Finite Group Databasebeta

← #376398

Subgroups — #376398

Order 512.

1073 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2093C211yesmaximal, characteristic
2#21037C211yesmaximal, characteristic, complemented
3#20879C211yesmaximal, characteristic
4#20455C211yesmaximal, characteristic, complemented
5#20416C211yesmaximal, characteristic
6#11365C211yesmaximal, characteristic, complemented
7#11210C211yesmaximal, characteristic, complemented
8#11127C211yesmaximal, characteristic
9#10958C211yesmaximal, characteristic
10#10943C211yesmaximal, characteristic
11#10910C211yesmaximal, characteristic, complemented
12#10909C211yesmaximal, characteristic, complemented
13#10891C211yesmaximal, characteristic, complemented
14#10870C211yesmaximal, characteristic, complemented
15#10749C211yesmaximal, characteristic
16C2×#518C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C2×#418C2211yescharacteristic
22C2×#418C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#100C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×#405C2211yescharacteristic
28#2316C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#384C2211yescharacteristic
31#2274C2211yescharacteristic
32#2274C2211yescharacteristic
33#2093C418yescomplemented
34#1921—22no—
35#1918—22no—
36#1864—22no—
37#1862—22no—
38#1776—22no—
39C4×#107—22no—
40#1762—22no—
41C4×#102—22no—
42C2×#343C2211yescharacteristic
43C2×#343C2211yescharacteristic
44C2×#343C2211yescharacteristic
45C2×#341C2211yescharacteristic
46#1736C2211yescharacteristic
47#1736C2211yescharacteristic
48#1736C2211yescharacteristic
49#1736C2211yescharacteristic
50#1736C2211yescharacteristic
51#1736C2211yescharacteristic
52#1735C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C23×D4C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67#518—22no—
68#518—22no—
69#518—22no—
70#518—41no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C22×C4⋊C4—41no—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—41no—
82C22×C22⋊C4—41no—
83C22×C42C2311yescharacteristic
84C2×#108—22no—
85C2×#108—22no—
86C2×#108—41no—
87#427—22no—
88#427—22no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#107—21no—
92C2×#107—21no—
93C2×#107—22no—
94C2×#107—22no—
95C2×#107C2×C414yescomplemented
96C2×#107C2×C414yescomplemented
97C2×#107D412yes—
98C2×#106D412yes—
99#418—22no—
100#418—22no—
101#418—41no—
102#418Q814yes—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—22no—
108C2×#102—22no—
109C2×#102—22no—
110C2×#102—22no—
111C2×#102C2×C414yes—
112C2×#102C2×C414yes—
113C2×#100—22no—
114C2×#100—22no—
115C2×#100—41no—
116#405—22no—
117#405—22no—
118#405—22no—
119#405—41no—
120C2×#97—21no—
121C2×#97—21no—
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124#390—41no—
125#390C2×C414yes—
126#389—22no—
127#389—22no—
128#384—41no—
129#384C2×C414yescomplemented
130#380—41no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C22⋊C4—41no—
136C4×C22⋊C4—41no—
137#343—41no—
138#343—41no—
139#343—41no—
140#341—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#340—22no—
146#340—22no—
147#340D412yes—
148#340D412yes—
149#339—22no—
150#339—22no—
151#339—41no—
152#339—41no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336C2×C414yescomplemented
158C25—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4C22×C412yes—
171C22×D4C2×Q812yes—
172C22×D4#4112yes—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—42no—
178C23×C4C2411yescharacteristic
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16C2×Q812yes—
202C2×M16#4112yes—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8C22×C412yes—
216C22×C8C22×C412yes—
217C22×C8C2×D411yescharacteristic
218C22×C8C2×D411yescharacteristic
219C22×C8C2×D411yescharacteristic
220C22×C8C2×D411yescharacteristic
221C22×C8#4112yes—
222C22×C8#4112yes—
223C4×D4—28no—
224C4×D4—44no—
225C4×D4—44no—
226C4×D4—44no—
227#117—21no—
228#117—21no—
229#117—21no—
230#117—21no—
231#117—21no—
232#117—21no—
233#117—21no—
234#117—21no—
235#117—21no—
236#117—21no—
237#117—21no—
238#117—21no—
239#117C2×Q812yes—
240#117#4112yes—
241#117#4112yes—
242#117#4112yes—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—84no—
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C22×C412yes—
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—84no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42C22×C412yes—
289#108—44no—
290#107—24no—
291#107—24no—
292#107—24no—
293#107—24no—
294#107—24no—
295#106—24no—
296#102—24no—
297#102—24no—
298#102—24no—
299#102—24no—
300#102—28no—
301#102—44no—
302#100—44no—
303#97—24no—
304#97—42no—
305C4×C8—24no—
306C4×C8—42no—
307#95—42no—
308#95—42no—
309#95—42no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—41no—
314C24—41no—
315C24—41no—
316C24—41no—
317C24—82no—
318C24—82no—
319C24H(5,𝔽2)11yescharacteristic
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4C4×Q814yes—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—48no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4—162no—
371C22×C4—162no—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×Q811yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4H(5,𝔽2)11yescharacteristic
384C22×C4H(5,𝔽2)11yescharacteristic
385C22×C42−1+411yescharacteristic
386M16—22no—
387M16—22no—
388M16—22no—
389M16—22no—
390M16—41no—
391M16—41no—
392M16—41no—
393M16—41no—
394M16—41no—
395M16—41no—
396M16—41no—
397M16—41no—
398M16—41no—
399M16—41no—
400M16—41no—
401M16—41no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8C4×D412yes—
431C2×C8C4×D412yes—
432C2×C8C4×D412yes—
433C2×C8C4×D412yes—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—88no—
453C4⋊C4—162no—
454C4⋊C4C22⋊Q814yes—
455C4⋊C4#12314yes—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—48no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—88no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—41no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—162no—
479C42—162no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—82no—
490C23—84no—
491C23—84no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—162no—
496C23—162no—
497C23—161no—
498C23—161no—
499C23#51811yescharacteristic
500C23#53811yescharacteristic
501C23#53911yescharacteristic
502C23D4211yescharacteristic
503C23#55111yescharacteristic
504C23#55411yescharacteristic
505C23#55911yescharacteristic
506D4—24no—
507D4—24no—
508D4—42no—
509D4—42no—
510D4—42no—
511D4—42no—
512D4—42no—
513D4—42no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—168no—
541C2×C4—161no—
542C2×C4—161no—
543C2×C4—161no—
544C2×C4C2×C4×D411yescharacteristic
545C2×C4C2×C4×D411yescharacteristic
546C2×C4C2×C4×Q811yescharacteristic
547C2×C4C4×#4111yescharacteristic
548C2×C4C4×#4111yescharacteristic
549C2×C4#52011yescharacteristic
550C2×C4C2×C22⋊Q811yescharacteristic
551C2×C4C2×#12311yescharacteristic
552C2×C4#52511yescharacteristic
553C2×C4#52511yescharacteristic
554C2×C4#52911yescharacteristic
555C2×C4#52911yescharacteristic
556C2×C4#52911yescharacteristic
557C2×C4#53311yescharacteristic
558C2×C4#54011yescharacteristic
559C2×C4#54211yescharacteristic
560C2×C4#54611yescharacteristic
561C2×C4#54611yescharacteristic
562C2×C4#54711yescharacteristic
563C2×C4D4×Q811yescharacteristic
564C2×C4#55011yescharacteristic
565C2×C4#55011yescharacteristic
566C2×C4#55111yescharacteristic
567C2×C4#55111yescharacteristic
568C2×C4#55211yescharacteristic
569C2×C4#55211yescharacteristic
570C2×C4#55411yescharacteristic
571C2×C4#55411yescharacteristic
572C8—22no—
573C8—22no—
574C8—22no—
575C8—22no—
576C8—42no—
577C8—42no—
578C8—81no—
579C8—81no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—44no—
589C22—84no—
590C22—84no—
591C22—84no—
592C22—82no—
593C22—82no—
594C22—162no—
595C22—162no—
596C22—162no—
597C22#233511yescharacteristic
598C22#236611yescharacteristic
599C22#236811yescharacteristic
600C22#241811yescharacteristic
601C22#244911yescharacteristic
602C22#257511yescharacteristic
603C22#258911yescharacteristic
604C22#328912yes—
605C22#329212yes—
606C4—44no—
607C4—44no—
608C4—44no—
609C4—82no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4C4×#12311yescharacteristic
615C4C4×C22⋊Q811yescharacteristic
616C4#235911yescharacteristic
617C4#236011yescharacteristic
618C4#242311yescharacteristic
619C4#245411yescharacteristic
620C4#247111yescharacteristic
621C4#247911yescharacteristic
622C2—21no—
623C2—21no—
624C2—21no—
625C2—21no—
626C2—84no—
627C2#1417811yescharacteristic
628C2#2320111yescharacteristic
629C2#2320311yescharacteristic
630C2#2492512yes—
631C2#2640012yes—
subgroupG/H
center, socleC23D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21037C2semidirect
4. #20455C2semidirect
6. #11365C2semidirect
7. #11210C2semidirect
11. #10910C2semidirect
12. #10909C2semidirect
13. #10891C2semidirect
14. #10870C2semidirect
33. #2093C4semidirect
95. C2×#107C2×C4semidirect
96. C2×#107C2×C4semidirect
129. #384C2×C4semidirect
157. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20932
#210372
#208792
#204552
#204162
#113652
#112102
#111272
#109582
#109432
#109102
#109092
#108912
#108702
#107492