Finite Group Databasebeta

← #365204

Subgroups — #365204

Order 512.

1190 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20846C211yesmaximal, characteristic
2C2×#1824C211yesmaximal, characteristic
3C2×#1821C211yesmaximal, characteristic
4#20379C211yesmaximal, characteristic
5C2×#1736C211yesmaximal, characteristic
6#11303C212yesmaximal, complemented
7#11290C212yesmaximal, complemented
8#10896C212yesmaximal, complemented
9#10827C212yesmaximal, complemented
10#10739C211yesmaximal, characteristic
11#10734C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#428C2211yescharacteristic
16C2×#428C2211yescharacteristic
17C2×#407C2211yescharacteristic
18C2×#407C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C2×#405C2212yes—
21C2×#380C2211yescharacteristic
22#1854—21no—
23#1854—21no—
24#1853—21no—
25#1852—21no—
26#1824—21no—
27#1824—21no—
28#1824—22no—
29#1821C418yescomplemented
30#1750—22no—
31#1749—21no—
32C4×C22⋊C8—21no—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#341C2212yes—
38C2×#341C2212yes—
39#1736—21no—
40#1736—21no—
41#1736C2212yes—
42#1736C2211yescharacteristic
43#1736C2211yescharacteristic
44#1736C2212yes—
45C2×#340—21no—
46C2×#340—21no—
47C2×#340C2212yes—
48C2×#340C2212yes—
49#1729C2212yes—
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C22×M16—21no—
53C22×M16—24no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—42no—
57C22×M16C2312yes—
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2312yes—
61C23×C8—21no—
62C23×C8C2312yes—
63C2×#117—21no—
64C2×#117—21no—
65C2×#117—24no—
66C2×#117—42no—
67C2×#117D411yescharacteristic
68C2×#117D411yescharacteristic
69C2×#117C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4C2×C414yes—
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C42—21no—
88#428—24no—
89#428—42no—
90C2×#107—21no—
91C2×#107—21no—
92C2×#106—21no—
93C2×#106—21no—
94#407—22no—
95#407—22no—
96#407C2×C414yes—
97#407C2×C414yes—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8D411yescharacteristic
105C2×C22⋊C8D411yescharacteristic
106C2×C22⋊C8D412yes—
107#405—24no—
108#405—42no—
109C2×#97—22no—
110C2×C4×C8—22no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115C2×#95—21no—
116#343—24no—
117#343—42no—
118#343C2×C418yescomplemented
119#343C2×C418yescomplemented
120#341—24no—
121#341—42no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#336—22no—
135#336—22no—
136C25#4111yescharacteristic
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—41no—
148C23×C4—41no—
149C23×C4—42no—
150C23×C4—82no—
151C23×C4—82no—
152C23×C4C22×C412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4#4111yescharacteristic
158C23×C4#4112yes—
159C23×C4#4112yes—
160C23×C4#4112yes—
161C23×C4C2411yescharacteristic
162C2×M16—24no—
163C2×M16—22no—
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—48no—
180C2×M16—48no—
181C2×M16—42no—
182C2×M16—42no—
183C2×M16C22×C412yes—
184C2×M16C22×C414yes—
185C2×M16C22×C412yes—
186C2×M16#4114yes—
187C22×C8—24no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—42no—
199C22×C8—42no—
200C22×C8C2×D412yes—
201C22×C8C2×D412yes—
202#117—22no—
203#117—22no—
204#117—22no—
205#117—22no—
206#117—22no—
207#117—44no—
208#117—44no—
209#117—41no—
210#117—41no—
211#117#4114yes—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4C22×C414yes—
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—82no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—44no—
261#107—42no—
262#107—42no—
263#106—42no—
264#106—42no—
265C22⋊C8—24no—
266C22⋊C8—24no—
267C22⋊C8—24no—
268C22⋊C8—24no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8—24no—
272#97—44no—
273C4×C8—44no—
274#95—41no—
275#95—41no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—41no—
281C24—41no—
282C24#11712yes—
283C24#12111yescharacteristic
284C24#12111yescharacteristic
285C24#12612yes—
286C24C2×#4111yescharacteristic
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—81no—
327C22×C4—81no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—88no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—88no—
336C22×C4—81no—
337C22×C4—81no—
338C22×C4—81no—
339C22×C4—81no—
340C22×C4#11712yes—
341C22×C4C4×D412yes—
342C22×C4C4×D412yes—
343C22×C4C4×D412yes—
344C22×C4C4×D412yes—
345C22×C4#12111yescharacteristic
346C22×C4#12111yescharacteristic
347C22×C4#12312yes—
348C22×C4#12312yes—
349C22×C4#12412yes—
350C22×C4#12412yes—
351C22×C4#12612yes—
352C22×C4C23×C411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4112yes—
358C22×C4C2×#4112yes—
359C22×C4C2×#4111yescharacteristic
360C22×C4H(5,𝔽2)12yes—
361C22×C42−1+412yes—
362M16—24no—
363M16—24no—
364M16—24no—
365M16—24no—
366M16—24no—
367M16—24no—
368M16—48no—
369M16—42no—
370M16—42no—
371M16—42no—
372M16—42no—
373M16—42no—
374M16—42no—
375C2×C8—22no—
376C2×C8—24no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—24no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—48no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—41no—
391C2×C8—42no—
392C2×C8—41no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4#12312yes—
419C4⋊C4#12312yes—
420C22⋊C4—24no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—82no—
425C22⋊C4—82no—
426C42—24no—
427C42—42no—
428C42—42no—
429C42—42no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—24no—
439C23—21no—
440C23—21no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—42no—
449C23—42no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23#38812yes—
455C23#38812yes—
456C23#47112yes—
457C23#47312yes—
458C23C2×#11711yescharacteristic
459C23C2×#12111yescharacteristic
460C23C2×#12611yescharacteristic
461C23#55112yes—
462C23#55212yes—
463C2×C4—24no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—84no—
495C2×C4—84no—
496C2×C4—88no—
497C2×C4—84no—
498C2×C4—88no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—84no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—162no—
509C2×C4—162no—
510C2×C4#38812yes—
511C2×C4#38812yes—
512C2×C4C2×C4×D411yescharacteristic
513C2×C4C2×C4×D411yescharacteristic
514C2×C4C4×#4112yes—
515C2×C4#52012yes—
516C2×C4C2×#12311yescharacteristic
517C2×C4C2×#12311yescharacteristic
518C2×C4#52511yescharacteristic
519C2×C4C2×#12411yescharacteristic
520C2×C4#52912yes—
521C2×C4#53411yescharacteristic
522C2×C4#53611yescharacteristic
523C2×C4#53912yes—
524C2×C4#54612yes—
525C2×C4#54712yes—
526C2×C4#55112yes—
527C2×C4#55112yes—
528C2×C4#55212yes—
529C2×C4#55212yes—
530C8—44no—
531C8—42no—
532C8—44no—
533C8—42no—
534C8—42no—
535C8—42no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—48no—
554C22—48no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22#193014yes—
560C22C2×#38811yescharacteristic
561C22#234312yes—
562C22#241211yescharacteristic
563C22#241311yescharacteristic
564C22#249312yes—
565C22C2×#47111yescharacteristic
566C22C2×#47311yescharacteristic
567C22#306811yescharacteristic
568C22#306811yescharacteristic
569C4—22no—
570C4—22no—
571C4—42no—
572C4—42no—
573C4—84no—
574C4—81no—
575C4—81no—
576C4—161no—
577C4—161no—
578C4C4×#12311yescharacteristic
579C4#232211yescharacteristic
580C4#235912yes—
581C4#241912yes—
582C4#250311yescharacteristic
583C4#256111yescharacteristic
584C2—22no—
585C2—24no—
586C2—22no—
587C2—42no—
588C2#1397211yescharacteristic
589C2C2×#193011yescharacteristic
590C2#2107811yescharacteristic
591C2#2457211yescharacteristic
592C2#2457311yescharacteristic
593C2#2462211yescharacteristic
594C2#2462311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11303C2semidirect
7. #11290C2semidirect
8. #10896C2semidirect
9. #10827C2semidirect
29. #1821C4semidirect
118. #343C2×C4semidirect
119. #343C2×C4semidirect

Maximal subgroups

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

subgroupindex
#208462
C2×#18242
C2×#18212
#203792
C2×#17362
#113032
#112902
#108962
#108272
#107392
#107342