Finite Group Databasebeta

← #480687

Subgroups — #480687

Order 512.

877 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35311C211yesmaximal, characteristic
2#17357C211yesmaximal, characteristic
3C2×#1464C211yesmaximal, characteristic
4#11556C214yesmaximal, complemented
5#11551C214yesmaximal, complemented
6#11515C212yesmaximal, complemented
7#11510C212yesmaximal, complemented
8C22×C22⋊Q8C411yescharacteristic
9C22×#117C2211yescharacteristic
10#2920—21no—
11C2×#432—21no—
12#2855—21no—
13#2855—21no—
14#2625—21no—
15#2625—21no—
16#2503C2212yes—
17#2502C412yes—
18#2499C412yes—
19#2499C2212yes—
20#2484—21no—
21#2484—21no—
22#2483—21no—
23#2483—21no—
24#2317C411yescharacteristic
25#2317C2211yescharacteristic
26#2310C411yescharacteristic
27#2310C2211yescharacteristic
28#2280C411yescharacteristic
29#1464—21no—
30#1464—21no—
31#1464—21no—
32#1464C2212yes—
33#1464C2212yes—
34#1464C2212yes—
35#1463C2212yes—
36#1462—21no—
37#1458—21no—
38#1458C2212yes—
39C2×#324C2212yes—
40C2×#323—21no—
41#1324C2218yescomplemented
42#1319C2218yescomplemented
43C23×Q8C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—41no—
48C2×C22⋊Q8—41no—
49C2×C22⋊Q8—41no—
50C2×C22⋊Q8—41no—
51C2×C22⋊Q8C2×C412yes—
52C2×C22⋊Q8C2×C412yes—
53C2×#117—41no—
54C2×#117—41no—
55C2×#117—41no—
56C2×#117—41no—
57C2×#117C2×C411yescharacteristic
58C2×#117C2×C411yescharacteristic
59C2×#117C2×C411yescharacteristic
60C2×#117C2×C411yescharacteristic
61C2×#117D411yescharacteristic
62C2×#117D411yescharacteristic
63C2×#117D411yescharacteristic
64C2×#117D411yescharacteristic
65C2×#117C2311yescharacteristic
66C2×#117C2311yescharacteristic
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4D411yescharacteristic
73C22×C22⋊C4D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76#432—22no—
77#432—22no—
78#432—22no—
79#432—41no—
80#432—41no—
81#432—41no—
82#431—22no—
83#431—41no—
84#423—41no—
85C2×C4⋊C8—41no—
86#407—22no—
87C2×C22⋊C8—22no—
88C4×M16—41no—
89C2×#97—41no—
90#400—22no—
91#400—22no—
92#398—22no—
93#398—22no—
94#397—22no—
95#397—22no—
96#397—22no—
97#397—22no—
98#391—21no—
99#391—21no—
100#391—21no—
101#391—21no—
102#391C2×C412yes—
103#391C2×C412yes—
104#391C2×C412yes—
105#391C2312yes—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#385—22no—
115#385—22no—
116#385—21no—
117#385—21no—
118#385—21no—
119#385—21no—
120#384C2×C412yes—
121#384C2×C412yes—
122#382C2×C412yes—
123#382C2312yes—
124#381—21no—
125#381—22no—
126#381—22no—
127#381—21no—
128#381—21no—
129#381—21no—
130#381—21no—
131#380—21no—
132#380—21no—
133#379—21no—
134C4×C22⋊C4C2×C412yes—
135C4×C22⋊C4C2×C412yes—
136#324—22no—
137#324—22no—
138#324C2314yes—
139#324C2314yes—
140#323—22no—
141#323—22no—
142#323—22no—
143#323—22no—
144C25C22⋊C411yescharacteristic
145C22×Q8—21no—
146C22×Q8—42no—
147C22×Q8—41no—
148C22×Q8—41no—
149C22×Q8—42no—
150C22×Q8C22×C412yes—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C2×M16—42no—
170C22×C8—42no—
171C22⋊Q8—42no—
172C22⋊Q8—42no—
173C22⋊Q8—41no—
174C22⋊Q8—41no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C22⋊Q8—42no—
180C22⋊Q8—42no—
181#117—41no—
182#117—41no—
183#117—41no—
184#117—41no—
185#117—41no—
186#117—41no—
187#117—41no—
188#117—41no—
189#117—41no—
190#117—41no—
191#117—41no—
192#117—41no—
193#117—41no—
194#117—41no—
195#117—41no—
196#117—41no—
197#117—42no—
198#117—42no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4C22⋊C411yescharacteristic
217C2×C4⋊C4C22⋊C411yescharacteristic
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2411yescharacteristic
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4C22⋊C411yescharacteristic
238C2×C22⋊C4C22⋊C411yescharacteristic
239C2×C22⋊C4C22⋊C411yescharacteristic
240C2×C22⋊C4C22⋊C411yescharacteristic
241C2×C22⋊C4C2×D411yescharacteristic
242C2×C22⋊C4C2×D411yescharacteristic
243C2×C22⋊C4C2×D411yescharacteristic
244C2×C22⋊C4C2×D411yescharacteristic
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22×C412yes—
258C2×C42C22×C412yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C4⋊C8—42no—
264C4⋊C8—42no—
265C4⋊C8—42no—
266C4⋊C8—42no—
267C22⋊C8—24no—
268C22⋊C8—24no—
269C22⋊C8—42no—
270C22⋊C8—42no—
271#97—42no—
272#97—42no—
273#97—42no—
274C4×C8—42no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289C24—21no—
290C24—21no—
291C24—22no—
292C24—41no—
293C24—42no—
294C24—81no—
295C24C2×C22⋊C411yescharacteristic
296C24C2×C22⋊C411yescharacteristic
297C24C2×C22⋊C411yescharacteristic
298C2×Q8—24no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—84no—
306C2×Q8—84no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—81no—
352C22×C4—81no—
353C22×C4—82no—
354C22×C4#9911yescharacteristic
355C22×C4#9911yescharacteristic
356C22×C4#9911yescharacteristic
357C22×C4#9911yescharacteristic
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×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C23×C411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370M16—84no—
371C2×C8—44no—
372C2×C8—44no—
373C2×C8—84no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—82no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—24no—
426C42—42no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—22no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—41no—
436C23—42no—
437C23—42no—
438C23—41no—
439C23—41no—
440C23—42no—
441C23—41no—
442C23—41no—
443C23—42no—
444C23—81no—
445C23—81no—
446C23—84no—
447C23—84no—
448C23C2×#9911yescharacteristic
449C23C2×#9911yescharacteristic
450C23#41312yes—
451C23C22×C22⋊C411yescharacteristic
452Q8—84no—
453Q8—82no—
454Q8—82no—
455C2×C4—24no—
456C2×C4—24no—
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—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
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—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4C2×#9911yescharacteristic
517C2×C4C2×#9911yescharacteristic
518C2×C4C2×#9911yescharacteristic
519C2×C4C2×#9911yescharacteristic
520C2×C4#56812yes—
521C8—84no—
522C8—84no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—22no—
530C22—21no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—84no—
541C22—81no—
542C22—81no—
543C22—84no—
544C22#212912yes—
545C22#287911yescharacteristic
546C22C22×#9911yescharacteristic
547C22C2×#41311yescharacteristic
548C4—22no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4—82no—
566C2—21no—
567C2—21no—
568C2—22no—
569C2—41no—
570C2—42no—
571C2—81no—
572C2#1739211yescharacteristic
573C2C2×#212911yescharacteristic
574C2#3336911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11556C2semidirect
5. #11551C2semidirect
6. #11515C2semidirect
7. #11510C2semidirect
41. #1324C22semidirect
42. #1319C22semidirect

Maximal subgroups

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

subgroupindex
#353112
#173572
C2×#14642
#115562
#115512
#115152
#115102