Finite Group Databasebeta

← #219824

Subgroups — #219824

Order 512.

1044 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34932C211yesmaximal, characteristic
2#22795C211yesmaximal, characteristic, complemented
3#21264C211yesmaximal, characteristic, complemented
4C2×#1482C212yesmaximal, complemented
5#9673C211yesmaximal, characteristic, complemented
6#9637C211yesmaximal, characteristic, complemented
7#9075C212yesmaximal, complemented
8#8938C212yesmaximal, complemented
9#8563C211yesmaximal, characteristic, complemented
10#8315C212yesmaximal, complemented
11C4×D4⋊C8C211yesmaximal, characteristic, complemented
12C2×#529C2212yes—
13C2×#431C2212yes—
14C2×#431C2212yes—
15#2553—22no—
16#2552C411yescharacteristic
17#2547C411yescharacteristic
18#2542C411yescharacteristic
19#2537C411yescharacteristic
20#2496—21no—
21#2489C412yes—
22#2427C412yes—
23#2422—21no—
24#2395—22no—
25C4×C4⋊Q8C2211yescharacteristic
26C4×C22⋊Q8C2212yes—
27C4×#121—21no—
28#2285—21no—
29C42×D4C2211yescharacteristic
30C42×D4C2211yescharacteristic
31#1954—21no—
32#1954—21no—
33#1939—21no—
34#1923—22no—
35#1923—22no—
36#1842C2211yescharacteristic
37#1797C2212yes—
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C22⋊C8C2212yes—
42C42×C8C2211yescharacteristic
43#1482—28no—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47C2×Q8⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49C2×D4⋊C8C2211yescharacteristic
50C2×D4⋊C8C2211yescharacteristic
51#1279C2212yes—
52#1279C2211yescharacteristic
53#1279C2211yescharacteristic
54C2×C4⋊Q8C2×C411yescharacteristic
55#529—24no—
56#529—24no—
57#529—42no—
58#529—42no—
59C2×#126—22no—
60C2×#125C2×C412yes—
61C2×#124C2×C412yes—
62C2×#123—22no—
63C2×C22⋊Q8C2×C412yes—
64C2×C22⋊Q8C2×C412yes—
65C2×#121—21no—
66C2×#121C2×C412yes—
67C2×C4×Q8C2312yes—
68C2×C4×D4—22no—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×#117—22no—
76C22×C42—21no—
77C22×C42C2312yes—
78#431—24no—
79#431—24no—
80#431—42no—
81#431—42no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2311yescharacteristic
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97C2×C4×C8C2312yes—
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#400—22no—
101#399—22no—
102#398—22no—
103#397—22no—
104#397—22no—
105#397—22no—
106#394—22no—
107#394—22no—
108#391C2×C412yes—
109#391C2×C412yes—
110#390—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389C2×C412yes—
115#388—22no—
116#387—22no—
117#386—22no—
118#385—22no—
119#384—21no—
120#384—21no—
121#384C2×C411yescharacteristic
122#382—22no—
123#380—21no—
124#380—21no—
125#380—21no—
126#380—22no—
127#380C2×C412yes—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4D411yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C4×C22⋊C4—22no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4C2×C412yes—
142C4×C22⋊C4D412yes—
143C4×C22⋊C4D412yes—
144C43C2311yescharacteristic
145#336—22no—
146#336—21no—
147#336—21no—
148#336—22no—
149Q8⋊C8—24no—
150Q8⋊C8—24no—
151Q8⋊C8—42no—
152Q8⋊C8—42no—
153D4⋊C8—24no—
154D4⋊C8—22no—
155D4⋊C8D414yescomplemented
156C22×Q8C22×C412yes—
157C22×D4—22no—
158C22×D4C22⋊C411yescharacteristic
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C22×C412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174#126—44no—
175#126—44no—
176#125—24no—
177#125—42no—
178#124—24no—
179#124—42no—
180#123—44no—
181#123—44no—
182C22⋊Q8—24no—
183C22⋊Q8—24no—
184C22⋊Q8—42no—
185C22⋊Q8—42no—
186#121—24no—
187#121—42no—
188#121—44no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—41no—
203C4×D4—42no—
204C4×D4—44no—
205C4×D4—44no—
206C4×D4C2×D412yes—
207C4×D4#4112yes—
208#117—44no—
209#117—44no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22⋊C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4C22⋊C412yes—
237C2×C22⋊C4C22⋊C412yes—
238C2×C22⋊C4C22⋊C412yes—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C412yes—
251C2×C42C22×C411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2411yescharacteristic
259C4⋊C8—22no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—22no—
264C4⋊C8—44no—
265C4⋊C8—42no—
266C4⋊C8—42no—
267C4⋊C8—81no—
268C4⋊C8C2×D412yes—
269C22⋊C8—44no—
270C22⋊C8—44no—
271C22⋊C8—44no—
272C22⋊C8—82no—
273C4×C8—22no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—22no—
278C4×C8—44no—
279C4×C8—42no—
280C4×C8—42no—
281C4×C8—42no—
282C4×C8C2×D412yes—
283#95—21no—
284#95—21no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—21no—
292#95C22⋊C412yes—
293#95C22⋊C412yes—
294#95C22⋊C412yes—
295C24—21no—
296C24C2×C22⋊C412yes—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—82no—
314C2×D4C4×D412yes—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—81no—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C23×C411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—41no—
375C2×C8—42no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—81no—
395C4⋊C4—82no—
396C4⋊C4C4×D412yes—
397C4⋊C4#13112yes—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—42no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—22no—
422C42—41no—
423C42—42no—
424C42—42no—
425C42—41no—
426C42—41no—
427C42—42no—
428C42—42no—
429C42#13112yes—
430C42#13112yes—
431C42#13512yes—
432C42#13512yes—
433C42C22×D411yescharacteristic
434C42C2×#4111yescharacteristic
435C42H(5,𝔽2)11yescharacteristic
436C42H(5,𝔽2)11yescharacteristic
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23C22×C22⋊C411yescharacteristic
452Q8—42no—
453Q8—42no—
454Q8—42no—
455Q8—42no—
456D4—22no—
457D4—22no—
458D4—41no—
459D4—41no—
460D4—44no—
461D4—88no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—84no—
510C2×C4—84no—
511C2×C4#40812yes—
512C2×C4#40812yes—
513C2×C4#41612yes—
514C2×C4#41612yes—
515C2×C4#42112yes—
516C2×C4C2×C4×D411yescharacteristic
517C2×C4#51811yescharacteristic
518C2×C4D4211yescharacteristic
519C2×C4D4211yescharacteristic
520C2×C4#54611yescharacteristic
521C2×C4#54611yescharacteristic
522C2×C4C2×#13111yescharacteristic
523C2×C4C2×#13111yescharacteristic
524C2×C4C2×#13111yescharacteristic
525C2×C4#56811yescharacteristic
526C2×C4C2×#13511yescharacteristic
527C2×C4C2×#13511yescharacteristic
528C2×C4#57711yescharacteristic
529C2×C4#57811yescharacteristic
530C8—42no—
531C8—42no—
532C8—42no—
533C8—82no—
534C8—82no—
535C8—81no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22D4×C22⋊C411yescharacteristic
551C22C2×#40811yescharacteristic
552C22#288011yescharacteristic
553C22C2×#41611yescharacteristic
554C22#289811yescharacteristic
555C22C2×#42111yescharacteristic
556C22#290111yescharacteristic
557C4—22no—
558C4—22no—
559C4—21no—
560C4—21no—
561C4—42no—
562C4—41no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—81no—
568C4—82no—
569C4—82no—
570C4#148214yescomplemented
571C4#293111yescharacteristic
572C4#293511yescharacteristic
573C4#297911yescharacteristic
574C4#298111yescharacteristic
575C4#328311yescharacteristic
576C4#328411yescharacteristic
577C4#330211yescharacteristic
578C4#330311yescharacteristic
579C2—82no—
580C2—82no—
581C2C2×#148211yescharacteristic
582C2#1746911yescharacteristic
583C2#1750811yescharacteristic
584C2#1752511yescharacteristic
585C2#2240211yescharacteristic
586C2#2241711yescharacteristic
587C2#2243511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#2537C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22795C2semidirect
3. #21264C2semidirect
4. C2×#1482C2semidirect
5. #9673C2semidirect
6. #9637C2semidirect
7. #9075C2semidirect
8. #8938C2semidirect
9. #8563C2semidirect
10. #8315C2semidirect
11. C4×D4⋊C8C2semidirect
155. D4⋊C8D4semidirect
570. C4#1482semidirect

Maximal subgroups

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

subgroupindex
#349322
#227952
#212642
C2×#14822
#96732
#96372
#90752
#89382
#85632
#83152
C4×D4⋊C82