#332C2211yescharacteristic
58C2×#331C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×C82C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×SD16—21no—
70#557—22no—
71#557—22no—
72#557—41no—
73#557C2×C414yescomplemented
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8C2311yescharacteristic
76C2×#127C2311yescharacteristic
77C2×#125—21no—
78C2×#125—41no—
79C2×C4×Q8—21no—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8C2311yescharacteristic
83C2×C4×Q8C2311yescharacteristic
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86#486—41no—
87#486C2×C414yes—
88#460—22no—
89#460—22no—
90#460C2×C414yes—
91#460C2×C414yes—
92C8×Q8—22no—
93C8×Q8—22no—
94C8×Q8—41no—
95C8×Q8—41no—
96C4×SD16—44no—
97C2×#106—21no—
98C2×#106—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103D411yescharacteristic
112C2×#103D411yescharacteristic
113C2×#103D411yescharacteristic
114C2×#103D411yescharacteristic
115C2×#102—21no—
116C2×#102—21no—
117C2×#102—21no—
118C2×#102—21no—
119C2×#102—21no—
120C2×#102—21no—
121C2×#102—41no—
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8—21no—
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129#391—21no—
130#390—21no—
131#390—21no—
132#389—21no—
133#389—41no—
134#387—41no—
135#386—41no—
136#384—21no—
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#382—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4C2311yescharacteristic
144C4×C4⋊C4C2311yescharacteristic
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—41no—
154#340D411yescharacteristic
155#340D411yescharacteristic
156#340D411yescharacteristic
157#340D411yescharacteristic
158#336—21no—
159#336—21no—
160#332C2×C414yescomplemented
161#331—41no—
162Q8⋊C8C2×C414yescomplemented
163Q8⋊C8C2×C414yescomplemented
164C82—41no—
165C22×Q8—21no—
166C22×Q8C2×D411yescharacteristic
167C22×Q8C2×D411yescharacteristic
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C23×C4—41no—
172C2×SD16—24no—
173C2×SD16—24no—
174C2×SD16—42no—
175C2×SD16—42no—
176C2×SD16—81no—
177C2×SD16—81no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C4⋊Q8—24no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—42no—
192C4⋊Q8C22×C412yes—
193#127—21no—
194#127—21no—
195#127—21no—
196#127C22×C412yes—
197#125—24no—
198#125—42no—
199#125—44no—
200#125—44no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—24no—
206C4×Q8—24no—
207C4×Q8—42no—
208C4×Q8—42no—
209C4×Q8—42no—
210C4×Q8—81no—
211C4×Q8C22×C412yes—
212C4×Q8C22×C412yes—
213C4×Q8#4112yes—
214C4×Q8#4112yes—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—81no—
218C4×D4—81no—
219C2×C4⋊C4—21no—
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⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C22⋊C4—41no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—41no—
248C2×C42—41no—
249C2×C42—41no—
250C2×C42C2411yescharacteristic
251#106—42no—
252#106—81no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—42no—
260C4⋊C8—42no—
261C4⋊C8C22×C412yes—
262C4⋊C8C22×C412yes—
263#103—22no—
264#103—22no—
265#103—22no—
266#103—42no—
267#103—42no—
268#103—42no—
269#103—81no—
270#103—81no—
271#103—81no—
272#103SD1614yescomplemented
273#102—24no—
274#102—24no—
275#102—44no—
276#102—42no—
277#102—42no—
278#102—81no—
279#102—81no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—42no—
286C4×C8—42no—
287C4×C8—42no—
288C4×C8—81no—
289C4×C8C22×C412yes—
290#95—21no—
291#95—21no—
292#95—41no—
293#95—41no—
294#95—41no—
295#95—41no—
296C24—41no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—22no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8C4×D412yes—
312C2×Q8C4×D412yes—
313C2×Q8C2×SD1612yes—
314C2×Q8#13512yes—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—81no—
330C2×D4—81no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—161no—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4H(5,𝔽2)11yescharacteristic
348SD16—48no—
349SD16—84no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—41no—
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×C8C2×SD1612yes—
376C2×C8#13512yes—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—82no—
405C4⋊C4—81no—
406C4⋊C4—81no—
407C4⋊C4—161no—
408C4⋊C4C4×D412yes—
409C4⋊C4C4×D412yes—
410C4⋊C4C2×SD1612yes—
411C4⋊C4#13512yes—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—161no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—44no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—82no—
425C42—161no—
426C42C23×C411yescharacteristic
427C42C2×#4111yescharacteristic
428C42C2×#4111yescharacteristic
429C42H(5,𝔽2)11yescharacteristic
430C23—42no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23D4211yescharacteristic
438Q8—22no—
439Q8—22no—
440Q8—22no—
441Q8—22no—
442Q8—22no—
443Q8—41no—
444Q8—41no—
445Q8—42no—
446Q8—42no—
447Q8—81no—
448Q8—81no—
449Q8C4×SD1614yes—
450D4—42no—
451D4—42no—
452D4—42no—
453D4—42no—
454D4—82no—
455D4—81no—
456D4—81no—
457D4—81no—
458D4—81no—
459D4—161no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
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—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—164no—
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C2×C4×D411yescharacteristic
498C2×C4#51811yescharacteristic
499C2×C4#54611yescharacteristic
500C2×C4#54611yescharacteristic
501C2×C4#55111yescharacteristic
502C2×C4C22×SD1611yescharacteristic
503C2×C4C2×#13511yescharacteristic
504C2×C4C2×#13511yescharacteristic
505C2×C4C2×#13511yescharacteristic
506C2×C4#57611yescharacteristic
507C2×C4#57711yescharacteristic
508C2×C4#57711yescharacteristic
509C2×C4#57811yescharacteristic
510C8—22no—
511C8—22no—
512C8—42no—
513C8—42no—
514C8—42no—
515C8—41no—
516C8—41no—
517C8—81no—
518C8—81no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—161no—
523C22—161no—
524C22—161no—
525C22—161no—
526C22#235811yescharacteristic
527C22D4×SD1611yescharacteristic
528C22#328211yescharacteristic
529C22#328511yescharacteristic
530C22#328711yescharacteristic
531C22#329011yescharacteristic
532C22#329011yescharacteristic
533C4—24no—
534C4—21no—
535C4—21no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—81no—
542C4—81no—
543C4—82no—
544C4—161no—
545C4—161no—
546C4C2×C4×SD1611yescharacteristic
547C4C4×#13511yescharacteristic
548C4#294611yescharacteristic
549C4#294711yescharacteristic
550C4#329811yescharacteristic
551C4#330111yescharacteristic
552C4#330111yescharacteristic
553C4#330511yescharacteristic
554C2—162no—
555C2#1997811yescharacteristic
556C2#1998511yescharacteristic
557C2#2000511yescharacteristic
558C2#2001511yescharacteristic
559C2#2287511yescharacteristic
560C2#2287711yescharacteristic
561C2#2289511yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22542C2semidirect
4. #22521C2semidirect
7. #9565C2semidirect
8. #9562C2semidirect
12. #8896C2semidirect
13. #8891C2semidirect
14. C8×#103C2semidirect
15. C8×#103C2semidirect
56. #1626C4semidirect
73. #557C2×C4semidirect
160. #332C2×C4semidirect
162. Q8⋊C8C2×C4semidirect
163. Q8⋊C8C2×C4semidirect
272. #103SD16semidirect

Maximal subgroups

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

subgroupindex
#230952
#229972
#225422
#225212
C2×#16262
#98222
#95652
#95622
#92222
#90572
#88962
#88912
C8×#1032