Finite Group Databasebeta

← #370630

Subgroups — #370630

Order 512.

1340 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2C2×C8×C4⋊C4C211yesmaximal, characteristic
3#13910C211yesmaximal, characteristic
4#11357C212yesmaximal
5#11207C212yesmaximal
6#10880C212yesmaximal
7#10849C211yesmaximal, characteristic
8#10849C211yesmaximal, characteristic
9#10841C212yesmaximal
10#10841C211yesmaximal, characteristic
11C4×#340C211yesmaximal, characteristic
12C22×#125—21no—
13C22×C4⋊C8C2211yescharacteristic
14C22×C4×C8C2212yes—
15C22×C4×C8C2211yescharacteristic
16#2552—24no—
17#2552C418yescomplemented
18C2×#398—22no—
19C2×#389—21no—
20C2×#389—21no—
21C2×#389—21no—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#384C2212yes—
25C2×#384C2212yes—
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C8×C4⋊C4—24no—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42#1444—22no—
43C4⋊#95C2212yes—
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C2×#125—28no—
48C2×#125—44no—
49C2×#125—44no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C42—21no—
68C22×C42—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8Q814yes—
74C2×C4×C8—22no—
75C2×C4×C8—22no—
76C2×C4×C8—24no—
77C2×C4×C8—24no—
78C2×C4×C8—42no—
79C2×C4×C8Q814yes—
80#398—216no—
81#389—28no—
82#389—24no—
83#389—44no—
84#389—44no—
85#389C2×C418yes—
86#384—24no—
87#384—24no—
88#384—22no—
89#384—22no—
90#384—22no—
91#384—42no—
92#384C2×C418yes—
93#384C2×C414yes—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4C2×C414yes—
98C4×C4⋊C4C2×C414yes—
99C4×C4⋊C4D414yes—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95D412yes—
106#340—22no—
107#340—22no—
108#340—24no—
109#340—22no—
110#340—24no—
111#340—22no—
112#340—42no—
113#340—42no—
114#340—42no—
115#340—42no—
116#336—24no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2411yescharacteristic
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8C2×Q812yes—
145C22×C8C2×Q814yes—
146C22×C8C2×Q812yes—
147C22×C8#4112yes—
148C22×C8#4112yes—
149#125—432no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
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—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
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—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—81no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—81no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—81no—
207C2×C4⋊C4—81no—
208C2×C4⋊C4C22⋊C414yes—
209C2×C4⋊C4C22⋊C414yes—
210C2×C4⋊C4D814yes—
211C2×C4⋊C4SD1614yes—
212C2×C4⋊C4SD1614yes—
213C2×C4⋊C4Q1614yes—
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C414yes—
217C2×C4⋊C4#4112yes—
218C2×C4⋊C4#4112yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—24no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42C22×C414yes—
230C2×C42C22×C412yes—
231C2×C42C2×D414yes—
232C2×C42C2×D412yes—
233C2×C42C2×Q812yes—
234C2×C42C2×Q812yes—
235C2×C42#4112yes—
236C2×C42#4112yes—
237C4⋊C8—28no—
238C4×C8—28no—
239C4×C8—48no—
240#95—24no—
241#95—28no—
242#95—42no—
243#95—44no—
244#95—44no—
245#95C22⋊C418yes—
246C24C22×D411yescharacteristic
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C414yes—
303C22×C4C2×C22⋊C414yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4#12412yes—
306C22×C4#12412yes—
307C22×C4C4⋊Q812yes—
308C22×C4C4⋊Q812yes—
309C22×C4C2×D814yes—
310C22×C4C2×D812yes—
311C22×C4C2×SD1614yes—
312C22×C4C2×SD1614yes—
313C22×C4C2×SD1612yes—
314C22×C4C2×SD1612yes—
315C22×C4C2×Q1614yes—
316C22×C4C2×Q1612yes—
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×Q811yescharacteristic
320C22×C4C22×Q811yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4H(5,𝔽2)11yescharacteristic
328C22×C4H(5,𝔽2)11yescharacteristic
329C22×C42−1+411yescharacteristic
330C22×C42−1+411yescharacteristic
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8#12514yescomplemented
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—41no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—416no—
373C4⋊C4—416no—
374C4⋊C4—48no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—88no—
380C4⋊C4—84no—
381C4⋊C4—88no—
382C4⋊C4—88no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C4⋊C4#102116yes—
386C4⋊C4#103116yes—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—24no—
391C42—48no—
392C42—84no—
393C42C4×Q814yes—
394C42C4×Q814yes—
395C42C22⋊Q814yes—
396C42C22⋊Q814yes—
397C23C22×C22⋊C411yescharacteristic
398C23C2×#12411yescharacteristic
399C23C2×C4⋊Q811yescharacteristic
400C23#54611yescharacteristic
401C23#54711yescharacteristic
402C23#54811yescharacteristic
403C23#55011yescharacteristic
404C23C22×D811yescharacteristic
405C23C22×SD1611yescharacteristic
406C23C22×SD1611yescharacteristic
407C23C22×Q1611yescharacteristic
408C23#57611yescharacteristic
409C23#57711yescharacteristic
410C23#57711yescharacteristic
411C23#57811yescharacteristic
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—82no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4#39114yes—
473C2×C4#39114yes—
474C2×C4C2×#10218yes—
475C2×C4C2×#10214yes—
476C2×C4C2×#10318yes—
477C2×C4C2×#10314yes—
478C2×C4#48614yes—
479C2×C4#48714yes—
480C2×C4#49814yes—
481C2×C4#50014yes—
482C2×C4C2×C4×Q811yescharacteristic
483C2×C4C2×C4×Q811yescharacteristic
484C2×C4C4×#4111yescharacteristic
485C2×C4C4×#4111yescharacteristic
486C2×C4#52011yescharacteristic
487C2×C4#52011yescharacteristic
488C2×C4C2×C22⋊Q811yescharacteristic
489C2×C4C2×C22⋊Q811yescharacteristic
490C2×C4C2×#12511yescharacteristic
491C2×C4#52911yescharacteristic
492C2×C4#52911yescharacteristic
493C2×C4#53311yescharacteristic
494C2×C4#54011yescharacteristic
495C2×C4#54111yescharacteristic
496C2×C4#54411yescharacteristic
497C2×C4#54411yescharacteristic
498C2×C4#54611yescharacteristic
499C2×C4#54711yescharacteristic
500C2×C4#54811yescharacteristic
501C2×C4#55011yescharacteristic
502C2×C4#55211yescharacteristic
503C2×C4#55211yescharacteristic
504C2×C4#55311yescharacteristic
505C2×C4#55311yescharacteristic
506C2×C4#55411yescharacteristic
507C2×C4#55411yescharacteristic
508C2×C4#55711yescharacteristic
509C2×C4#55711yescharacteristic
510C8—24no—
511C8—42no—
512C8—44no—
513C22C2×#39111yescharacteristic
514C22#233911yescharacteristic
515C22#234111yescharacteristic
516C22#248911yescharacteristic
517C22#249011yescharacteristic
518C22#255011yescharacteristic
519C22#255211yescharacteristic
520C22C22×#10211yescharacteristic
521C22C22×#10311yescharacteristic
522C22#289711yescharacteristic
523C22#289811yescharacteristic
524C22C2×#48611yescharacteristic
525C22C2×#48711yescharacteristic
526C22#313611yescharacteristic
527C22#313711yescharacteristic
528C22C2×#49811yescharacteristic
529C22C2×#50011yescharacteristic
530C22#316311yescharacteristic
531C22#316411yescharacteristic
532C22#329411yescharacteristic
533C22#329511yescharacteristic
534C22#329811yescharacteristic
535C22#329911yescharacteristic
536C22#333411yescharacteristic
537C22#333511yescharacteristic
538C22#333511yescharacteristic
539C22#333811yescharacteristic
540C22#335811yescharacteristic
541C22#335911yescharacteristic
542C22#336211yescharacteristic
543C22#336311yescharacteristic
544C22#339111yescharacteristic
545C22#339211yescharacteristic
546C22#339211yescharacteristic
547C22#339511yescharacteristic
548C4—24no—
549C4—28no—
550C4—28no—
551C4—82no—
552C4—84no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4—162no—
557C4#199018yes—
558C4C4×#12511yescharacteristic
559C4#233011yescharacteristic
560C4#237011yescharacteristic
561C4#237011yescharacteristic
562C4#244111yescharacteristic
563C4#244211yescharacteristic
564C4#245311yescharacteristic
565C4#245411yescharacteristic
566C2#1405411yescharacteristic
567C2C2×#199011yescharacteristic
568C2#2132611yescharacteristic
569C2#2248511yescharacteristic
570C2#2248711yescharacteristic
571C2#2254011yescharacteristic
572C2#2254211yescharacteristic
573C2#2772711yescharacteristic
574C2#2772911yescharacteristic
575C2#2774711yescharacteristic
576C2#2774911yescharacteristic
577C2#2878911yescharacteristic
578C2#2879111yescharacteristic
579C2#2882711yescharacteristic
580C2#2882911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #2552C4semidirect
351. C2×C8#125semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×C8×C4⋊C42
#139102
#113572
#112072
#108802
#108492
#108412
C4×#3402