Finite Group Databasebeta

← #207194

Subgroups — #207194

Order 512.

828 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic
2#21978C211yesmaximal, characteristic
3#20431C211yesmaximal, characteristic
4#9907C211yesmaximal, characteristic, complemented
5#9906C211yesmaximal, characteristic, complemented
6#9904C211yesmaximal, characteristic, complemented
7#9903C211yesmaximal, characteristic, complemented
8#9901C211yesmaximal, characteristic, complemented
9#9900C211yesmaximal, characteristic, complemented
10#9899C211yesmaximal, characteristic, complemented
11#9896C211yesmaximal, characteristic, complemented
12#8045C211yesmaximal, characteristic
13#8044C211yesmaximal, characteristic
14#8042C211yesmaximal, characteristic
15#8041C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#423C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C2×#405—21no—
21#2868C2211yescharacteristic
22#2859C2211yescharacteristic
23#2859C2211yescharacteristic
24#2859C2211yescharacteristic
25#2629—21no—
26#2628—21no—
27#2570—21no—
28#2570C412yes—
29#2550—21no—
30#2550C412yes—
31#2441—21no—
32#2441C412yes—
33#2439—21no—
34#2439C412yes—
35#2331—21no—
36#2330—21no—
37#2285C2211yescharacteristic
38#2285C2211yescharacteristic
39C4×#117C2211yescharacteristic
40#1940—21no—
41#1940—21no—
42#1851—21no—
43#1851—21no—
44#1842C2211yescharacteristic
45#1842C2211yescharacteristic
46#1831C2211yescharacteristic
47#1831C2211yescharacteristic
48#1768—21no—
49#1768—21no—
50#1767C2211yescharacteristic
51#1767C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1729—21no—
55#1296C2211yescharacteristic
56#1296C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C22×M16—21no—
72C22×M16—21no—
73C23×C8—21no—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8C2×C412yes—
77C2×#125—21no—
78C2×#125—21no—
79C2×#125C2×C412yes—
80C2×#117—21no—
81C2×#117C2×C412yes—
82C2×#117D411yescharacteristic
83C2×#117D411yescharacteristic
84C22×C42C2311yescharacteristic
85#432—22no—
86#432—22no—
87#423—22no—
88#423—22no—
89#423—22no—
90#423Q814yescomplemented
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8Q814yescomplemented
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C22⋊C8—21no—
109#405—44no—
110C2×#97—21no—
111C2×#97—21no—
112C2×#97—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115#400—22no—
116#399—22no—
117#398—22no—
118#398—22no—
119#396—22no—
120#396—22no—
121#395—22no—
122#393—22no—
123#391—22no—
124#391—22no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#389—21no—
132#389—21no—
133#389—22no—
134#389—22no—
135#389C2×C412yes—
136#389C2×C412yes—
137#389C2×C412yes—
138#389C2×C412yes—
139#387—22no—
140#387—22no—
141#387—22no—
142#387—22no—
143#384—21no—
144#384—21no—
145#384C2×C412yes—
146#384C2311yescharacteristic
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#384C2311yescharacteristic
150#382—21no—
151#382—21no—
152#382—21no—
153#382C2×C412yes—
154#382C2×C412yes—
155#380—21no—
156#380—21no—
157#380—21no—
158#380C2×C412yes—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4C2×C412yes—
163C4×C4⋊C4C2×C411yescharacteristic
164C4×C4⋊C4C2×C411yescharacteristic
165C4×C4⋊C4D411yescharacteristic
166C4×C4⋊C4D411yescharacteristic
167C4×C22⋊C4C2×C411yescharacteristic
168C4×C22⋊C4C2×C411yescharacteristic
169C4×C22⋊C4D411yescharacteristic
170C4×C22⋊C4D411yescharacteristic
171#376D411yescharacteristic
172#376D411yescharacteristic
173C43C2311yescharacteristic
174C43C2311yescharacteristic
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182#336—21no—
183C22×Q8—22no—
184C23×C4C22×C411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C2×M16—24no—
188C2×M16—42no—
189C2×M16—42no—
190C2×M16—42no—
191C2×M16—81no—
192C2×M16—81no—
193C22×C8—24no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—42no—
203C22×C8—42no—
204C4⋊Q8—42no—
205C4⋊Q8—42no—
206C4⋊Q8—81no—
207C4⋊Q8—81no—
208C4⋊Q8—82no—
209#125—42no—
210#125—42no—
211#125—81no—
212#125—81no—
213#125—82no—
214#117—22no—
215#117—22no—
216#117—41no—
217#117—41no—
218#117—81no—
219#117—81no—
220#117—82no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4C22⋊C411yescharacteristic
239C2×C4⋊C4C22⋊C411yescharacteristic
240C2×C4⋊C4C22⋊C411yescharacteristic
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22×C412yes—
243C2×C4⋊C4C22×C412yes—
244C2×C4⋊C4C22×C412yes—
245C2×C4⋊C4C22×C412yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4C22⋊C411yescharacteristic
249C2×C22⋊C4C22⋊C411yescharacteristic
250C2×C22⋊C4C22⋊C411yescharacteristic
251C2×C22⋊C4C22⋊C411yescharacteristic
252C2×C42—22no—
253C2×C42—22no—
254C2×C42C22×C411yescharacteristic
255C2×C42C22×C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×Q812yes—
270C2×C42C2×Q812yes—
271C2×C42#4112yes—
272C2×C42#4112yes—
273C2×C42C2411yescharacteristic
274C4⋊C8—24no—
275C4⋊C8—24no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—42no—
289C4⋊C8—81no—
290C4⋊C8—81no—
291C4⋊C8—81no—
292C4⋊C8—81no—
293C4⋊C8C2×Q812yes—
294C4⋊C8C2×Q812yes—
295C4⋊C8C2×Q812yes—
296C4⋊C8C2×Q812yes—
297C22⋊C8—42no—
298#97—42no—
299#97—42no—
300#97—42no—
301C4×C8—42no—
302C4×C8—42no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95C22⋊C411yescharacteristic
314#95C22⋊C411yescharacteristic
315#95C22⋊C411yescharacteristic
316#95C22⋊C411yescharacteristic
317#95C22⋊C411yescharacteristic
318#95C22⋊C411yescharacteristic
319#95C22⋊C411yescharacteristic
320#95C22⋊C411yescharacteristic
321C24C2×C22⋊C411yescharacteristic
322C2×Q8—44no—
323C2×Q8—82no—
324C2×Q8—82no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C4×Q812yes—
351C22×C4C4×Q812yes—
352C22×C4C22⋊Q812yes—
353C22×C4C22⋊Q812yes—
354C22×C4#12412yes—
355C22×C4C4⋊Q812yes—
356C22×C4C23×C411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359M16—48no—
360M16—84no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—48no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
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×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
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—44no—
412C4⋊C4—44no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—82no—
424C22⋊C4—82no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—82no—
444C42—82no—
445C42C4×Q812yes—
446C42C4×Q812yes—
447C42C22⋊Q812yes—
448C42C22⋊Q812yes—
449C42#12412yes—
450C42C4⋊Q812yes—
451C42AGL(1,ℤ/8)12yes—
452C42#13712yes—
453C42C22×Q811yescharacteristic
454C42C22×Q811yescharacteristic
455C42C2×#4111yescharacteristic
456C42C2×#4111yescharacteristic
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23#39112yes—
464C23C22×C22⋊C411yescharacteristic
465Q8—164no—
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—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4#39112yes—
519C2×C4#39112yes—
520C2×C4#39112yes—
521C2×C4#41712yes—
522C2×C4#41712yes—
523C2×C4C2×C4×Q811yescharacteristic
524C2×C4C2×C4×Q811yescharacteristic
525C2×C4C2×C22⋊Q811yescharacteristic
526C2×C4C2×C22⋊Q811yescharacteristic
527C2×C4C2×#12411yescharacteristic
528C2×C4C2×C4⋊Q811yescharacteristic
529C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
530C2×C4C2×#13711yescharacteristic
531C2×C4#57511yescharacteristic
532C2×C4#57511yescharacteristic
533C2×C4#57611yescharacteristic
534C2×C4#57711yescharacteristic
535C2×C4#57711yescharacteristic
536C2×C4#57811yescharacteristic
537C8—42no—
538C8—42no—
539C8—42no—
540C8—42no—
541C8—81no—
542C8—81no—
543C8—81no—
544C8—81no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22C2×#39111yescharacteristic
553C22#288811yescharacteristic
554C22#288911yescharacteristic
555C22C2×#41711yescharacteristic
556C22#289611yescharacteristic
557C22#289711yescharacteristic
558C22#289811yescharacteristic
559C4—22no—
560C4—22no—
561C4—21no—
562C4—21no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4#307711yescharacteristic
577C4#307711yescharacteristic
578C4#307811yescharacteristic
579C4#307811yescharacteristic
580C4#313611yescharacteristic
581C4#313711yescharacteristic
582C4#316311yescharacteristic
583C4#316411yescharacteristic
584C2—42no—
585C2#1776411yescharacteristic
586C2#1776511yescharacteristic
587C2#1776611yescharacteristic
588C2#1776711yescharacteristic
589C2#2131211yescharacteristic
590C2#2132611yescharacteristic
591C2#2133811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9907C2semidirect
5. #9906C2semidirect
6. #9904C2semidirect
7. #9903C2semidirect
8. #9901C2semidirect
9. #9900C2semidirect
10. #9899C2semidirect
11. #9896C2semidirect
90. #423Q8semidirect
99. C2×C4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#352942
#219782
#204312
#99072
#99062
#99042
#99032
#99012
#99002
#98992
#98962
#80452
#80442
#80422
#80412