Finite Group Databasebeta

← #496410

Subgroups — #496410

Order 512.

835 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35668C211yesmaximal, characteristic
2#20730C211yesmaximal, characteristic
3#20730C211yesmaximal, characteristic
4#13303C212yesmaximal, complemented
5#13303C212yesmaximal, complemented
6#13302C212yesmaximal, complemented
7#13301C212yesmaximal, complemented
8#12862C212yesmaximal
9#12860C212yesmaximal
10C2×#544C411yescharacteristic
11C2×#519C2211yescharacteristic
12#2921—21no—
13#2921—21no—
14#2568C411yescharacteristic
15#2568C2211yescharacteristic
16#2567C411yescharacteristic
17#2567C2211yescharacteristic
18#2565C411yescharacteristic
19#2565C411yescharacteristic
20#2565C2211yescharacteristic
21#2565C2211yescharacteristic
22#2453—21no—
23#2453—21no—
24#2451—21no—
25#2451—21no—
26#2438—21no—
27#2438—21no—
28#2438—21no—
29#2438—21no—
30#2317C411yescharacteristic
31#2310C2211yescharacteristic
32#2296C411yescharacteristic
33C2×#391C2211yescharacteristic
34#2285C411yescharacteristic
35#1822—21no—
36#1822—21no—
37#1801—21no—
38#1801—21no—
39#1801C2212yes—
40#1801C2212yes—
41#1784C2212yes—
42#1784C2212yes—
43C4×#101—21no—
44C4×#101—21no—
45C4×#101C2212yes—
46C4×#101C2212yes—
47#1394C2214yes—
48#1394C2214yes—
49#1394C2214yes—
50#1394C2214yes—
51C23×Q8C2311yescharacteristic
52#544—44no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125—21no—
57C2×#125C2×C411yescharacteristic
58C2×#125C2×C411yescharacteristic
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61#519—44no—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117D411yescharacteristic
75C2×#117D411yescharacteristic
76C2×#117D411yescharacteristic
77C2×#117D411yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80#433—44no—
81#433—44no—
82#423—22no—
83#423—22no—
84C2×#101—22no—
85C2×#101—22no—
86C2×#101C2314yes—
87C2×#101C2314yes—
88#407—22no—
89#407—22no—
90C4×M16—22no—
91C4×M16—22no—
92#400—21no—
93#400—21no—
94#400—21no—
95#400—21no—
96#398—21no—
97#398—21no—
98#398—21no—
99#398—21no—
100#398—21no—
101#398—21no—
102#398—21no—
103#398—21no—
104#395—21no—
105#395—21no—
106#395—21no—
107#395—21no—
108#391—21no—
109#391—21no—
110#391—21no—
111#391—21no—
112#391—21no—
113#391—21no—
114#391—21no—
115#391—21no—
116#391C2×C411yescharacteristic
117#391C2×C411yescharacteristic
118#391C2×C411yescharacteristic
119#391C2×C411yescharacteristic
120#391C2311yescharacteristic
121#391C2311yescharacteristic
122#391C2311yescharacteristic
123#391C2311yescharacteristic
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#389—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#385—21no—
141#385—21no—
142#385—21no—
143#385—21no—
144#384C2×C411yescharacteristic
145#384C2×C411yescharacteristic
146#384C2×C411yescharacteristic
147#384C2×C411yescharacteristic
148#382C2×C411yescharacteristic
149#382C2×C411yescharacteristic
150#382C2×C411yescharacteristic
151#382C2×C411yescharacteristic
152#381—21no—
153#381—21no—
154#380—21no—
155#380—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4C2×C411yescharacteristic
159C4×C4⋊C4C2×C411yescharacteristic
160C4×C4⋊C4C2×C411yescharacteristic
161C4×C4⋊C4C2×C411yescharacteristic
162C2×#95—21no—
163C2×#95—21no—
164#343—22no—
165#343—22no—
166#343—22no—
167#343—22no—
168C22×Q8—24no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×Q8C22×C411yescharacteristic
174C22×Q8C2×D411yescharacteristic
175C22×Q8C2×D411yescharacteristic
176C22×Q8C2411yescharacteristic
177C23×C4—21no—
178C23×C4—21no—
179C23×C4C22×C411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C2×M16—24no—
183C2×M16—24no—
184C4⋊Q8—42no—
185C4⋊Q8—42no—
186C4⋊Q8—44no—
187#125—42no—
188#125—42no—
189#125—44no—
190C22⋊Q8—44no—
191C22⋊Q8—44no—
192C4×Q8—42no—
193C4×Q8—42no—
194C4×Q8—42no—
195C4×Q8—42no—
196C4×Q8—42no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8—42no—
200#117—24no—
201#117—24no—
202#117—24no—
203#117—42no—
204#117—42no—
205#117—42no—
206#117—42no—
207#117—42no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
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—42no—
229C2×C4⋊C4C22⋊C411yescharacteristic
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⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4C22⋊C411yescharacteristic
248C2×C22⋊C4C22⋊C411yescharacteristic
249C2×C22⋊C4C22⋊C411yescharacteristic
250C2×C22⋊C4C22⋊C411yescharacteristic
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—42no—
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C4⋊C8—44no—
271C4⋊C8—44no—
272#101—28no—
273#101—28no—
274#101—44no—
275#101—44no—
276C22⋊C8—44no—
277C22⋊C8—44no—
278#97—42no—
279#97—42no—
280C4×C8—42no—
281C4×C8—42no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
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—
310C24C2×C22⋊C411yescharacteristic
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—416no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—81no—
326C2×Q8—81no—
327C2×Q8—81no—
328C2×Q8—81no—
329C2×Q82−1+414yes—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—81no—
351C22×C4—81no—
352C22×C4—81no—
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C23×C411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367M16—44no—
368M16—44no—
369M16—44no—
370M16—44no—
371C2×C8—44no—
372C2×C8—44no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—81no—
402C4⋊C4—81no—
403C4⋊C4—81no—
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—42no—
413C22⋊C4—42no—
414C22⋊C4#9914yes—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—42no—
432C42—42no—
433C23—21no—
434C23—21no—
435C23—41no—
436C23—41no—
437C23C2×#9912yes—
438C23C22×C22⋊C411yescharacteristic
439Q8—44no—
440Q8—44no—
441Q8—44no—
442Q8—44no—
443Q8—162no—
444Q8—162no—
445C2×C4—28no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—84no—
501C2×C4—162no—
502C2×C4C2×#9912yes—
503C2×C4C2×#9912yes—
504C2×C4#41012yes—
505C2×C4#51912yes—
506C2×C4#53612yes—
507C2×C4#53612yes—
508C8—84no—
509C8—84no—
510C22—21no—
511C22—21no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—42no—
516C22—42no—
517C22#231011yescharacteristic
518C22C22×#9911yescharacteristic
519C22C2×#41011yescharacteristic
520C4—22no—
521C4—22no—
522C4—22no—
523C4—22no—
524C4—22no—
525C4—22no—
526C4—42no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—81no—
536C4—81no—
537C4—81no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—81no—
542C2—21no—
543C2—21no—
544C2—42no—
545C2#2182511yescharacteristic
546C2#3337011yescharacteristic
547C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13303C2semidirect
5. #13303C2semidirect
6. #13302C2semidirect
7. #13301C2semidirect

Maximal subgroups

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

subgroupindex
#356682
#207302
#133032
#133022
#133012
#128622
#128602