Finite Group Databasebeta

← #224561

Subgroups — #224561

Order 512.

1074 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34934C211yesmaximal, characteristic
2#22722C211yesmaximal, characteristic, complemented
3C4×C8×Q8C211yesmaximal, characteristic, complemented
4#9678C212yesmaximal, complemented
5#9674C211yesmaximal, characteristic, complemented
6#9638C211yesmaximal, characteristic, complemented
7#9638C212yesmaximal, complemented
8#8567C212yesmaximal, complemented
9#8563C211yesmaximal, characteristic, complemented
10#8177C212yesmaximal, complemented
11C4×D4⋊C8C211yesmaximal, characteristic, complemented
12C2×#529—21no—
13C2×#446—21no—
14C2×C8×Q8—21no—
15#2554—22no—
16#2552C411yescharacteristic
17#2548C411yescharacteristic
18#2541C411yescharacteristic
19#2538C411yescharacteristic
20#2512C412yes—
21#2490—21no—
22#2445C412yes—
23#2426—21no—
24#2392—22no—
25C4×#127C2211yescharacteristic
26C4×#127C2212yes—
27C4×#121—21no—
28#2293C2212yes—
29C42×Q8C2211yescharacteristic
30C42×D4C2211yescharacteristic
31#1940—22no—
32#1939—21no—
33#1842C2211yescharacteristic
34#1842C2212yes—
35C8×C4⋊C4—21no—
36C8×C4⋊C4—22no—
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41C42×C8C2211yescharacteristic
42C42×C8C2212yes—
43C2×D4⋊C8C2211yescharacteristic
44C2×D4⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46C2×D4⋊C8C2211yescharacteristic
47C2×D4⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49C2×D4⋊C8C2212yes—
50#1279C2212yes—
51#1279C2211yescharacteristic
52#1279C2211yescharacteristic
53C2×#127C2×C411yescharacteristic
54C2×#127C2×C412yes—
55#529—44no—
56#529—44no—
57C2×#126—22no—
58C2×#125—21no—
59C2×#124—21no—
60C2×#123—22no—
61C2×C22⋊Q8—21no—
62C2×#121—21no—
63C2×#121—21no—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×D4—22no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—22no—
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C2×C4×D4C2312yes—
75C2×#117—22no—
76C22×C42—21no—
77#446—44no—
78C8×Q8—28no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2311yescharacteristic
88C2×#97—22no—
89C2×C4×C8—21no—
90C2×C4×C8—22no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93C2×C4×C8—22no—
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2312yes—
96C2×C4×C8C2311yescharacteristic
97#398—22no—
98#397—22no—
99#396—22no—
100#396—22no—
101#394—22no—
102#394—22no—
103#394—22no—
104#392—22no—
105#390—21no—
106#390—21no—
107#390—22no—
108#390—22no—
109#390—21no—
110#390C2×C412yes—
111#390C2×C412yes—
112#390C2×C412yes—
113#390C2×C412yes—
114#390C2×C412yes—
115#389—21no—
116#389C2×C412yes—
117#389C2×C412yes—
118#388—22no—
119#388—22no—
120#388—22no—
121#386—22no—
122#384C2×C412yes—
123#384C2×C411yescharacteristic
124#380—21no—
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4D412yes—
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4D412yes—
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C43C2311yescharacteristic
139C43C2312yes—
140#336—22no—
141#336—21no—
142#336—21no—
143#336—22no—
144D4⋊C8—24no—
145D4⋊C8—22no—
146D4⋊C8—22no—
147D4⋊C8—24no—
148D4⋊C8—42no—
149D4⋊C8—42no—
150D4⋊C8—42no—
151C22×Q8C22⋊C411yescharacteristic
152C22×D4—21no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4C22×C412yes—
157C22×D4C22×C411yescharacteristic
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168#127—24no—
169#127—22no—
170#127—22no—
171#127—42no—
172#126—48no—
173#125—44no—
174#124—44no—
175#123—48no—
176C22⋊Q8—44no—
177#121—44no—
178#121—44no—
179C4×Q8—24no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8D814yes—
183C4×Q8SD1614yes—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—21no—
188C4×D4—22no—
189C4×D4—21no—
190C4×D4—22no—
191C4×D4—21no—
192C4×D4—22no—
193C4×D4—21no—
194C4×D4—22no—
195C4×D4—41no—
196C4×D4—42no—
197C4×D4—44no—
198C4×D4—42no—
199C4×D4—44no—
200C4×D4—81no—
201C4×D4—82no—
202C4×D4—81no—
203C4×D4—82no—
204#117—44no—
205#117—44no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C411yescharacteristic
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C412yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C412yes—
243C2×C42C22×C412yes—
244C2×C42C22×C411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D412yes—
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D412yes—
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D411yescharacteristic
253C2×C42C2411yescharacteristic
254C4⋊C8—28no—
255C4⋊C8—24no—
256C4⋊C8—21no—
257C4⋊C8—22no—
258C4⋊C8—21no—
259C4⋊C8—22no—
260C4⋊C8—21no—
261C4⋊C8—42no—
262C4⋊C8—44no—
263C4⋊C8—82no—
264C4⋊C8—81no—
265C4⋊C8#4112yes—
266#97—44no—
267C4×C8—28no—
268C4×C8—24no—
269C4×C8—22no—
270C4×C8—21no—
271C4×C8—22no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—42no—
275C4×C8—44no—
276C4×C8—42no—
277C4×C8#4112yes—
278#95—21no—
279#95—21no—
280#95—22no—
281#95—22no—
282#95—21no—
283#95C22⋊C412yes—
284#95C22⋊C412yes—
285#95C22⋊C412yes—
286C24—21no—
287C24—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8#10214yes—
291C2×D4—21no—
292C2×D4—22no—
293C2×D4—21no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—21no—
301C2×D4—22no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—44no—
307C2×D4—42no—
308C2×D4—44no—
309C2×D4—42no—
310C2×D4—82no—
311C2×D4—81no—
312C2×D4—82no—
313C2×D4—81no—
314C2×D4#13112yes—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—81no—
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C23×C411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—41no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—41no—
363C2×C8—42no—
364C2×C8—41no—
365C2×C8—42no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—82no—
382C4⋊C4—81no—
383C4⋊C4#10214yes—
384C4⋊C4#10218yes—
385C4⋊C4#10414yes—
386C22⋊C4—42no—
387C22⋊C4—41no—
388C22⋊C4—42no—
389C22⋊C4—41no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C42—21no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—21no—
407C42—22no—
408C42—21no—
409C42—22no—
410C42—21no—
411C42—22no—
412C42—24no—
413C42—22no—
414C42—41no—
415C42—42no—
416C42—41no—
417C42—41no—
418C42—42no—
419C42—42no—
420C42#13112yes—
421C42#13112yes—
422C42C2×D812yes—
423C42C2×D814yes—
424C42C2×SD1612yes—
425C42C2×SD1614yes—
426C42C2×#4111yescharacteristic
427C42C2×#4111yescharacteristic
428C42H(5,𝔽2)11yescharacteristic
429C422−1+411yescharacteristic
430C23—22no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23C22×C22⋊C411yescharacteristic
445Q8—28no—
446D4—22no—
447D4—22no—
448D4—42no—
449D4—41no—
450D4—42no—
451D4—41no—
452D4—42no—
453D4—42no—
454D4—88no—
455D4—88no—
456D4—84no—
457D4—84no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—84no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—81no—
496C2×C4—82no—
497C2×C4—84no—
498C2×C4#40812yes—
499C2×C4#40812yes—
500C2×C4C2×#10212yes—
501C2×C4C2×#10214yes—
502C2×C4C2×#10214yes—
503C2×C4C2×#10212yes—
504C2×C4C2×#10414yes—
505C2×C4C2×#10412yes—
506C2×C4C4×#4111yescharacteristic
507C2×C4#52011yescharacteristic
508C2×C4#54811yescharacteristic
509C2×C4#54811yescharacteristic
510C2×C4#55011yescharacteristic
511C2×C4#55011yescharacteristic
512C2×C4C2×#13111yescharacteristic
513C2×C4C2×#13111yescharacteristic
514C2×C4C2×#13111yescharacteristic
515C2×C4#56811yescharacteristic
516C2×C4C22×D811yescharacteristic
517C2×C4C22×SD1611yescharacteristic
518C2×C4#57611yescharacteristic
519C2×C4#57711yescharacteristic
520C8—24no—
521C8—44no—
522C8—42no—
523C8—42no—
524C8—81no—
525C8—82no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22#234111yescharacteristic
541C22C2×#40811yescharacteristic
542C22#288011yescharacteristic
543C22C22×#10211yescharacteristic
544C22#289711yescharacteristic
545C22C22×#10411yescharacteristic
546C22#290111yescharacteristic
547C4—21no—
548C4—21no—
549C4—24no—
550C4—22no—
551C4—22no—
552C4—24no—
553C4—24no—
554C4—42no—
555C4—41no—
556C4—81no—
557C4—82no—
558C4—81no—
559C4—81no—
560C4#149214yes—
561C4#296611yescharacteristic
562C4#296911yescharacteristic
563C4#299311yescharacteristic
564C4#299411yescharacteristic
565C4#335811yescharacteristic
566C4#336211yescharacteristic
567C4#339111yescharacteristic
568C4#339211yescharacteristic
569C2—82no—
570C2—162no—
571C2C2×#149211yescharacteristic
572C2#1748311yescharacteristic
573C2#1750711yescharacteristic
574C2#1751711yescharacteristic
575C2#2253111yescharacteristic
576C2#2254011yescharacteristic
577C2#2255011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#2541C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22722C2semidirect
3. C4×C8×Q8C2semidirect
4. #9678C2semidirect
5. #9674C2semidirect
6. #9638C2semidirect
7. #9638C2semidirect
8. #8567C2semidirect
9. #8563C2semidirect
10. #8177C2semidirect
11. C4×D4⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#349342
#227222
C4×C8×Q82
#96782
#96742
#96382
#85672
#85632
#81772
C4×D4⋊C82