Finite Group Databasebeta

← #214632

Subgroups — #214632

Order 512.

884 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35669C211yesmaximal, characteristic
2#20730C211yesmaximal, characteristic, complemented
3#20691C211yesmaximal, characteristic, complemented
4#9466C212yesmaximal
5#9462C212yesmaximal, complemented
6#9446C212yesmaximal
7#9444C212yesmaximal, complemented
8#8740C212yesmaximal, complemented
9#8519C212yesmaximal
10C2×#542C2211yescharacteristic
11C2×#519C2211yescharacteristic
12#2921C2212yes—
13#2568C411yescharacteristic, complemented
14#2567C411yescharacteristic, complemented
15#2545C411yescharacteristic, complemented
16#2545C411yescharacteristic, complemented
17#2500C2211yescharacteristic
18#2499C2211yescharacteristic
19#2474—21no—
20#2474—21no—
21#2470—21no—
22#2470—21no—
23#2469—21no—
24#2469—21no—
25#2465—21no—
26#2465—21no—
27#2436C2211yescharacteristic
28#2434C2211yescharacteristic
29#2327C411yescharacteristic, complemented
30#2326C411yescharacteristic, complemented
31#2310C2212yes—
32C4×C4⋊Q8C411yescharacteristic, complemented
33C4×#124C411yescharacteristic, complemented
34#2298C2211yescharacteristic
35#1822C2212yes—
36#1820—21no—
37#1820—21no—
38#1820C2212yes—
39#1820C2212yes—
40#1801—21no—
41#1801—21no—
42#1784C2212yes—
43C4×#101—21no—
44C4×#101—21no—
45#1740—21no—
46#1740—21no—
47#1511C2214yes—
48#1510C2214yes—
49#1284C2214yes—
50#1284C2214yes—
51C23×Q8C2311yescharacteristic
52#542—44no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#126—21no—
57C2×#124—21no—
58C2×#124C2×C411yescharacteristic
59C2×#124C2×C411yescharacteristic
60C2×#124C2×C411yescharacteristic
61C2×#123—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8C2311yescharacteristic
64C2×#121C2311yescharacteristic
65#519—44no—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8D411yescharacteristic
74C2×C4×Q8D411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×#117—22no—
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C2×#117D411yescharacteristic
80C2×#117D411yescharacteristic
81C2×#117C2311yescharacteristic
82C2×#117C2311yescharacteristic
83#433—44no—
84#423—22no—
85C2×#101—22no—
86C2×#101—22no—
87#407C2314yes—
88C4×M16—22no—
89#400—21no—
90#400—21no—
91#397—21no—
92#397—21no—
93#396—21no—
94#396—21no—
95#396—21no—
96#396—21no—
97#395—21no—
98#395—21no—
99#395—21no—
100#395—21no—
101#394—21no—
102#394—21no—
103#393—21no—
104#393—21no—
105#391—21no—
106#391—21no—
107#391—44no—
108#391C2×C411yescharacteristic
109#391C2×C411yescharacteristic
110#391C2×C411yescharacteristic
111#391C2×C411yescharacteristic
112#391C2×C411yescharacteristic
113#391C2×C411yescharacteristic
114#390C2×C411yescharacteristic
115#390C2×C411yescharacteristic
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#388—21no—
121#388—21no—
122#388—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#388—21no—
129#388—21no—
130#388—21no—
131#388—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#387—21no—
136#387—21no—
137#387—21no—
138#387—21no—
139#387—21no—
140#386—21no—
141#386—21no—
142#385—21no—
143#385—21no—
144#385—21no—
145#385—21no—
146#384C2×C411yescharacteristic
147#384C2×C411yescharacteristic
148#382C2×C411yescharacteristic
149#382C2×C411yescharacteristic
150#381—22no—
151#381C2314yes—
152C4×C4⋊C4—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4C2×C411yescharacteristic
155C4×C4⋊C4C2×C411yescharacteristic
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4C2311yescharacteristic
159C4×C22⋊C4C2311yescharacteristic
160#376C2×C411yescharacteristic
161C43C2×C411yescharacteristic
162#343—22no—
163#343—22no—
164#342—22no—
165#342—22no—
166#342—22no—
167#342—22no—
168#328—44no—
169#327—44no—
170C22×Q8—21no—
171C22×Q8—24no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8C22×C411yescharacteristic
175C22×Q8C22×C411yescharacteristic
176C22×Q8C22×C411yescharacteristic
177C22×Q8C2×D411yescharacteristic
178C22×Q8C2×D411yescharacteristic
179C22×D4—21no—
180C22×D4C22×C411yescharacteristic
181C23×C4—21no—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2411yescharacteristic
185C2×M16—24no—
186C4⋊Q8—42no—
187C4⋊Q8—42no—
188C4⋊Q8—42no—
189#126—44no—
190#124—42no—
191#124—44no—
192#124—42no—
193#124—42no—
194#123—44no—
195C22⋊Q8—24no—
196C22⋊Q8—44no—
197#121—24no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×Q8—42no—
206C4×Q8—42no—
207C4×D4—42no—
208#117—24no—
209#117—24no—
210#117—42no—
211#117—42no—
212#117—42no—
213#117—42no—
214#117—44no—
215#117—44no—
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—42no—
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C411yescharacteristic
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⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
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⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4C2×D411yescharacteristic
261C2×C22⋊C4C2×D411yescharacteristic
262C2×C22⋊C4C2×D411yescharacteristic
263C2×C22⋊C4C2×D411yescharacteristic
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—42no—
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C22×C411yescharacteristic
278C2×C42C22×C411yescharacteristic
279C2×C42C22×C411yescharacteristic
280C2×C42C22×C411yescharacteristic
281C2×C42C22×C411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C4⋊C8—44no—
285#101—44no—
286#101—44no—
287C22⋊C8—24no—
288C22⋊C8—24no—
289#97—42no—
290C4×C8—42no—
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#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—24no—
314#95—24no—
315#95—44no—
316C24—21no—
317C24C22×D411yescharacteristic
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—48no—
332C2×Q8—48no—
333C2×Q8—42no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—81no—
337C2×Q8—81no—
338C2×D4—22no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—42no—
342C2×D4—81no—
343C2×D4—81no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—81no—
362C22×C4—81no—
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×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C23×C411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4H(5,𝔽2)14yes—
378M16—44no—
379M16—44no—
380C2×C8—44no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
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—84no—
404C4⋊C4—81no—
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—42no—
414C22⋊C4—42no—
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—42no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—81no—
428C22⋊C4—81no—
429C22⋊C4—81no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—42no—
447C42—42no—
448C42—84no—
449C42—81no—
450C23—21no—
451C23—21no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23C22×C22⋊C411yescharacteristic
462C23#53412yes—
463Q8—44no—
464Q8—44no—
465Q8—44no—
466Q8—44no—
467Q8—82no—
468Q8—162no—
469D4—82no—
470D4—162no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—44no—
497C2×C4—41no—
498C2×C4—44no—
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—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—84no—
527C2×C4—84no—
528C2×C4—84no—
529C2×C4—162no—
530C2×C4#41012yes—
531C2×C4#41012yes—
532C2×C4#51812yes—
533C2×C4#53412yes—
534C8—84no—
535C22—21no—
536C22—21no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22#230811yescharacteristic
550C22C2×#41011yescharacteristic
551C22C2×#41011yescharacteristic
552C4—22no—
553C4—22no—
554C4—22no—
555C4—22no—
556C4—42no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—164no—
574C2—21no—
575C2—21no—
576C2—42no—
577C2—161no—
578C2#2174811yescharacteristic
579C2#2178311yescharacteristic
580C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20730C2semidirect
3. #20691C2semidirect
5. #9462C2semidirect
7. #9444C2semidirect
8. #8740C2semidirect
13. #2568C4semidirect
14. #2567C4semidirect
15. #2545C4semidirect
16. #2545C4semidirect
29. #2327C4semidirect
30. #2326C4semidirect
32. C4×C4⋊Q8C4semidirect
33. C4×#124C4semidirect

Maximal subgroups

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

subgroupindex
#356692
#207302
#206912
#94662
#94622
#94462
#94442
#87402
#85192