Finite Group Databasebeta

← #194112

Subgroups — #194112

Order 512.

1180 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3249C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic
3#21400C211yesmaximal, characteristic
4#21312C211yesmaximal, characteristic
5C2×#1493C211yesmaximal, characteristic
6#8987C212yesmaximal, complemented
7#8986C212yesmaximal, complemented
8#8975C212yesmaximal, complemented
9#8974C212yesmaximal, complemented
10#8045C211yesmaximal, characteristic
11C2×#1279C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#533C2211yescharacteristic
14#3249—24no—
15#3249C418yescomplemented
16C2×#477—21no—
17C2×#477C2212yes—
18C2×#467—22no—
19C2×#462—21no—
20C2×#462C2212yes—
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24#2370—22no—
25#2369—21no—
26#2365—21no—
27#2355—22no—
28#2355C2212yes—
29Q8×C4⋊C4C2212yes—
30#2329—22no—
31#2317—21no—
32C4×C22⋊Q8—22no—
33#2285C2211yescharacteristic
34#2274—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1989—22no—
39#1988C2212yes—
40#1987C2212yes—
41#1956C2211yescharacteristic
42#1956C2211yescharacteristic
43#1892—22no—
44C4×#103—22no—
45#1736—21no—
46#1736—21no—
47#1493—24no—
48C2×Q8⋊C8C2212yes—
49C2×Q8⋊C8C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C22×Q16—22no—
55C22×M16—21no—
56C23×C8—21no—
57C23×C8—21no—
58#533—22no—
59#533—22no—
60#533—22no—
61#533—22no—
62#533—22no—
63#533C2×C414yes—
64#533C2×C414yes—
65#533D414yes—
66C2×C4⋊Q8D411yescharacteristic
67C2×C4⋊Q8D411yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×#125—21no—
71C2×#125—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8D412yes—
76C2×C4×Q8—22no—
77C2×C4×Q8—21no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8C2312yes—
81C2×#117—21no—
82C2×#117—21no—
83C2×#117—21no—
84C22×C4⋊C4D411yescharacteristic
85C22×C4⋊C4D411yescharacteristic
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88#477—28no—
89#477—24no—
90#477C2×C418yescomplemented
91#467—216no—
92#462—28no—
93#462—24no—
94#462C2×C418yescomplemented
95C2×#107—22no—
96#423—22no—
97#423—22no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8C2×C414yes—
102C2×C4⋊C8C2312yes—
103C2×C4⋊C8C2312yes—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—22no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8—21no—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#391—24no—
114#391—22no—
115#389—22no—
116#389—22no—
117#389—24no—
118#389—22no—
119#387—24no—
120#387—24no—
121#387—24no—
122#385—24no—
123#384—22no—
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#382—22no—
127#382—21no—
128#380—21no—
129#380D412yes—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4D412yes—
140C4×C4⋊C4D412yes—
141C4×C4⋊C4C2312yes—
142C4×C22⋊C4—24no—
143C2×#95—21no—
144#340—22no—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150Q8⋊C8—24no—
151Q8⋊C8—24no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8C2×D412yes—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C2×Q16—28no—
162C2×Q16—82no—
163C2×Q16—82no—
164C2×M16—42no—
165C2×M16—42no—
166C2×M16—42no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—24no—
171C22×C8—22no—
172C22×C8—42no—
173C22×C8—42no—
174C22×C8—42no—
175C22×C8—81no—
176C22×C8—81no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8C22⋊C414yes—
186C4⋊Q8C22⋊C414yes—
187C4⋊Q8C22×C412yes—
188C4⋊Q8C22×C412yes—
189#125—24no—
190#125—24no—
191#125—24no—
192#125—42no—
193C22⋊Q8—24no—
194C22⋊Q8—24no—
195C22⋊Q8—24no—
196C22⋊Q8—24no—
197C22⋊Q8—24no—
198C22⋊Q8—42no—
199C22⋊Q8—42no—
200C22⋊Q8C22⋊C418yes—
201C4×Q8—24no—
202C4×Q8—24no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—24no—
206C4×Q8—24no—
207C4×Q8—42no—
208C4×Q8—42no—
209C4×Q8—82no—
210C4×Q8—82no—
211C4×Q8C22×C414yes—
212C4×Q8C2×D414yes—
213#117—24no—
214#117—24no—
215#117—24no—
216#117—42no—
217#117—42no—
218#117—42no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4Q1614yescomplemented
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—41no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—41no—
257C2×C42—81no—
258C2×C42—81no—
259C2×C42C22×C412yes—
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D412yes—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42#4112yes—
265C2×C42#4112yes—
266C2×C42C2411yescharacteristic
267#107—28no—
268C4⋊C8—24no—
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8—22no—
272C4⋊C8—42no—
273C4⋊C8C22×C414yes—
274#103—28no—
275#103—28no—
276#103—44no—
277#103—44no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—42no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291C24#12011yescharacteristic
292C2×Q8—22no—
293C2×Q8—24no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—81no—
309C2×Q8—81no—
310C2×Q8C2×C22⋊C414yes—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—81no—
328C22×C4—81no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4C2×C22⋊C412yes—
332C22×C4C4×D412yes—
333C22×C4C4×D412yes—
334C22×C4#12011yescharacteristic
335C22×C4#12011yescharacteristic
336C22×C4#12011yescharacteristic
337C22×C4#12112yes—
338C22×C4#12112yes—
339C22×C4#12312yes—
340C22×C4C2×Q1612yes—
341C22×C4#13512yes—
342C22×C4#13712yes—
343C22×C4#13712yes—
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347Q16—88no—
348M16—84no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—48no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—84no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—28no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—84no—
396C4⋊C4—84no—
397C4⋊C4C2×C22⋊C414yes—
398C4⋊C4C2×Q1614yes—
399C22⋊C4—28no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C42—24no—
403C42—24no—
404C42—42no—
405C42—42no—
406C42—44no—
407C42—44no—
408C42—82no—
409C42—84no—
410C42—84no—
411C42C2×C22⋊C412yes—
412C42C4×D412yes—
413C42C4×D412yes—
414C42#12112yes—
415C42#12112yes—
416C42#12312yes—
417C42C23×C411yescharacteristic
418C42C22×D411yescharacteristic
419C42C2×#4111yescharacteristic
420C42C2×#4111yescharacteristic
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23#38612yes—
426C23#45112yes—
427C23#45212yes—
428C23C2×#12011yescharacteristic
429Q8—24no—
430Q8—24no—
431Q8—44no—
432Q8—42no—
433Q8—42no—
434Q8—42no—
435Q8—42no—
436Q8—84no—
437Q8—82no—
438Q8—81no—
439Q8—81no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—84no—
473C2×C4—84no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4#38612yes—
479C2×C4#38612yes—
480C2×C4#38612yes—
481C2×C4C4×Q1614yes—
482C2×C4#44114yes—
483C2×C4#45112yes—
484C2×C4#45212yes—
485C2×C4#46214yes—
486C2×C4#46414yes—
487C2×C4C22×C22⋊C411yescharacteristic
488C2×C4C2×C4×D411yescharacteristic
489C2×C4C2×C4×D411yescharacteristic
490C2×C4C2×#12111yescharacteristic
491C2×C4C2×#12111yescharacteristic
492C2×C4C2×#12311yescharacteristic
493C2×C4C22×Q1611yescharacteristic
494C2×C4C2×#13511yescharacteristic
495C2×C4C2×#13711yescharacteristic
496C2×C4C2×#13711yescharacteristic
497C2×C4#57611yescharacteristic
498C2×C4#57711yescharacteristic
499C2×C4#57711yescharacteristic
500C2×C4#57811yescharacteristic
501C8—42no—
502C8—42no—
503C8—44no—
504C8—82no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22#187814yes—
512C22C2×#38611yescharacteristic
513C22C2×#45111yescharacteristic
514C22C2×#45211yescharacteristic
515C22#301211yescharacteristic
516C22#301311yescharacteristic
517C22#301811yescharacteristic
518C22#301811yescharacteristic
519C4—22no—
520C4—21no—
521C4—21no—
522C4—22no—
523C4—44no—
524C4—44no—
525C4—44no—
526C4—82no—
527C4—82no—
528C4—82no—
529C4—82no—
530C4—82no—
531C4#187814yes—
532C4#289711yescharacteristic
533C4#289811yescharacteristic
534C4C2×C4×Q1611yescharacteristic
535C4C2×#44111yescharacteristic
536C4C2×#46211yescharacteristic
537C4C2×#46411yescharacteristic
538C4#309811yescharacteristic
539C4#309911yescharacteristic
540C2—24no—
541C2#1866511yescharacteristic
542C2#1866711yescharacteristic
543C2#1867711yescharacteristic
544C2#1867911yescharacteristic
545C2#2091511yescharacteristic
546C2C2×#187811yescharacteristic
547C2#2094211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8987C2semidirect
7. #8986C2semidirect
8. #8975C2semidirect
9. #8974C2semidirect
15. #3249C4semidirect
90. #477C2×C4semidirect
94. #462C2×C4semidirect
241. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32492
#342132
#214002
#213122
C2×#14932
#89872
#89862
#89752
#89742
#80452
C2×#12792