Finite Group Databasebeta

← #370629

Subgroups — #370629

Order 512.

1148 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic
2C2×C8×C4⋊C4C211yesmaximal, characteristic
3#13911C211yesmaximal, characteristic
4#11358C212yesmaximal
5#11208C212yesmaximal
6#10880C212yesmaximal
7#10849C211yesmaximal, characteristic
8#10849C211yesmaximal, characteristic
9#10842C212yesmaximal
10#10841C211yesmaximal, characteristic
11C4×#340C211yesmaximal, characteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C4×C8C2212yes—
14C22×C4×C8C2211yescharacteristic
15#2571—24no—
16#2571C418yescomplemented
17C2×#400—22no—
18C2×#389—21no—
19C2×#389C2212yes—
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382—21no—
23C2×#382C2212yes—
24C2×#382C2212yes—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C8×C4⋊C4—24no—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40#1442—22no—
41C4⋊#95—21no—
42C4⋊#95—21no—
43C4⋊#95C2212yes—
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C42—21no—
61C22×C42—21no—
62C22×C42C2312yes—
63C22×C42C2311yescharacteristic
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8Q814yes—
67C2×C4×C8—22no—
68C2×C4×C8—22no—
69C2×C4×C8—24no—
70C2×C4×C8—24no—
71C2×C4×C8—42no—
72C2×C4×C8Q814yes—
73#400—216no—
74#389—28no—
75#389—24no—
76#389C2×C418yes—
77#384—22no—
78#384—22no—
79#384—22no—
80#384C2×C414yes—
81#382—28no—
82#382—24no—
83#382—24no—
84#382—42no—
85#382C2×C418yes—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4C2×C414yes—
90C4×C4⋊C4C2×C414yes—
91C4×C4⋊C4D414yes—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95D412yes—
102#340—22no—
103#340—22no—
104#340—24no—
105#340—22no—
106#340—24no—
107#340—22no—
108#340—42no—
109#340—42no—
110#340—42no—
111#340—42no—
112#336—24no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4C2×D412yes—
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2411yescharacteristic
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8C2×Q812yes—
141C22×C8C2×Q814yes—
142C22×C8C2×Q812yes—
143C22×C8#4112yes—
144C22×C8#4112yes—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
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—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—81no—
190C2×C4⋊C4—81no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—81no—
194C2×C4⋊C4—81no—
195C2×C4⋊C4C22⋊C414yes—
196C2×C4⋊C4C22⋊C414yes—
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C414yes—
200C2×C4⋊C4#4112yes—
201C2×C4⋊C4#4112yes—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—24no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42C22×C414yes—
213C2×C42C22×C412yes—
214C2×C42C2×D414yes—
215C2×C42C2×D412yes—
216C2×C42C2×Q812yes—
217C2×C42C2×Q812yes—
218C2×C42#4112yes—
219C2×C42#4112yes—
220C4⋊C8—28no—
221C4×C8—28no—
222C4×C8—48no—
223#95—24no—
224#95—28no—
225#95—28no—
226#95—28no—
227#95—44no—
228#95—44no—
229#95—42no—
230#95—82no—
231#95—82no—
232#95C22⋊C418yes—
233C24C22×D411yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4C2×C22⋊C414yes—
293C22×C4C2×C22⋊C414yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4#12412yes—
297C22×C4#12412yes—
298C22×C4C4⋊Q812yes—
299C22×C4C4⋊Q812yes—
300C22×C4#13512yes—
301C22×C4#13512yes—
302C22×C4#13512yes—
303C22×C4#13512yes—
304C22×C4C23×C411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×Q811yescharacteristic
307C22×C4C22×Q811yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4H(5,𝔽2)11yescharacteristic
315C22×C4H(5,𝔽2)11yescharacteristic
316C22×C42−1+411yescharacteristic
317C22×C42−1+411yescharacteristic
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—24no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8#12514yes—
339C4⋊C4—28no—
340C4⋊C4—28no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—48no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—84no—
367C4⋊C4—88no—
368C4⋊C4—84no—
369C4⋊C4—84no—
370C42—24no—
371C42—24no—
372C42—24no—
373C42—24no—
374C42—48no—
375C42—84no—
376C42C4×Q814yes—
377C42C4×Q814yes—
378C42C22⋊Q814yes—
379C42C22⋊Q814yes—
380C23C22×C22⋊C411yescharacteristic
381C23C2×#12411yescharacteristic
382C23C2×C4⋊Q811yescharacteristic
383C23#54611yescharacteristic
384C23#54711yescharacteristic
385C23#54811yescharacteristic
386C23#55011yescharacteristic
387C23C2×#13511yescharacteristic
388C23C2×#13511yescharacteristic
389C23C2×#13511yescharacteristic
390C23C2×#13511yescharacteristic
391C23#57611yescharacteristic
392C23#57711yescharacteristic
393C23#57711yescharacteristic
394C23#57811yescharacteristic
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—82no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4#39114yes—
458C2×C4#39114yes—
459C2×C4#41614yes—
460C2×C4#41614yes—
461C2×C4#48814yes—
462C2×C4#48814yes—
463C2×C4#49914yes—
464C2×C4#49914yes—
465C2×C4C2×C4×Q811yescharacteristic
466C2×C4C2×C4×Q811yescharacteristic
467C2×C4C4×#4111yescharacteristic
468C2×C4C4×#4111yescharacteristic
469C2×C4#52011yescharacteristic
470C2×C4#52011yescharacteristic
471C2×C4C2×C22⋊Q811yescharacteristic
472C2×C4C2×C22⋊Q811yescharacteristic
473C2×C4C2×#12511yescharacteristic
474C2×C4#52911yescharacteristic
475C2×C4#52911yescharacteristic
476C2×C4#53311yescharacteristic
477C2×C4#54011yescharacteristic
478C2×C4#54111yescharacteristic
479C2×C4#54411yescharacteristic
480C2×C4#54411yescharacteristic
481C2×C4#54611yescharacteristic
482C2×C4#54711yescharacteristic
483C2×C4#54811yescharacteristic
484C2×C4#55011yescharacteristic
485C2×C4#55211yescharacteristic
486C2×C4#55211yescharacteristic
487C2×C4#55311yescharacteristic
488C2×C4#55311yescharacteristic
489C2×C4#55411yescharacteristic
490C2×C4#55411yescharacteristic
491C2×C4#55711yescharacteristic
492C2×C4#55711yescharacteristic
493C8—24no—
494C8—42no—
495C8—44no—
496C22C2×#39111yescharacteristic
497C22#233911yescharacteristic
498C22#234111yescharacteristic
499C22#248911yescharacteristic
500C22#249011yescharacteristic
501C22#255011yescharacteristic
502C22#255211yescharacteristic
503C22C2×#41611yescharacteristic
504C22C2×#41611yescharacteristic
505C22#289711yescharacteristic
506C22#289811yescharacteristic
507C22C2×#48811yescharacteristic
508C22C2×#48811yescharacteristic
509C22#313611yescharacteristic
510C22#313711yescharacteristic
511C22C2×#49911yescharacteristic
512C22C2×#49911yescharacteristic
513C22#316311yescharacteristic
514C22#316411yescharacteristic
515C22#330211yescharacteristic
516C22#330311yescharacteristic
517C22#330411yescharacteristic
518C22#330511yescharacteristic
519C22#334011yescharacteristic
520C22#334011yescharacteristic
521C22#334111yescharacteristic
522C22#334211yescharacteristic
523C22#336011yescharacteristic
524C22#336111yescharacteristic
525C22#336411yescharacteristic
526C22#336511yescharacteristic
527C22#339311yescharacteristic
528C22#339411yescharacteristic
529C22#339411yescharacteristic
530C22#339611yescharacteristic
531C4—24no—
532C4—44no—
533C4—44no—
534C4—82no—
535C4—84no—
536C4—82no—
537C4—82no—
538C4—82no—
539C4—162no—
540C4#199118yes—
541C4C4×#12511yescharacteristic
542C4#233011yescharacteristic
543C4#237011yescharacteristic
544C4#237011yescharacteristic
545C4#244111yescharacteristic
546C4#244211yescharacteristic
547C4#245311yescharacteristic
548C4#245411yescharacteristic
549C2#1405411yescharacteristic
550C2C2×#199111yescharacteristic
551C2#2132611yescharacteristic
552C2#2248911yescharacteristic
553C2#2249011yescharacteristic
554C2#2254111yescharacteristic
555C2#2254311yescharacteristic
556C2#2772811yescharacteristic
557C2#2773011yescharacteristic
558C2#2774811yescharacteristic
559C2#2775011yescharacteristic
560C2#2879011yescharacteristic
561C2#2879211yescharacteristic
562C2#2882811yescharacteristic
563C2#2883011yescharacteristic
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
16. #2571C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
C2×C8×C4⋊C42
#139112
#113582
#112082
#108802
#108492
#108422
#108412
C4×#3402