Finite Group Databasebeta

← #163927

Subgroups — #163927

Order 512.

1124 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34250C211yesmaximal, characteristic
2#22473C211yesmaximal, characteristic
3#22473C211yesmaximal, characteristic
4#7646C212yesmaximal, complemented
5#7632C212yesmaximal, complemented
6#7610C212yesmaximal, complemented
7#7607C212yesmaximal, complemented
8#7598C211yesmaximal, characteristic, complemented
9#7576C211yesmaximal, characteristic, complemented
10#7573C211yesmaximal, characteristic, complemented
11#7564C211yesmaximal, characteristic, complemented
12C2×#520C411yescharacteristic
13C2×#520C411yescharacteristic
14C2×C4×#41C2211yescharacteristic
15C2×#436—21no—
16C2×#436—21no—
17#2351—21no—
18Q8×C4⋊C4—21no—
19#2349—21no—
20#2349—22no—
21D4×C4⋊C4—21no—
22D4×C4⋊C4—22no—
23#2317C412yes—
24#2317C411yescharacteristic
25#2316C412yes—
26#2316C411yescharacteristic
27#2296C2211yescharacteristic
28#2296C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2212yes—
31#1984—21no—
32#1984—21no—
33#1870—22no—
34#1870—22no—
35#1817C2212yes—
36#1817C2212yes—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C4×C22⋊C8C2211yescharacteristic
40C4×C22⋊C8C2211yescharacteristic
41#1728C2212yes—
42#1728C2212yes—
43#1271C2211yescharacteristic
44#1271C2211yescharacteristic
45#1271C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1271C2212yes—
51#1271C2212yes—
52#1271C2212yes—
53C22×#41C2×C411yescharacteristic
54C22×M16—22no—
55C22×M16—22no—
56#520—24no—
57#520—24no—
58#520—44no—
59C4×#41—44no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×D4—22no—
65C2×C4×D4—21no—
66C2×C4×D4—22no—
67C2×C4×D4—21no—
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D412yes—
71C2×C4×D4D412yes—
72C2×#117C2×C412yes—
73C2×#117C2×C411yescharacteristic
74C2×#117C2×C412yes—
75C2×#117C2×C411yescharacteristic
76C2×#117C2×C412yes—
77C2×#117C2×C411yescharacteristic
78C22×C4⋊C4—22no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2×C412yes—
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4C2×C412yes—
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2312yes—
86#436—44no—
87#436—44no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C4×C8—21no—
97C2×C4×C8—21no—
98#389—21no—
99#389—22no—
100#389—21no—
101#389—22no—
102#389—21no—
103#389—22no—
104#389—21no—
105#389—22no—
106#389C2×C411yescharacteristic
107#389C2×C412yes—
108#389C2×C412yes—
109#389C2×C412yes—
110#389C2×C412yes—
111#389C2×C411yescharacteristic
112#389C2×C412yes—
113#389C2312yes—
114#389C2311yescharacteristic
115#389C2311yescharacteristic
116#385—22no—
117#385—21no—
118#385—22no—
119#385—21no—
120#385—22no—
121#385—22no—
122#385—22no—
123#384—21no—
124#384—22no—
125#382—21no—
126#382—22no—
127#380—22no—
128#380—22no—
129#380—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—22no—
134C2×#95—22no—
135C2×#95—21no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#336C2312yes—
141#336C2312yes—
142#336C2311yescharacteristic
143#336C2311yescharacteristic
144#336C2311yescharacteristic
145#336C2311yescharacteristic
146C2×#41—24no—
147C2×#41—24no—
148C2×#41—22no—
149C2×#41—22no—
150C22×Q8C22⋊C411yescharacteristic
151C22×D4C22⋊C412yes—
152C22×D4C22⋊C411yescharacteristic
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4C22×C412yes—
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C2×M16—44no—
164C2×M16—44no—
165C2×M16—44no—
166C2×M16—44no—
167C2×M16—82no—
168C2×M16—82no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C4×Q8—28no—
176C4×Q8—42no—
177C4×Q8—42no—
178C4×Q8—44no—
179C4×D4—216no—
180C4×D4—28no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—44no—
184C4×D4—44no—
185C4×D4—48no—
186C4×D4—44no—
187#117—22no—
188#117—24no—
189#117—24no—
190#117—44no—
191#117—42no—
192#117—44no—
193#117—42no—
194#117M1614yescomplemented
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—81no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—81no—
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C412yes—
222C2×C4⋊C4M1614yescomplemented
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4C22⋊C412yes—
234C2×C22⋊C4C22⋊C412yes—
235C2×C22⋊C4C22⋊C412yes—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—21no—
240C2×C42—42no—
241C2×C42—44no—
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C412yes—
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2411yescharacteristic
249C4⋊C8—42no—
250C4⋊C8—42no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253C22⋊C8—44no—
254C22⋊C8—44no—
255C22⋊C8—81no—
256C22⋊C8—81no—
257C4×C8—42no—
258C4×C8—42no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—21no—
263#95—22no—
264#95—21no—
265#95—22no—
266#95—21no—
267#95—22no—
268#95—21no—
269C24C2×C22⋊C411yescharacteristic
270C24C2×C22⋊C412yes—
271#41—216no—
272#41—216no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8#10114yes—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—22no—
279C2×D4—24no—
280C2×D4—22no—
281C2×D4#9914yes—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—28no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—81no—
315C22×C4—82no—
316C22×C4—81no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—81no—
320C22×C4—82no—
321C22×C4—81no—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×M1612yes—
329C22×C4C2×M1612yes—
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333M16—88no—
334M16—88no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—41no—
352C2×C8—41no—
353C4⋊C4—28no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—44no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—42no—
371C4⋊C4—41no—
372C4⋊C4—42no—
373C4⋊C4—82no—
374C4⋊C4—81no—
375C4⋊C4—82no—
376C4⋊C4—81no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C4⋊C4—82no—
380C4⋊C4C2×M1612yes—
381C4⋊C4C2×M1614yes—
382C4⋊C4C2×M1612yes—
383C22⋊C4—28no—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—44no—
390C42—28no—
391C42—24no—
392C42—22no—
393C42—22no—
394C42—41no—
395C42—41no—
396C42—42no—
397C42—42no—
398C42—42no—
399C42—44no—
400C42—42no—
401C23—22no—
402C23—21no—
403C23—21no—
404C23—22no—
405C23—21no—
406C23—22no—
407C23—21no—
408C23—22no—
409C23—21no—
410C23—21no—
411C23—22no—
412C23—22no—
413C23#40712yes—
414C23C2×#9914yes—
415C23C22×C22⋊C411yescharacteristic
416Q8—28no—
417D4—216no—
418D4—28no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—84no—
470C2×C4—88no—
471C2×C4—164no—
472C2×C4—162no—
473C2×C4#40712yes—
474C2×C4C2×#9912yes—
475C2×C4C2×#10114yes—
476C2×C4C2×#10112yes—
477C2×C4C22×M1611yescharacteristic
478C2×C4C22×M1611yescharacteristic
479C2×C4#57611yescharacteristic
480C2×C4#57711yescharacteristic
481C2×C4#57711yescharacteristic
482C2×C4#57811yescharacteristic
483C8—42no—
484C8—42no—
485C8—81no—
486C8—81no—
487C8—82no—
488C8—82no—
489C22—22no—
490C22—22no—
491C22—24no—
492C22—24no—
493C22—22no—
494C22—24no—
495C22—41no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—41no—
500C22—42no—
501C22—41no—
502C22—41no—
503C22C2×#40711yescharacteristic
504C22C22×#9911yescharacteristic
505C22C22×#10111yescharacteristic
506C22#289711yescharacteristic
507C22#289811yescharacteristic
508C22#290111yescharacteristic
509C22#290111yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—24no—
517C4—44no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—41no—
522C4—42no—
523C4—41no—
524C4—81no—
525C4—82no—
526C4—81no—
527C4#146314yes—
528C2—42no—
529C2—44no—
530C2C2×#146311yescharacteristic
531C2#1745311yescharacteristic
532C2#1745411yescharacteristic
533C2#1760311yescharacteristic
534C2#1760811yescharacteristic
535C2#1769211yescharacteristic
536C2#1777711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7646C2semidirect
5. #7632C2semidirect
6. #7610C2semidirect
7. #7607C2semidirect
8. #7598C2semidirect
9. #7576C2semidirect
10. #7573C2semidirect
11. #7564C2semidirect
194. #117M16semidirect
222. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
#342502
#224732
#76462
#76322
#76102
#76072
#75982
#75762
#75732
#75642