Finite Group Databasebeta

← #313601

Subgroups — #313601

Order 512.

766 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27813C211yesmaximal, characteristic
2#26046C211yesmaximal, characteristic, complemented
3#25760C211yesmaximal, characteristic
4Q8×#102C211yesmaximal, characteristic
5#10616C211yesmaximal, characteristic
6#10538C211yesmaximal, characteristic, complemented
7#10421C211yesmaximal, characteristic
8#10350C211yesmaximal, characteristic, complemented
9#9825C211yesmaximal, characteristic, complemented
10#9566C211yesmaximal, characteristic, complemented
11#9554C211yesmaximal, characteristic
12#9218C211yesmaximal, characteristic, complemented
13#8987C211yesmaximal, characteristic, complemented
14#8447C211yesmaximal, characteristic
15C8×#103C211yesmaximal, characteristic, complemented
16C2×D4×Q8C2211yescharacteristic
17C2×#494C2211yescharacteristic
18C2×#477—21no—
19C2×#462C2211yescharacteristic
20C2×#451—21no—
21C2×#451—21no—
22C2×C8×Q8C2211yescharacteristic
23#2551C2211yescharacteristic
24#2542C2211yescharacteristic
25Q8×C4⋊C4C2211yescharacteristic
26#2099—21no—
27#2087—21no—
28#2057—21no—
29#2023—21no—
30#2001—21no—
31#2001—21no—
32#1990—21no—
33#1990—21no—
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1988—21no—
39#1988C2211yescharacteristic
40#1988C2211yescharacteristic
41#1981C2211yescharacteristic
42#1954C2211yescharacteristic
43#1947C2211yescharacteristic
44#1939C2211yescharacteristic
45#1934—21no—
46#1863—21no—
47#1845C2211yescharacteristic
48#1803C2211yescharacteristic
49#1802—21no—
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C4×#103C2211yescharacteristic
54C4×#102—21no—
55C4×#102C2211yescharacteristic
56C2×#332C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×Q8—21no—
68C22×Q16—21no—
69C22×Q16—21no—
70C22×Q16—21no—
71D4×Q8—22no—
72D4×Q8—22no—
73D4×Q8—22no—
74D4×Q8—41no—
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8—21no—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8C2311yescharacteristic
79C2×C4⋊Q8C2311yescharacteristic
80C2×C22⋊Q8—21no—
81C2×C22⋊Q8—21no—
82C2×C22⋊Q8—21no—
83C2×C22⋊Q8—41no—
84C2×C4×Q8—21no—
85C2×C4×Q8C2311yescharacteristic
86C2×C4×D4—21no—
87C2×C4×D4C2311yescharacteristic
88#494—22no—
89#494—41no—
90#477—44no—
91#462—22no—
92#462—22no—
93#451—44no—
94#451—44no—
95C8×Q8—22no—
96C8×Q8—41no—
97C2×#107—21no—
98C2×#107—21no—
99C2×#107—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#103D411yescharacteristic
111C2×#103D411yescharacteristic
112C2×#102—21no—
113C2×#102—21no—
114C2×#102D411yescharacteristic
115C2×#102D411yescharacteristic
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123#399—41no—
124#395—41no—
125#390—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—41no—
130#384—21no—
131#384—21no—
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134C4×C4⋊C4—21no—
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—41no—
150#340—41no—
151#340D411yescharacteristic
152#340D411yescharacteristic
153#340D411yescharacteristic
154#340D411yescharacteristic
155#336—21no—
156#336—21no—
157#332—22no—
158#332—41no—
159Q8⋊C8—22no—
160D4⋊C8—41no—
161C82—41no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—41no—
165C22×Q8—41no—
166C22×Q8—42no—
167C22×Q8—42no—
168C22×Q8—42no—
169C22×Q8—42no—
170C22×Q8—81no—
171C22×Q8—81no—
172C22×Q8C2×D411yescharacteristic
173C22×D4C2×D411yescharacteristic
174C23×C4—21no—
175C23×C4—41no—
176C2×Q16—42no—
177C2×Q16—42no—
178C2×Q16—42no—
179C2×Q16—42no—
180C2×Q16—42no—
181C2×Q16—42no—
182C2×Q16—42no—
183C2×Q16—81no—
184C2×Q16—81no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C4⋊Q8—24no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—42no—
203C4⋊Q8—42no—
204C4⋊Q8—42no—
205C4⋊Q8—81no—
206C4⋊Q8—81no—
207C4⋊Q8#4112yes—
208C22⋊Q8—44no—
209C22⋊Q8—42no—
210C22⋊Q8—44no—
211C22⋊Q8—42no—
212C22⋊Q8—42no—
213C22⋊Q8—42no—
214C22⋊Q8—81no—
215C22⋊Q8—81no—
216C4×Q8—21no—
217C4×Q8—21no—
218C4×Q8—42no—
219C4×Q8—81no—
220C4×Q8#4112yes—
221C4×Q8#4112yes—
222C4×D4—24no—
223C4×D4—21no—
224C4×D4—21no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—42no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—41no—
260C2×C42C2411yescharacteristic
261#107—42no—
262#107—42no—
263#107—81no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—42no—
273C4⋊C8—42no—
274#103—42no—
275#103—42no—
276#103—42no—
277#103—42no—
278#103D814yes—
279#103SD1614yes—
280#102—41no—
281#102—42no—
282#102—81no—
283#102Q1614yescomplemented
284C22⋊C8—42no—
285C22⋊C8—42no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—42no—
293C4×C8—81no—
294#95—21no—
295#95—21no—
296#95—41no—
297#95—41no—
298C24—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—22no—
303C2×Q8—44no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—82no—
316C2×Q8—82no—
317C2×Q8—82no—
318C2×Q8—82no—
319C2×Q8—84no—
320C2×Q8—84no—
321C2×Q8—84no—
322C2×Q8—84no—
323C2×Q8—161no—
324C2×Q8#12412yes—
325C2×Q8C2×D812yes—
326C2×Q8C2×SD1612yes—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4C2×Q1612yes—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—44no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4H(5,𝔽2)11yescharacteristic
356Q16—84no—
357Q16—84no—
358Q16—84no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—22no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8C2×D812yes—
384C2×C8C2×SD1612yes—
385C2×C8C2×Q1612yes—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—44no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4—81no—
418C4⋊C4—81no—
419C4⋊C4—82no—
420C4⋊C4—161no—
421C4⋊C4#12412yes—
422C4⋊C4C2×D812yes—
423C4⋊C4C2×SD1612yes—
424C4⋊C4C2×Q1612yes—
425C22⋊C4—44no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—82no—
429C22⋊C4—82no—
430C42—22no—
431C42—22no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—82no—
437C42C2×#4111yescharacteristic
438C42C2×#4111yescharacteristic
439C42C2×#4111yescharacteristic
440C42H(5,𝔽2)11yescharacteristic
441C23—22no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23D4211yescharacteristic
449Q8—22no—
450Q8—22no—
451Q8—22no—
452Q8—42no—
453Q8—42no—
454Q8—88no—
455Q8—81no—
456Q8—81no—
457Q8—81no—
458Q8—81no—
459Q8—81no—
460Q8—81no—
461Q8—164no—
462Q8#48614yes—
463D4—22no—
464D4—22no—
465D4—41no—
466D4—41no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
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—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—88no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4#52511yescharacteristic
508C2×C4C2×#12411yescharacteristic
509C2×C4#53811yescharacteristic
510C2×C4#54611yescharacteristic
511C2×C4#54611yescharacteristic
512C2×C4#55111yescharacteristic
513C2×C4C22×D811yescharacteristic
514C2×C4C22×SD1611yescharacteristic
515C2×C4C22×Q1611yescharacteristic
516C2×C4#57511yescharacteristic
517C2×C4#57611yescharacteristic
518C2×C4#57711yescharacteristic
519C2×C4#57711yescharacteristic
520C2×C4#57811yescharacteristic
521C8—22no—
522C8—22no—
523C8—22no—
524C8—42no—
525C8—41no—
526C8—81no—
527C8—81no—
528C8—81no—
529C8—81no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22#247211yescharacteristic
538C22#327511yescharacteristic
539C22D4×D811yescharacteristic
540C22D4×SD1611yescharacteristic
541C22D4×Q1611yescharacteristic
542C22#329011yescharacteristic
543C22#329311yescharacteristic
544C4—24no—
545C4—21no—
546C4—21no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—42no—
551C4—81no—
552C4—82no—
553C4—81no—
554C4—161no—
555C4—161no—
556C4—161no—
557C4#310411yescharacteristic
558C4#312111yescharacteristic
559C4C2×#48611yescharacteristic
560C4#313611yescharacteristic
561C4#329411yescharacteristic
562C4#329511yescharacteristic
563C4#329911yescharacteristic
564C4#331311yescharacteristic
565C2—82no—
566C2#1988611yescharacteristic
567C2#1989411yescharacteristic
568C2D8×Q1611yescharacteristic
569C2SD16×Q1611yescharacteristic
570C2#2722811yescharacteristic
571C2#2725411yescharacteristic
572C2#2728711yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #26046C2semidirect
6. #10538C2semidirect
8. #10350C2semidirect
9. #9825C2semidirect
10. #9566C2semidirect
12. #9218C2semidirect
13. #8987C2semidirect
15. C8×#103C2semidirect
283. #102Q16semidirect

Maximal subgroups

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

subgroupindex
#278132
#260462
#257602
Q8×#1022
#106162
#105382
#104212
#103502
#98252
#95662
#95542
#92182
#89872
#84472
C8×#1032