Finite Group Databasebeta

← #214629

Subgroups — #214629

Order 512.

928 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35618C211yesmaximal, characteristic
2#20731C211yesmaximal, characteristic, complemented
3#20725C211yesmaximal, characteristic, complemented
4#9465C212yesmaximal, complemented
5#9460C212yesmaximal, complemented
6#9446C212yesmaximal, complemented
7#9444C212yesmaximal, complemented
8#8739C212yesmaximal, complemented
9#8517C212yesmaximal
10C2×#537C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2921C2212yes—
13#2504C411yescharacteristic, complemented
14#2504C2211yescharacteristic
15#2503C411yescharacteristic, complemented
16#2503C2211yescharacteristic
17#2500C411yescharacteristic, complemented
18#2500C2211yescharacteristic
19#2499C411yescharacteristic, complemented
20#2499C2211yescharacteristic
21#2444—21no—
22#2444—21no—
23#2444—21no—
24#2444—21no—
25#2421—21no—
26#2421—21no—
27#2420—21no—
28#2420—21no—
29#2317C411yescharacteristic, complemented
30#2313C2212yes—
31#2311C411yescharacteristic, complemented
32#2297C411yescharacteristic, complemented
33#2297C2211yescharacteristic
34C2×#382C411yescharacteristic, complemented
35#1821C2212yes—
36#1820C2212yes—
37#1801—21no—
38#1800—21no—
39#1799—21no—
40#1799—21no—
41#1785C2212yes—
42#1782C2212yes—
43C4×#101—21no—
44C4×#100—21no—
45C4×#99—21no—
46C4×#99—21no—
47#1511C2214yes—
48#1510C2214yes—
49#1284C2214yes—
50#1284C2214yes—
51C22×#41C2×C411yescharacteristic
52C22×#41C2311yescharacteristic
53#537—44no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8C2×C411yescharacteristic
56C2×C22⋊Q8C2311yescharacteristic
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121C2×C411yescharacteristic
61C2×#121C2311yescharacteristic
62#520—44no—
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117C2311yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C42C2×C411yescharacteristic
80#433—44no—
81#423—22no—
82C2×#101—22no—
83C2×#100—22no—
84C2×#99—22no—
85C2×#99—22no—
86#407C2314yes—
87C4×M16—22no—
88#400—21no—
89#400—21no—
90#400—21no—
91#400—21no—
92#397—21no—
93#397—21no—
94#397—21no—
95#397—21no—
96#397—21no—
97#397—21no—
98#397—21no—
99#397—21no—
100#394—21no—
101#394—21no—
102#394—21no—
103#394—21no—
104#391—21no—
105#391C2×C411yescharacteristic
106#391C2×C411yescharacteristic
107#391C2×C411yescharacteristic
108#391C2×C411yescharacteristic
109#390—21no—
110#390C2×C411yescharacteristic
111#390C2×C411yescharacteristic
112#390C2×C411yescharacteristic
113#390C2×C411yescharacteristic
114#389—21no—
115#389—21no—
116#388—21no—
117#388—21no—
118#388—21no—
119#388—21no—
120#387—21no—
121#387—21no—
122#387—21no—
123#387—21no—
124#387—44no—
125#386—21no—
126#386—21no—
127#386—21no—
128#386—21no—
129#386—21no—
130#386—21no—
131#386—21no—
132#386—21no—
133#386—21no—
134#386—21no—
135#385—21no—
136#385—21no—
137#384C2×C411yescharacteristic
138#384C2×C411yescharacteristic
139#382—21no—
140#382—21no—
141#382C2×C411yescharacteristic
142#382C2×C411yescharacteristic
143#382C2×C411yescharacteristic
144#382C2×C411yescharacteristic
145#382C2×C411yescharacteristic
146#382C2×C411yescharacteristic
147#381—21no—
148#381—21no—
149#381—21no—
150#381—21no—
151#381C2314yes—
152#380—21no—
153#380—21no—
154#380—22no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—22no—
158C4×C22⋊C4C2×C411yescharacteristic
159C4×C22⋊C4C2×C411yescharacteristic
160C4×C22⋊C4C2311yescharacteristic
161C4×C22⋊C4C2311yescharacteristic
162C2×#95—21no—
163C2×#95—21no—
164#343—22no—
165#343—22no—
166#342—22no—
167#342—22no—
168#328—44no—
169#327—44no—
170C2×#41—24no—
171C2×#41—42no—
172C2×#41—42no—
173C2×#41—42no—
174C2×#41—42no—
175C2×#41—42no—
176C22×Q8C22×C411yescharacteristic
177C22×Q8C2×D411yescharacteristic
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4C22×C411yescharacteristic
182C22×D4C2×D411yescharacteristic
183C22×D4C2×D411yescharacteristic
184C22×D4C2×D411yescharacteristic
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2411yescharacteristic
197C2×M16—24no—
198C22⋊Q8—24no—
199C22⋊Q8—42no—
200C22⋊Q8—44no—
201#121—24no—
202#121—42no—
203#121—44no—
204#121—44no—
205#121—44no—
206C4×Q8—42no—
207C4×Q8—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214#117—24no—
215#117—42no—
216#117—42no—
217#117—42no—
218C2×C4⋊C4—24no—
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—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
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⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—24no—
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⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4C22⋊C411yescharacteristic
262C2×C22⋊C4C22⋊C411yescharacteristic
263C2×C22⋊C4C22⋊C411yescharacteristic
264C2×C22⋊C4C22⋊C411yescharacteristic
265C2×C22⋊C4C2×D411yescharacteristic
266C2×C22⋊C4C2×D411yescharacteristic
267C2×C42—21no—
268C2×C42—42no—
269C2×C42—42no—
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×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C4⋊C8—44no—
281#101—44no—
282#100—44no—
283#99—44no—
284#99—44no—
285C22⋊C8—24no—
286C22⋊C8—24no—
287#97—42no—
288C4×C8—42no—
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#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—24no—
314#95—24no—
315#95—42no—
316#95—42no—
317C24—21no—
318C24C2×C22⋊C411yescharacteristic
319C24C2×C22⋊C411yescharacteristic
320C24C22×D411yescharacteristic
321#41—416no—
322#41—88no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—81no—
347C2×D4—81no—
348C2×D4—81no—
349C2×D4—81no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C23×C411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4H(5,𝔽2)14yes—
400M16—44no—
401M16—44no—
402C2×C8—44no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—81no—
420C4⋊C4—81no—
421C4⋊C4—81no—
422C4⋊C4—81no—
423C4⋊C4—162no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—81no—
438C22⋊C4—81no—
439C22⋊C4—81no—
440C22⋊C4—81no—
441C22⋊C4—81no—
442C22⋊C4—81no—
443C22⋊C4—81no—
444C22⋊C4—81no—
445C22⋊C4—81no—
446C22⋊C4—81no—
447C22⋊C4—81no—
448C22⋊C4—81no—
449C42—22no—
450C42—22no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—42no—
460C42—162no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23#41012yes—
482C23C22×C22⋊C411yescharacteristic
483C23#53412yes—
484Q8—44no—
485Q8—82no—
486D4—44no—
487D4—44no—
488D4—44no—
489D4—82no—
490D4—84no—
491D4—162no—
492D4—162no—
493C2×C4—28no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—44no—
512C2×C4—44no—
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—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—84no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4—161no—
550C2×C4—161no—
551C2×C4—161no—
552C2×C4—161no—
553C2×C4—162no—
554C2×C4#41012yes—
555C2×C4#51812yes—
556C2×C4#53412yes—
557C8—84no—
558C22—21no—
559C22—21no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—42no—
565C22—42no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22#230811yescharacteristic
583C22C2×#41011yescharacteristic
584C22C2×#41011yescharacteristic
585C4—22no—
586C4—22no—
587C4—22no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4—81no—
604C4—164no—
605C2—21no—
606C2—21no—
607C2—42no—
608C2—42no—
609C2—81no—
610C2—161no—
611C2#2174711yescharacteristic
612C2#2178411yescharacteristic
613C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#537C22
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. #20731C2semidirect
3. #20725C2semidirect
4. #9465C2semidirect
5. #9460C2semidirect
6. #9446C2semidirect
7. #9444C2semidirect
8. #8739C2semidirect
13. #2504C4semidirect
15. #2503C4semidirect
17. #2500C4semidirect
19. #2499C4semidirect
29. #2317C4semidirect
31. #2311C4semidirect
32. #2297C4semidirect
34. C2×#382C4semidirect

Maximal subgroups

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

subgroupindex
#356182
#207312
#207252
#94652
#94602
#94462
#94442
#87392
#85172