Finite Group Databasebeta

← #400106

Subgroups — #400106

Order 512.

1604 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic
2C2×#2016C211yesmaximal, characteristic
3C2×#1927C211yesmaximal, characteristic
4C2×#1878C212yesmaximal
5C2×#1865C212yesmaximal
6#15193C211yesmaximal, characteristic
7#11346C211yesmaximal, characteristic
8#11067C212yesmaximal
9#11024C212yesmaximal
10#11010C211yesmaximal, characteristic
11#10806C211yesmaximal, characteristic
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C22×#106C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C22×#103C2212yes—
20C22×#103C2212yes—
21#2365—24no—
22#2365—24no—
23C2×#391C2212yes—
24C2×#391C2211yescharacteristic
25C2×#391C2212yes—
26C2×#389—21no—
27C2×#389C2212yes—
28C2×#387—22no—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#2016—24no—
33#1927—24no—
34#1878—28no—
35#1865—28no—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1445—22no—
44#1445—21no—
45#1444—21no—
46C4⋊#95C2211yescharacteristic
47C23×Q8C2311yescharacteristic
48C23×Q8C2312yes—
49C22×Q16—24no—
50C22×Q16—24no—
51C22×Q16—24no—
52C22×Q16—24no—
53C22×Q16—24no—
54C22×Q16—42no—
55C22×Q16—42no—
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C23×C8C2311yescharacteristic
59C2×C4⋊Q8—24no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8D414yes—
63C2×C4×Q8—24no—
64C2×C4×Q8—24no—
65C2×C4×Q8—24no—
66C2×C4×Q8—24no—
67C2×C4×Q8—42no—
68C2×C4×Q8—42no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C42—22no—
80C22×C42—21no—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83C2×#106—24no—
84C2×#106—22no—
85C2×#103—22no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—22no—
89C2×#103—24no—
90C2×#103—22no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—22no—
97C2×#103—42no—
98C2×#103—42no—
99#391—24no—
100#391—24no—
101#391—24no—
102#391—22no—
103#391—22no—
104#391—42no—
105#389—28no—
106#389—24no—
107#389—42no—
108#387—216no—
109C4×C4⋊C4—28no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4D414yes—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95D412yes—
125#340—22no—
126#340—42no—
127#340D418yes—
128#336—24no—
129#336—24no—
130#336—22no—
131#336—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—21no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—21no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8C2×D414yes—
153C22×Q8#4112yes—
154C22×Q8#4114yes—
155C22×Q8#4114yes—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C2×Q16—216no—
168C2×Q16—48no—
169C2×Q16—48no—
170C2×Q16—48no—
171C2×Q16—48no—
172C2×Q16—48no—
173C2×Q16—48no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8C2×D414yes—
190C22×C8#4112yes—
191C4⋊Q8—28no—
192C4⋊Q8—28no—
193C4⋊Q8—216no—
194C4×Q8—48no—
195C4×Q8—48no—
196C4×Q8—48no—
197C4×Q8—48no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D414yes—
231C2×C4⋊C4C2×D414yes—
232C2×C4⋊C4C2×D412yes—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—28no—
236C2×C42—24no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—44no—
244C2×C42—44no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249#106—44no—
250#103—216no—
251#103—28no—
252#103—48no—
253#103—48no—
254#103—44no—
255#95—28no—
256#95—28no—
257#95—24no—
258#95—24no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—42no—
263#95—42no—
264#95—42no—
265#95—81no—
266C24C22×D411yescharacteristic
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—22no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—41no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—41no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—42no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8#12318yes—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4#12012yes—
369C22×C4#12012yes—
370C22×C4#13512yes—
371C22×C4#13714yes—
372C22×C4#13712yes—
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D412yes—
376C22×C4C2×#4112yes—
377C22×C4C2×#4112yes—
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4H(5,𝔽2)12yes—
381C22×C4H(5,𝔽2)11yescharacteristic
382C22×C42−1+411yescharacteristic
383Q16—432no—
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—42no—
390C2×C8—41no—
391C2×C8—42no—
392C2×C8—41no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—42no—
400C2×C8—42no—
401C4⋊C4—28no—
402C4⋊C4—216no—
403C4⋊C4—216no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—416no—
411C4⋊C4—42no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—88no—
420C4⋊C4#12714yes—
421C42—24no—
422C42—22no—
423C42—22no—
424C42—416no—
425C42—42no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—48no—
430C23C2×#12011yescharacteristic
431C23#52512yes—
432C23D4212yes—
433C23#54811yescharacteristic
434C23#55011yescharacteristic
435C23C2×#13511yescharacteristic
436C23C2×#13712yes—
437C23C2×#13711yescharacteristic
438C23#57711yescharacteristic
439C23#57811yescharacteristic
440C23#57812yes—
441Q8—28no—
442Q8—28no—
443Q8—44no—
444Q8—44no—
445Q8—44no—
446Q8—44no—
447Q8—44no—
448Q8—44no—
449Q8—42no—
450Q8—42no—
451Q8—44no—
452Q8—81no—
453Q8—82no—
454Q8—81no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—82no—
507C2×C4—81no—
508C2×C4—82no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4#45214yes—
515C2×C4C2×#12312yes—
516C2×C4#52511yescharacteristic
517C2×C4#52511yescharacteristic
518C2×C4#52511yescharacteristic
519C2×C4#52512yes—
520C2×C4#52512yes—
521C2×C4C2×#12711yescharacteristic
522C2×C4#53511yescharacteristic
523C2×C4#53711yescharacteristic
524C2×C4#53812yes—
525C2×C4#53912yes—
526C2×C4D4212yes—
527C2×C4#54611yescharacteristic
528C2×C4#54612yes—
529C2×C4#54711yescharacteristic
530C2×C4#54812yes—
531C2×C4#55112yes—
532C2×C4#55312yes—
533C8—42no—
534C8—44no—
535C8—81no—
536C22#240711yescharacteristic
537C22#243311yescharacteristic
538C22#245612yes—
539C22#246312yes—
540C22#284911yescharacteristic
541C22C2×#45211yescharacteristic
542C22#300411yescharacteristic
543C22#301111yescharacteristic
544C22#301311yescharacteristic
545C22#310812yes—
546C22#311312yes—
547C22#312112yes—
548C22#312712yes—
549C22#326612yes—
550C22#327012yes—
551C22#327612yes—
552C22#328312yes—
553C22#336411yescharacteristic
554C22#336911yescharacteristic
555C22#337412yes—
556C22#339411yescharacteristic
557C22#339811yescharacteristic
558C22#339812yes—
559C4—24no—
560C4—44no—
561C4—44no—
562C4—48no—
563C4—42no—
564C4—82no—
565C4—82no—
566C4—82no—
567C4—84no—
568C4—161no—
569C4#239711yescharacteristic
570C4#241812yes—
571C4#244312yes—
572C4#254011yescharacteristic
573C4#259811yescharacteristic
574C4#263611yescharacteristic
575C2#1479611yescharacteristic
576C2#2418911yescharacteristic
577C2#2420311yescharacteristic
578C2#2575811yescharacteristic
579C2#2576611yescharacteristic
580C2#2658912yes—
581C2#2660712yes—
582C2#2689512yes—
583C2#2691112yes—
584C2#3300811yescharacteristic
585C2#3301911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#23652
C2×#20162
C2×#19272
C2×#18782
C2×#18652
#151932
#113462
#110672
#110242
#110102
#108062