Finite Group Databasebeta

← #215067

Subgroups — #215067

Order 512.

860 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35712C211yesmaximal, characteristic
2#20731C211yesmaximal, characteristic
3#17368C211yesmaximal, characteristic
4#9493C212yesmaximal, complemented
5#9390C212yesmaximal, complemented
6#9189C212yesmaximal, complemented
7#9029C212yesmaximal, complemented
8#8775C212yesmaximal, complemented
9#8374C212yesmaximal, complemented
10C2×#557C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2973C2212yes—
13#2921C2212yes—
14#2633C411yescharacteristic
15#2633C2211yescharacteristic
16#2621C411yescharacteristic
17#2611C411yescharacteristic
18#2611C2211yescharacteristic
19#2599C411yescharacteristic
20#2599C2211yescharacteristic
21#2580C411yescharacteristic
22#2547C411yescharacteristic
23#2473—21no—
24#2473—21no—
25#2460—21no—
26#2460—21no—
27#2446—21no—
28#2432—21no—
29#2398C411yescharacteristic
30#2398C2211yescharacteristic
31#2396C411yescharacteristic
32#2367—21no—
33#2360—21no—
34#2356C2211yescharacteristic
35#1821—21no—
36#1801—21no—
37#1800—21no—
38#1785—21no—
39C4×#101C2212yes—
40C4×#100C2212yes—
41#1528C2214yes—
42#1491C2214yes—
43#1467C2212yes—
44#1465—21no—
45#1464C2212yes—
46#1463—21no—
47#1459—21no—
48C2×#324—21no—
49#1300C2214yes—
50#1288C2214yes—
51C22×#41D411yescharacteristic
52#557—44no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#125—21no—
56C2×#125—21no—
57C2×#125C2×C411yescharacteristic
58C2×#125C2311yescharacteristic
59C2×#124—21no—
60C2×C22⋊Q8—21no—
61#520—44no—
62C2×C4×Q8—21no—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×C4×Q8C2311yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C2×#117C2×C411yescharacteristic
75C2×#117C2×C411yescharacteristic
76C2×#117C2311yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80#435—44no—
81#433—44no—
82#432C2314yes—
83#423—22no—
84C2×#101—22no—
85C2×#100—22no—
86#407—22no—
87#407—22no—
88C2×C22⋊C8—22no—
89C4×M16C2314yes—
90#400—21no—
91#400—21no—
92#400C2×C411yescharacteristic
93#399C2×C411yescharacteristic
94#399C2×C411yescharacteristic
95#399C2×C411yescharacteristic
96#398—21no—
97#398—21no—
98#398—21no—
99#398—21no—
100#398C2×C411yescharacteristic
101#398C2×C411yescharacteristic
102#397—21no—
103#397C2×C411yescharacteristic
104#397C2×C411yescharacteristic
105#397C2×C411yescharacteristic
106#397C2×C411yescharacteristic
107#397C2×C411yescharacteristic
108#397C2×C411yescharacteristic
109#396—21no—
110#395C2×C411yescharacteristic
111#394—21no—
112#394—21no—
113#394—21no—
114#394—21no—
115#393C2×C411yescharacteristic
116#393C2×C411yescharacteristic
117#393C2×C411yescharacteristic
118#391—21no—
119#391—21no—
120#391—21no—
121#390—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#388—21no—
129#387—21no—
130#387—21no—
131#387—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#387—21no—
136#387—21no—
137#385—21no—
138#385—21no—
139#385—21no—
140#385—21no—
141#385—21no—
142#385—21no—
143#385—21no—
144#385—21no—
145#385—21no—
146#385—21no—
147#383C2×C411yescharacteristic
148#383C2311yescharacteristic
149#382C2×C411yescharacteristic
150#380—21no—
151#380—21no—
152C4×C4⋊C4—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—
165Q8⋊C8—44no—
166#324—22no—
167#324—22no—
168#324—22no—
169#323—22no—
170C2×#41—42no—
171C2×#41—42no—
172C2×#41—42no—
173C22×Q8—21no—
174C22×Q8C22×C411yescharacteristic
175C22×Q8C2×D411yescharacteristic
176C22×D4C22⋊C411yescharacteristic
177C22×D4C22⋊C411yescharacteristic
178C22×D4C2×D411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C2×M16—24no—
189C2×M16—42no—
190C22×C8—42no—
191C4⋊Q8—42no—
192C4⋊Q8—44no—
193#125—24no—
194#125—42no—
195#125—44no—
196#125—44no—
197#124—44no—
198C22⋊Q8—44no—
199C4×Q8—24no—
200C4×Q8—24no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—44no—
205C4×D4—24no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211#117—24no—
212#117—42no—
213#117—42no—
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—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
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⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4C22⋊C411yescharacteristic
259C2×C22⋊C4C22⋊C411yescharacteristic
260C2×C22⋊C4C22×C411yescharacteristic
261C2×C22⋊C4C22×C411yescharacteristic
262C2×C22⋊C4C22×C411yescharacteristic
263C2×C22⋊C4C2×D411yescharacteristic
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2411yescharacteristic
276C4⋊C8—24no—
277C4⋊C8—44no—
278#101—44no—
279#100—44no—
280C22⋊C8—24no—
281C22⋊C8—42no—
282C22⋊C8—44no—
283C22⋊C8—42no—
284#97—24no—
285#97—24no—
286C4×C8—24no—
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
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C411yescharacteristic
314C24C2×C22⋊C411yescharacteristic
315#41—416no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—41no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—81no—
357C22×C4—81no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4—81no—
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—
376M16—162no—
377C2×C8—44no—
378C2×C8—44no—
379C2×C8—162no—
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—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—81no—
406C4⋊C4—81no—
407C4⋊C4—81no—
408C4⋊C4—81no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—81no—
425C22⋊C4—81no—
426C22⋊C4—81no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—41no—
431C42—41no—
432C42—42no—
433C42—42no—
434C42—42no—
435C42—42no—
436C42—81no—
437C42—81no—
438C42H(5,𝔽2)14yes—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23#42112yes—
454C23C22×C22⋊C411yescharacteristic
455Q8—24no—
456Q8—44no—
457Q8—42no—
458Q8—84no—
459D4—44no—
460D4—44no—
461D4—44no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
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—41no—
512C2×C4—41no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—161no—
516C2×C4—161no—
517C2×C4—161no—
518C2×C4—161no—
519C2×C4#42112yes—
520C2×C4#51812yes—
521C2×C4#53412yes—
522C2×C4#53412yes—
523C8—84no—
524C8—84no—
525C22—21no—
526C22—21no—
527C22—22no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22#230811yescharacteristic
543C22C2×#42111yescharacteristic
544C22C2×#42111yescharacteristic
545C4—22no—
546C4—22no—
547C4—22no—
548C4—22no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—42no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4—81no—
565C2—21no—
566C2—21no—
567C2—42no—
568C2—81no—
569C2—81no—
570C2#2177411yescharacteristic
571C2#2178011yescharacteristic
572C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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
4. #9493C2semidirect
5. #9390C2semidirect
6. #9189C2semidirect
7. #9029C2semidirect
8. #8775C2semidirect
9. #8374C2semidirect

Maximal subgroups

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

subgroupindex
#357122
#207312
#173682
#94932
#93902
#91892
#90292
#87752
#83742