Finite Group Databasebeta

← #215075

Subgroups — #215075

Order 512.

855 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35738C211yesmaximal, characteristic
2#20730C211yesmaximal, characteristic, complemented
3#17371C211yesmaximal, characteristic, complemented
4#9494C212yesmaximal
5#9390C212yesmaximal, complemented
6#9190C212yesmaximal, complemented
7#9032C212yesmaximal, complemented
8#8774C212yesmaximal, complemented
9#8372C212yesmaximal, complemented
10C2×#554C2211yescharacteristic
11C2×#519C2211yescharacteristic
12#2983C2212yes—
13#2921C2212yes—
14#2634C411yescharacteristic
15#2634C2211yescharacteristic
16#2628C411yescharacteristic
17#2611C411yescharacteristic
18#2611C2211yescharacteristic
19#2598C411yescharacteristic
20#2598C2211yescharacteristic
21#2568C411yescharacteristic
22#2536C411yescharacteristic
23#2477—21no—
24#2477—21no—
25#2465—21no—
26#2465—21no—
27#2421—21no—
28#2409—21no—
29#2398C411yescharacteristic
30#2398C2211yescharacteristic
31#2392C411yescharacteristic
32#2360—21no—
33#2356C2211yescharacteristic
34#2352—21no—
35#1822—21no—
36#1801—21no—
37#1801—21no—
38#1784—21no—
39C4×#101C2212yes—
40C4×#101C2212yes—
41#1528C2214yes—
42#1490C2214yes—
43#1467—21no—
44#1467C2212yes—
45#1464—21no—
46#1464C2212yes—
47#1461—21no—
48#1458—21no—
49#1300C2214yes—
50#1288C2214yes—
51C23×Q8D411yescharacteristic
52#554—44no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125C2×C411yescharacteristic
55C2×#125C2311yescharacteristic
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60#519—44no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8C2311yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117C2×C411yescharacteristic
75C2×#117D411yescharacteristic
76C2×#117D411yescharacteristic
77C2×#117D411yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C4⋊C4—21no—
80#446—44no—
81#433—44no—
82#432—22no—
83#432C2314yes—
84#431—22no—
85#423—22no—
86C2×#101—22no—
87C2×#101—22no—
88#407—22no—
89C4×M16C2314yes—
90#400—21no—
91#400—21no—
92#400C2×C411yescharacteristic
93#400C2×C411yescharacteristic
94#400C2×C411yescharacteristic
95#400C2×C411yescharacteristic
96#400C2×C411yescharacteristic
97#400C2×C411yescharacteristic
98#399C2×C411yescharacteristic
99#399C2×C411yescharacteristic
100#398C2×C411yescharacteristic
101#398C2×C411yescharacteristic
102#398C2×C411yescharacteristic
103#397—21no—
104#397—21no—
105#397C2×C411yescharacteristic
106#396—21no—
107#395—21no—
108#395—21no—
109#395—21no—
110#394—21no—
111#394C2×C411yescharacteristic
112#394C2×C411yescharacteristic
113#394C2×C411yescharacteristic
114#393—21no—
115#393—21no—
116#393—21no—
117#392C2×C411yescharacteristic
118#391—21no—
119#391—21no—
120#391—21no—
121#391—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#388—21no—
129#388—21no—
130#388—21no—
131#387—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#387—21no—
136#387—21no—
137#387—21no—
138#387—21no—
139#386—21no—
140#386—21no—
141#386—21no—
142#386—21no—
143#385—21no—
144#385—21no—
145#385—21no—
146#385—21no—
147#383C2×C411yescharacteristic
148#383C2311yescharacteristic
149#382—21no—
150#382C2×C411yescharacteristic
151#380—21no—
152#380—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4C2×C411yescharacteristic
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160#376C2×C411yescharacteristic
161#376C2311yescharacteristic
162#343—22no—
163#343—22no—
164#330—44no—
165D4⋊C8—44no—
166#324—22no—
167#324—22no—
168#324—22no—
169#323—22no—
170C22×Q8—21no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8C22⋊C411yescharacteristic
175C22×Q8C22⋊C411yescharacteristic
176C22×Q8C2×D411yescharacteristic
177C22×Q8C2×D411yescharacteristic
178C22×D4—21no—
179C22×D4C22×C411yescharacteristic
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C2×M16—24no—
187C4⋊Q8—42no—
188#125—24no—
189#125—42no—
190C22⋊Q8—44no—
191C22⋊Q8—44no—
192C22⋊Q8—44no—
193C22⋊Q8—44no—
194C4×Q8—24no—
195C4×Q8—24no—
196C4×Q8—42no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×D4—24no—
203C4×D4—42no—
204C4×D4—42no—
205C4×D4—44no—
206#117—24no—
207#117—42no—
208#117—42no—
209#117—42no—
210#117—42no—
211#117—42no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4C22⋊C411yescharacteristic
254C2×C22⋊C4C22⋊C411yescharacteristic
255C2×C22⋊C4C22×C411yescharacteristic
256C2×C22⋊C4C2×D411yescharacteristic
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2411yescharacteristic
272C4⋊C8—24no—
273C4⋊C8—44no—
274C4⋊C8—42no—
275C4⋊C8—42no—
276#101—44no—
277#101—44no—
278C22⋊C8—24no—
279C22⋊C8—44no—
280#97—24no—
281#97—24no—
282#97—42no—
283C4×C8—24no—
284C4×C8—42no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95C22×C411yescharacteristic
309#95C22×C411yescharacteristic
310#95C22×C411yescharacteristic
311#95C22×C411yescharacteristic
312C24—21no—
313C24C2×C22⋊C411yescharacteristic
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—42no—
327C2×Q8—416no—
328C2×Q8—81no—
329C2×Q8—81no—
330C2×D4—22no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—81no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—81no—
357C22×C4—81no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C23×C411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374M16—44no—
375M16—44no—
376C2×C8—44no—
377C2×C8—44no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—81no—
403C4⋊C4—81no—
404C4⋊C4—81no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—81no—
420C22⋊C4—81no—
421C22⋊C4—81no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—42no—
434C42—81no—
435C42—81no—
436C42H(5,𝔽2)14yes—
437C23—22no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23C22×C22⋊C411yescharacteristic
450Q8—44no—
451Q8—44no—
452Q8—44no—
453Q8—44no—
454Q8—162no—
455D4—24no—
456D4—42no—
457D4—84no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
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—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—84no—
512C2×C4—162no—
513C2×C4—162no—
514C2×C4#42112yes—
515C2×C4#42112yes—
516C2×C4#51812yes—
517C2×C4#53412yes—
518C2×C4#53412yes—
519C8—84no—
520C8—84no—
521C22—21no—
522C22—21no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—41no—
527C22—42no—
528C22—41no—
529C22—41no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22#230811yescharacteristic
536C22C2×#42111yescharacteristic
537C22C2×#42111yescharacteristic
538C4—22no—
539C4—22no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4—81no—
556C4—81no—
557C4—81no—
558C4—81no—
559C2—21no—
560C2—21no—
561C2—81no—
562C2—82no—
563C2#2177411yescharacteristic
564C2#2178011yescharacteristic
565C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#392C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20730C2semidirect
3. #17371C2semidirect
5. #9390C2semidirect
6. #9190C2semidirect
7. #9032C2semidirect
8. #8774C2semidirect
9. #8372C2semidirect

Maximal subgroups

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

subgroupindex
#357382
#207302
#173712
#94942
#93902
#91902
#90322
#87742
#83722