Finite Group Databasebeta

← #224437

Subgroups — #224437

Order 512.

999 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36545C211yesmaximal, characteristic
2Q8×C22⋊C8C211yesmaximal, characteristic
3#13931C211yesmaximal, characteristic
4#9913C212yesmaximal
5#9909C211yesmaximal, characteristic
6#9886C212yesmaximal
7#9885C211yesmaximal, characteristic
8#8574C212yesmaximal
9#8572C211yesmaximal, characteristic
10#8221C212yesmaximal
11C4×#328C211yesmaximal, characteristic
12C2×#557C411yescharacteristic, complemented
13C22×C4×Q8C2211yescharacteristic
14C2×C8×Q8—21no—
15#2857C2211yescharacteristic
16#2851C2212yes—
17#2584C412yescomplemented
18#2582C411yescharacteristic, complemented
19#2581C412yescomplemented
20#2546C411yescharacteristic, complemented
21#2473—22no—
22#2462—22no—
23#2446—21no—
24#2434—21no—
25#2365—21no—
26#2360—21no—
27#2355C411yescharacteristic, complemented
28C4×C22⋊Q8C2211yescharacteristic
29C4×C22⋊Q8C2212yes—
30C2×#391—21no—
31#1984—21no—
32#1984—22no—
33#1797C2211yescharacteristic
34#1797C2212yes—
35C4×C22⋊C8C2211yescharacteristic
36C4×C22⋊C8C2212yes—
37C2×#328C2211yescharacteristic
38C2×#328C2212yes—
39C2×#328C2211yescharacteristic
40#1445—22no—
41C4⋊#95—21no—
42C4⋊#95C2211yescharacteristic
43#1434C2212yes—
44#1434C2212yes—
45C4×#95C2211yescharacteristic
46#1294C2211yescharacteristic
47#1294C2211yescharacteristic
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C23×Q8D411yescharacteristic
54#557—44no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#125—22no—
59C2×#125C2×C412yes—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8C2311yescharacteristic
62C2×C4×Q8—21no—
63C2×C4×Q8—44no—
64C2×C4×Q8C2×C412yes—
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C412yes—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8D412yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D412yes—
75C22×C42—21no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78C8×Q8—44no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81C2×C22⋊C8C2312yes—
82C2×C22⋊C8C2311yescharacteristic
83C2×C22⋊C8C2311yescharacteristic
84C2×C4×C8—21no—
85C2×C4×C8—22no—
86#400—22no—
87#398—22no—
88#398—22no—
89#398—22no—
90#395—22no—
91#395—22no—
92#391—21no—
93#391—21no—
94#391—21no—
95#391—21no—
96#391—44no—
97#391C2×C411yescharacteristic
98#391C2×C412yes—
99#389—21no—
100#389—22no—
101#389—22no—
102#389—21no—
103#389C2×C412yes—
104#389C2×C412yes—
105#389C2×C412yes—
106#389C2×C411yescharacteristic
107#389C2×C412yes—
108#387—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#387—21no—
115#385—22no—
116#385—22no—
117#385—22no—
118#385—21no—
119#384C2×C411yescharacteristic
120#382C2×C412yes—
121#380—21no—
122#380C2311yescharacteristic
123#380C2312yes—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C22⋊C4—21no—
130C4×C22⋊C4—22no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—21no—
137C2×#95C2311yescharacteristic
138C2×#95C2312yes—
139C2×#95C2311yescharacteristic
140#336—22no—
141#336—21no—
142#336—22no—
143#336—21no—
144#328—22no—
145#328—24no—
146#328Q814yescomplemented
147C22×Q8—24no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—42no—
152C22×Q8C22⋊C411yescharacteristic
153C22×Q8C22⋊C412yes—
154C22×Q8C22⋊C411yescharacteristic
155C22×Q8C22×C411yescharacteristic
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2411yescharacteristic
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C4⋊Q8—42no—
168C4⋊Q8—44no—
169C4⋊Q8—44no—
170#125—44no—
171#125—48no—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174C22⋊Q8—42no—
175C22⋊Q8—42no—
176C22⋊Q8C2×Q812yes—
177C22⋊Q8#4112yes—
178C4×Q8—24no—
179C4×Q8—24no—
180C4×Q8—44no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×Q8—44no—
184C4×Q8—44no—
185C4×Q8—42no—
186C4×Q8—44no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C412yes—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—44no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42C22×C412yes—
230C2×C42C22×C411yescharacteristic
231C2×C42C22×C412yes—
232C2×C42C22×C411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D411yescharacteristic
237C4⋊C8—44no—
238C4⋊C8—42no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—21no—
244C22⋊C8C2×Q812yes—
245C4×C8—44no—
246C4×C8—42no—
247#95—21no—
248#95—22no—
249#95—21no—
250#95—22no—
251#95—22no—
252#95—21no—
253#95—22no—
254#95—21no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—21no—
261#95—22no—
262#95—21no—
263#95—21no—
264#95—22no—
265#95—21no—
266#95—22no—
267#95—42no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—82no—
272#95—81no—
273#95C2×Q812yes—
274C24C22×D411yescharacteristic
275C2×Q8—21no—
276C2×Q8—24no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—24no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—42no—
284C2×Q8—48no—
285C2×Q8—48no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8#9914yes—
292C2×Q8C4×Q812yes—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4#13512yes—
345C22×C4#13512yes—
346C22×C4C23×C411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×Q811yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C42−1+411yescharacteristic
351C22×C42−1+411yescharacteristic
352C2×C8—22no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—42no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—41no—
360C2×C8—42no—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—41no—
372C4⋊C4—42no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—44no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—81no—
385C4⋊C4—81no—
386C4⋊C4—82no—
387C4⋊C4—84no—
388C4⋊C4C4×Q812yes—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C42—24no—
396C42—22no—
397C42—24no—
398C42—22no—
399C42—41no—
400C42—42no—
401C42—41no—
402C42—42no—
403C42—44no—
404C42—42no—
405C42—42no—
406C42—82no—
407C42—84no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23C22×C22⋊C411yescharacteristic
413C23#54811yescharacteristic
414C23D4×Q811yescharacteristic
415C23C2×#13511yescharacteristic
416C23C2×#13511yescharacteristic
417C23#57711yescharacteristic
418C23#57811yescharacteristic
419Q8—28no—
420Q8—48no—
421Q8—42no—
422Q8—42no—
423Q8—44no—
424Q8—84no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4C2×#9914yes—
500C2×C4C2×#9912yes—
501C2×C4#41612yes—
502C2×C4#41612yes—
503C2×C4#42112yes—
504C2×C4C2×C4×Q811yescharacteristic
505C2×C4#51911yescharacteristic
506C2×C4#54811yescharacteristic
507C2×C4D4×Q811yescharacteristic
508C8—42no—
509C8—82no—
510C8—81no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22Q8×C22⋊C411yescharacteristic
524C22C22×#9911yescharacteristic
525C22#288411yescharacteristic
526C22C2×#41611yescharacteristic
527C22#289811yescharacteristic
528C22C2×#42111yescharacteristic
529C22#290111yescharacteristic
530C22#336411yescharacteristic
531C22#336511yescharacteristic
532C22#338111yescharacteristic
533C22#338311yescharacteristic
534C4—22no—
535C4—24no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—41no—
544C4—42no—
545C4—41no—
546C4—42no—
547C4—82no—
548C4—82no—
549C4—82no—
550C4—81no—
551C4—81no—
552C4—82no—
553C4—162no—
554C4#151614yes—
555C2—21no—
556C2—21no—
557C2—21no—
558C2—21no—
559C2C2×#151611yescharacteristic
560C2#1760111yescharacteristic
561C2#1760611yescharacteristic
562C2#1761611yescharacteristic
563C2Q8×#9911yescharacteristic
564C2#2252111yescharacteristic
565C2#2252911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. C2×#557C4semidirect
17. #2584C4semidirect
18. #2582C4semidirect
19. #2581C4semidirect
20. #2546C4semidirect
27. #2355C4semidirect
146. #328Q8semidirect

Maximal subgroups

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

subgroupindex
#365452
Q8×C22⋊C82
#139312
#99132
#99092
#98862
#98852
#85742
#85722
#82212
C4×#3282