Finite Group Databasebeta

← #415189

Subgroups — #415189

Order 512.

1068 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1984C211yesmaximal, characteristic
2C2×#1947C211yesmaximal, characteristic
3#13926C211yesmaximal, characteristic
4#13926C211yesmaximal, characteristic
5#11379C212yesmaximal
6#11315C212yesmaximal
7#11315C212yesmaximal
8#11256C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10841C211yesmaximal, characteristic
11C4×#340C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#391—21no—
17C2×#391—21no—
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1984—24no—
23#1947—24no—
24C2×#340C2212yes—
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1441—22no—
36#1441—22no—
37C4⋊#95—21no—
38C4⋊#95—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C23×Q8D411yescharacteristic
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D412yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C42—21no—
66C22×C42—21no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69C2×#107—24no—
70C2×#107—24no—
71C2×#107—42no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8D414yes—
75C2×C4×C8—22no—
76C2×C4×C8—22no—
77C2×C4×C8—22no—
78#391—44no—
79#391—44no—
80#384—22no—
81#384—22no—
82#384—22no—
83#384—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98#340—24no—
99#340—22no—
100#340—24no—
101#340—22no—
102#340—22no—
103#340—24no—
104#340—42no—
105#340—42no—
106#340Q814yescomplemented
107#336—24no—
108#336—24no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8D814yes—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2411yescharacteristic
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8C2×D414yes—
146C22×C8C2×Q812yes—
147C22×C8#4112yes—
148C4×Q8—44no—
149C4×Q8—44no—
150C4×Q8—44no—
151C4×Q8—44no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—81no—
199C2×C4⋊C4—81no—
200C2×C4⋊C4—81no—
201C2×C4⋊C4—81no—
202C2×C4⋊C4C2×Q812yes—
203C2×C4⋊C4C2×Q812yes—
204C2×C4⋊C4#4112yes—
205C2×C4⋊C4#4112yes—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42C2×D412yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42#4112yes—
223C2×C42#4112yes—
224#107—48no—
225C4⋊C8—28no—
226C4×C8—44no—
227#95—42no—
228#95—42no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—82no—
234#95—82no—
235#95—82no—
236#95—82no—
237C24C22×D411yescharacteristic
238C2×Q8—28no—
239C2×Q8—28no—
240C2×Q8—28no—
241C2×Q8—44no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—44no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4#12112yes—
305C22×C4#12112yes—
306C22×C4#12312yes—
307C22×C4#12312yes—
308C22×C4#12312yes—
309C22×C4#12312yes—
310C22×C4C2×D814yes—
311C22×C4C2×D812yes—
312C22×C4#13512yes—
313C22×C4#13712yes—
314C22×C4#13712yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×Q811yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4H(5,𝔽2)11yescharacteristic
326C22×C42−1+411yescharacteristic
327C22×C42−1+411yescharacteristic
328C22×C42−1+411yescharacteristic
329C2×C8—24no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—41no—
349C2×C8—42no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—48no—
367C4⋊C4—44no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C4⋊C4—84no—
380C42—42no—
381C42—42no—
382C42—42no—
383C42—42no—
384C42—84no—
385C42—84no—
386C42#12314yes—
387C42#12414yes—
388C42#12614yes—
389C42#12614yes—
390C23C2×#12111yescharacteristic
391C23C2×#12311yescharacteristic
392C23C2×#12311yescharacteristic
393C23#54811yescharacteristic
394C23#54811yescharacteristic
395C23D4×Q811yescharacteristic
396C23#55011yescharacteristic
397C23C22×D811yescharacteristic
398C23C2×#13511yescharacteristic
399C23C2×#13711yescharacteristic
400C23C2×#13711yescharacteristic
401C23#57611yescharacteristic
402C23#57711yescharacteristic
403C23#57711yescharacteristic
404C23#57811yescharacteristic
405Q8—416no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4#39714yes—
469C2×C4#39714yes—
470C2×C4#46614yes—
471C2×C4#47014yes—
472C2×C4#48014yes—
473C2×C4#48014yes—
474C2×C4#48514yes—
475C2×C4#48514yes—
476C2×C4C2×#12311yescharacteristic
477C2×C4C2×#12411yescharacteristic
478C2×C4C2×#12611yescharacteristic
479C2×C4C2×#12611yescharacteristic
480C2×C4#52911yescharacteristic
481C2×C4#52911yescharacteristic
482C2×C4#52911yescharacteristic
483C2×C4#53311yescharacteristic
484C2×C4#53311yescharacteristic
485C2×C4#53311yescharacteristic
486C2×C4#54111yescharacteristic
487C2×C4#54111yescharacteristic
488C2×C4#54111yescharacteristic
489C2×C4#54211yescharacteristic
490C2×C4#54211yescharacteristic
491C2×C4#54211yescharacteristic
492C2×C4#54611yescharacteristic
493C2×C4#54711yescharacteristic
494C2×C4#54811yescharacteristic
495C2×C4D4×Q811yescharacteristic
496C2×C4#55211yescharacteristic
497C2×C4#55211yescharacteristic
498C2×C4#55411yescharacteristic
499C2×C4#55611yescharacteristic
500C2×C4#55611yescharacteristic
501C2×C4#55611yescharacteristic
502C2×C4#55711yescharacteristic
503C2×C4#55911yescharacteristic
504C8—44no—
505C8—42no—
506C8—42no—
507C22C2×#39711yescharacteristic
508C22#242111yescharacteristic
509C22#242311yescharacteristic
510C22#242611yescharacteristic
511C22#242911yescharacteristic
512C22#244611yescharacteristic
513C22#244811yescharacteristic
514C22C2×#46611yescharacteristic
515C22C2×#47011yescharacteristic
516C22#306211yescharacteristic
517C22#306311yescharacteristic
518C22C2×#48011yescharacteristic
519C22C2×#48011yescharacteristic
520C22C2×#48511yescharacteristic
521C22C2×#48511yescharacteristic
522C22#309611yescharacteristic
523C22#309711yescharacteristic
524C22#309811yescharacteristic
525C22#309911yescharacteristic
526C22#335811yescharacteristic
527C22#335811yescharacteristic
528C22#336111yescharacteristic
529C22#336411yescharacteristic
530C22#336911yescharacteristic
531C22#337211yescharacteristic
532C22#337411yescharacteristic
533C22#337511yescharacteristic
534C22Q8×D811yescharacteristic
535C22#338111yescharacteristic
536C22#338511yescharacteristic
537C22#338711yescharacteristic
538C22#339111yescharacteristic
539C22#339411yescharacteristic
540C22#339811yescharacteristic
541C22#340011yescharacteristic
542C4—24no—
543C4—44no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4—82no—
549C4—162no—
550C4—162no—
551C4#207518yes—
552C4#244611yescharacteristic
553C4#244711yescharacteristic
554C4#248911yescharacteristic
555C4#249211yescharacteristic
556C4#251111yescharacteristic
557C4#251411yescharacteristic
558C4#264511yescharacteristic
559C4#286311yescharacteristic
560C2#1518611yescharacteristic
561C2C2×#207511yescharacteristic
562C2#2364311yescharacteristic
563C2#2499011yescharacteristic
564C2#2499511yescharacteristic
565C2#2505611yescharacteristic
566C2#2506111yescharacteristic
567C2#2529211yescharacteristic
568C2#2530211yescharacteristic
569C2#2546911yescharacteristic
570C2#2548011yescharacteristic
571C2#2627611yescharacteristic
572C2#2627911yescharacteristic
573C2#2631611yescharacteristic
574C2#2631911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
106. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19842
C2×#19472
#139262
#113792
#113152
#112562
#108492
#108412
C4×#3402