Finite Group Databasebeta

← #403840

Subgroups — #403840

Order 512.

1398 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic
2C2×#2043C211yesmaximal, characteristic
3C2×#2024C212yesmaximal
4C2×#1981C211yesmaximal, characteristic
5#15180C211yesmaximal, characteristic
6#11228C211yesmaximal, characteristic
7#11202C211yesmaximal, characteristic
8#11067C212yesmaximal
9#10932C212yesmaximal
10#10932C212yesmaximal
11#10807C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19#2369—24no—
20#2369—24no—
21C2×#391—21no—
22C2×#391—21no—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#389—21no—
26C2×#389C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#2043—24no—
34#2024—28no—
35#1981—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1442—21no—
44#1442—21no—
45#1441—21no—
46#1441—21no—
47C4⋊#95C2211yescharacteristic
48C23×Q8—21no—
49C23×Q8C2312yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—42no—
61C2×C4⋊Q8D414yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C42—21no—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×C4⋊C8—22no—
81C2×C4⋊C8—42no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—42no—
94#391—28no—
95#391—28no—
96#391—24no—
97#391—24no—
98#391—24no—
99#391—42no—
100#389—28no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389D414yes—
105C4×C4⋊C4—28no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4D414yes—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95D411yescharacteristic
126C2×#95D411yescharacteristic
127#340—24no—
128#340—24no—
129#340—24no—
130#340—42no—
131#336—24no—
132#336—22no—
133#336—42no—
134C22×Q8—24no—
135C22×Q8—24no—
136C22×Q8—24no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—82no—
151C22×Q8—82no—
152C22×Q8#4114yes—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2411yescharacteristic
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C4⋊Q8—28no—
181C4⋊Q8—28no—
182C4⋊Q8—28no—
183C4⋊Q8—28no—
184C4⋊Q8—28no—
185C4⋊Q8—48no—
186C4⋊Q8—44no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4Q1614yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4#4112yes—
228C2×C4⋊C4#4112yes—
229C2×C4⋊C4#4114yes—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—24no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—82no—
246C2×C42—82no—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42#4112yes—
251C4⋊C8—44no—
252#103—48no—
253#103—48no—
254#103—48no—
255#103—48no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—22no—
262#95—22no—
263#95—44no—
264#95—44no—
265#95—42no—
266#95—44no—
267#95—44no—
268#95—42no—
269C24C22×D411yescharacteristic
270C2×Q8—216no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—88no—
300C2×Q8—88no—
301C2×Q8—84no—
302C2×Q8—84no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4#12012yes—
361C22×C4#12012yes—
362C22×C4#12112yes—
363C22×C4#12112yes—
364C22×C4C2×Q1612yes—
365C22×C4C2×Q1612yes—
366C22×C4C2×Q1612yes—
367C22×C4#13712yes—
368C22×C4#13712yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4112yes—
376C22×C4C2×#4112yes—
377C22×C4H(5,𝔽2)12yes—
378C22×C42−1+412yes—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C4⋊C4—216no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—28no—
403C4⋊C4—24no—
404C4⋊C4—44no—
405C4⋊C4—48no—
406C4⋊C4—48no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C4⋊C4#12114yes—
416C4⋊C4#12114yes—
417C4⋊C4#12414yes—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—24no—
425C42—48no—
426C42—41no—
427C42—41no—
428C42—42no—
429C42—84no—
430C42—84no—
431C23C2×#12011yescharacteristic
432C23C2×#12111yescharacteristic
433C23#52511yescharacteristic
434C23#54612yes—
435C23#54812yes—
436C23C22×Q1611yescharacteristic
437C23C2×#13711yescharacteristic
438C23C2×#13711yescharacteristic
439C23#57511yescharacteristic
440C23#57712yes—
441C23#57712yes—
442Q8—44no—
443Q8—44no—
444Q8—44no—
445Q8—44no—
446Q8—816no—
447Q8—82no—
448Q8—82no—
449Q8—82no—
450Q8—82no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4#45114yes—
516C2×C4#46214yes—
517C2×C4C2×#12111yescharacteristic
518C2×C4C2×#12111yescharacteristic
519C2×C4#52511yescharacteristic
520C2×C4#52511yescharacteristic
521C2×C4#52511yescharacteristic
522C2×C4C2×#12411yescharacteristic
523C2×C4#52912yes—
524C2×C4#52912yes—
525C2×C4#53511yescharacteristic
526C2×C4#53611yescharacteristic
527C2×C4#54212yes—
528C2×C4#54212yes—
529C2×C4#54612yes—
530C2×C4#54812yes—
531C2×C4#54812yes—
532C2×C4#55012yes—
533C2×C4#55212yes—
534C2×C4#55912yes—
535C8—82no—
536C8—82no—
537C22#240411yescharacteristic
538C22#242612yes—
539C22#243611yescharacteristic
540C22#245812yes—
541C22#284711yescharacteristic
542C22C2×#45111yescharacteristic
543C22#300711yescharacteristic
544C22#300911yescharacteristic
545C22#300911yescharacteristic
546C22C2×#46211yescharacteristic
547C22#303711yescharacteristic
548C22#304212yes—
549C22#310411yescharacteristic
550C22#311311yescharacteristic
551C22#312312yes—
552C22#329912yes—
553C22#331612yes—
554C22#332112yes—
555C22#332912yes—
556C22#336312yes—
557C22#336912yes—
558C22#337212yes—
559C22#337612yes—
560C4—24no—
561C4—24no—
562C4—44no—
563C4—42no—
564C4—44no—
565C4—44no—
566C4—42no—
567C4—42no—
568C4—84no—
569C4—81no—
570C4—82no—
571C4—162no—
572C4#239811yescharacteristic
573C4#242612yes—
574C4#245812yes—
575C4#253611yescharacteristic
576C4#254811yescharacteristic
577C4#263211yescharacteristic
578C2#1488311yescharacteristic
579C2#2397011yescharacteristic
580C2#2397711yescharacteristic
581C2#2527812yes—
582C2#2528712yes—
583C2#2593611yescharacteristic
584C2#2594511yescharacteristic
585C2#2669412yes—
586C2#2671112yes—
587C2#3290611yescharacteristic
588C2#3291411yescharacteristic
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×#23692
C2×#20432
C2×#20242
C2×#19812
#151802
#112282
#112022
#110672
#109322
#108072