Finite Group Databasebeta

← #208176

Subgroups — #208176

Order 512.

808 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35291C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic
3#20431C211yesmaximal, characteristic
4#9906C211yesmaximal, characteristic, complemented
5#9901C211yesmaximal, characteristic, complemented
6#9709C211yesmaximal, characteristic, complemented
7#9697C211yesmaximal, characteristic, complemented
8#9139C211yesmaximal, characteristic
9#9135C211yesmaximal, characteristic
10#9131C211yesmaximal, characteristic
11#9127C211yesmaximal, characteristic
12#8236C211yesmaximal, characteristic, complemented
13#8235C211yesmaximal, characteristic, complemented
14C4×#332C211yesmaximal, characteristic, complemented
15C4×#333C211yesmaximal, characteristic, complemented
16C2×#423C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C2×#405—21no—
19#2862C2211yescharacteristic
20#2859C2211yescharacteristic
21#2857C2211yescharacteristic
22#2857C2211yescharacteristic
23#2645—21no—
24#2625—21no—
25#2514C412yes—
26#2509—21no—
27#2508—21no—
28#2495—21no—
29#2494—21no—
30#2489C412yes—
31#2453C412yes—
32#2421C412yes—
33#2323—21no—
34#2320—21no—
35C4×C4⋊Q8C2211yescharacteristic
36C4×C4⋊Q8C2211yescharacteristic
37C4×#117C2211yescharacteristic
38#1923—21no—
39#1923—21no—
40#1916—21no—
41#1916—21no—
42#1842C2211yescharacteristic
43#1831C2211yescharacteristic
44#1797C2211yescharacteristic
45#1797C2211yescharacteristic
46#1768—21no—
47#1768—21no—
48#1767C2211yescharacteristic
49#1767C2211yescharacteristic
50C4×C4⋊C8C2211yescharacteristic
51C4×C4⋊C8C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1729—21no—
55C2×#333C2211yescharacteristic
56C2×#333C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C22×M16—21no—
72C23×C8—21no—
73C2×C4⋊Q8C2311yescharacteristic
74C2×C4⋊Q8C2311yescharacteristic
75C2×#125C2×C412yes—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8C2×C412yes—
78C2×C4×Q8—21no—
79C2×C4×Q8C2×C412yes—
80C2×#117D411yescharacteristic
81C2×#117D411yescharacteristic
82C22×C42C2311yescharacteristic
83#423—22no—
84#423—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101#405—44no—
102C2×#97—21no—
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108#401—22no—
109#400—22no—
110#400—22no—
111#400—22no—
112#400—22no—
113#398—22no—
114#397—22no—
115#397—22no—
116#391—21no—
117#391—21no—
118#391—21no—
119#391—21no—
120#391C2×C412yes—
121#391C2×C412yes—
122#391C2×C412yes—
123#391C2×C412yes—
124#389—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—21no—
134#389—21no—
135#389C2×C412yes—
136#389C2×C412yes—
137#388—22no—
138#388—22no—
139#387—22no—
140#387—22no—
141#387—22no—
142#384C2311yescharacteristic
143#384C2311yescharacteristic
144#383—22no—
145#382—21no—
146#382—21no—
147#382—22no—
148#382C2×C412yes—
149#380—21no—
150#380—21no—
151#380—21no—
152#380C2×C412yes—
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4C2×C412yes—
157C4×C4⋊C4C2×C411yescharacteristic
158C4×C4⋊C4C2×C411yescharacteristic
159C4×C4⋊C4D411yescharacteristic
160C4×C4⋊C4D411yescharacteristic
161C4×C22⋊C4C2×C411yescharacteristic
162C4×C22⋊C4C2×C411yescharacteristic
163C4×C22⋊C4D411yescharacteristic
164C4×C22⋊C4D411yescharacteristic
165#376D411yescharacteristic
166#376D411yescharacteristic
167C43C2311yescharacteristic
168C43C2311yescharacteristic
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#333—22no—
180#333—22no—
181#333—22no—
182#333—22no—
183#332—22no—
184#332—22no—
185#332—22no—
186#332—22no—
187C22×Q8—21no—
188C22×Q8C22×C412yes—
189C23×C4C22×C411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C2×M16—24no—
193C2×M16—42no—
194C2×M16—42no—
195C22×C8—24no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—42no—
205C22×C8—42no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214#125—82no—
215C22⋊Q8—42no—
216C22⋊Q8—42no—
217C22⋊Q8—82no—
218C4×Q8—42no—
219C4×Q8—42no—
220C4×Q8—82no—
221#117—22no—
222#117—22no—
223#117—22no—
224#117—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
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—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×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4C22⋊C411yescharacteristic
248C2×C22⋊C4C22⋊C411yescharacteristic
249C2×C22⋊C4C22⋊C411yescharacteristic
250C2×C22⋊C4C22⋊C411yescharacteristic
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2411yescharacteristic
273C4⋊C8—24no—
274C4⋊C8—24no—
275C4⋊C8—21no—
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—81no—
288C4⋊C8—81no—
289C4⋊C8#4112yes—
290C4⋊C8#4112yes—
291C4⋊C8#4112yes—
292C4⋊C8#4112yes—
293C22⋊C8—81no—
294C22⋊C8—81no—
295#97—42no—
296#97—42no—
297C4×C8—42no—
298C4×C8—42no—
299C4×C8—81no—
300C4×C8—81no—
301#95—22no—
302#95—22no—
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#95—22no—
314#95—22no—
315#95C22⋊C411yescharacteristic
316#95C22⋊C411yescharacteristic
317#95C22⋊C411yescharacteristic
318#95C22⋊C411yescharacteristic
319#95C22⋊C411yescharacteristic
320#95C22⋊C411yescharacteristic
321#95C22⋊C411yescharacteristic
322#95C22⋊C411yescharacteristic
323C24C2×C22⋊C411yescharacteristic
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4C2×C22⋊C411yescharacteristic
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×C4#13512yes—
364C22×C4#13512yes—
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368M16—48no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—48no—
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—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—42no—
401C4⋊C4—42no—
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—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—82no—
419C4⋊C4—82no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—44no—
427C22⋊C4—82no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—44no—
453C42—82no—
454C42AGL(1,ℤ/8)12yes—
455C42#13712yes—
456C42C2×#4111yescharacteristic
457C42C2×#4111yescharacteristic
458C42C2×#4111yescharacteristic
459C42C2×#4111yescharacteristic
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23#41612yes—
467C23C22×C22⋊C411yescharacteristic
468Q8—82no—
469Q8—82no—
470Q8—81no—
471Q8—81no—
472Q8—81no—
473Q8—81no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
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—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4#41612yes—
529C2×C4#41712yes—
530C2×C4#41712yes—
531C2×C4C4×#4111yescharacteristic
532C2×C4C4×#4111yescharacteristic
533C2×C4#52511yescharacteristic
534C2×C4#52511yescharacteristic
535C2×C4#53211yescharacteristic
536C2×C4#53211yescharacteristic
537C2×C4C2×#13511yescharacteristic
538C2×C4C2×#13511yescharacteristic
539C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
540C2×C4C2×#13711yescharacteristic
541C2×C4#57711yescharacteristic
542C2×C4#57711yescharacteristic
543C2×C4#57811yescharacteristic
544C2×C4#57811yescharacteristic
545C8—42no—
546C8—42no—
547C8—42no—
548C8—42no—
549C8—81no—
550C8—81no—
551C8—81no—
552C8—81no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22#229711yescharacteristic
561C22#288811yescharacteristic
562C22#288911yescharacteristic
563C22C2×#41611yescharacteristic
564C22C2×#41711yescharacteristic
565C22#289811yescharacteristic
566C22#289811yescharacteristic
567C4—22no—
568C4—22no—
569C4—22no—
570C4—21no—
571C4—21no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4#310511yescharacteristic
585C4#310511yescharacteristic
586C4#310811yescharacteristic
587C4#310811yescharacteristic
588C4#317411yescharacteristic
589C4#317711yescharacteristic
590C4#317811yescharacteristic
591C4#317811yescharacteristic
592C2—42no—
593C2#1772511yescharacteristic
594C2#1772611yescharacteristic
595C2#1776311yescharacteristic
596C2#1776411yescharacteristic
597C2#2138811yescharacteristic
598C2#2141311yescharacteristic
599C2#2142711yescharacteristic
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. #9906C2semidirect
5. #9901C2semidirect
6. #9709C2semidirect
7. #9697C2semidirect
12. #8236C2semidirect
13. #8235C2semidirect
14. C4×#332C2semidirect
15. C4×#333C2semidirect

Maximal subgroups

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

subgroupindex
#352912
#219222
#204312
#99062
#99012
#97092
#96972
#91392
#91352
#91312
#91272
#82362
#82352
C4×#3322
C4×#3332