Finite Group Databasebeta

← #374790

Subgroups — #374790

Order 512.

1170 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1990C211yesmaximal, characteristic
2C2×#1987C211yesmaximal, characteristic
3C2×C8×C4⋊C4C211yesmaximal, characteristic
4#14161C211yesmaximal, characteristic
5#14151C211yesmaximal, characteristic
6#11316C212yesmaximal
7#11293C212yesmaximal
8#10880C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10820C212yesmaximal
11C4×#340C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4×C8C2212yes—
15C22×C4×C8C2211yescharacteristic
16C2×#389—21no—
17C2×#389C2212yes—
18C2×#384C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1990C418yescomplemented
23#1987—24no—
24C8×C4⋊C4—24no—
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#336C2212yes—
36#1446—22no—
37#1441—22no—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95—21no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C23×Q8—21no—
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C414yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42—21no—
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8D414yes—
73C2×C4×C8—22no—
74C2×C4×C8—22no—
75C2×C4×C8—24no—
76C2×C4×C8—24no—
77C2×C4×C8—42no—
78C2×C4×C8C2×C414yes—
79#389—24no—
80#389—42no—
81#389—44no—
82#384—22no—
83#384C2×C414yes—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—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—
98C2×#95—22no—
99C2×#95D412yes—
100#340—24no—
101#340—24no—
102#340—22no—
103#340—22no—
104#340—24no—
105#340—22no—
106#340—42no—
107#340C2×C418yes—
108#340C2×C418yes—
109#340Q814yescomplemented
110#336—24no—
111C22×Q8—24no—
112C22×Q8—42no—
113C22×Q8—42no—
114C22×Q8—81no—
115C22×Q8—81no—
116C22×Q8—81no—
117C22×Q8—81no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2411yescharacteristic
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8C22×C412yes—
146C22×C8C22×C412yes—
147C22×C8C2×D414yes—
148C22×C8C2×Q812yes—
149C22×C8#4112yes—
150C4⋊Q8—28no—
151C4⋊Q8—28no—
152C4⋊Q8—44no—
153C4⋊Q8—44no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4—81no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4SD1614yes—
201C2×C4⋊C4Q1614yes—
202C2×C4⋊C4C22×C414yes—
203C2×C4⋊C4C22×C414yes—
204C2×C4⋊C4C2×Q812yes—
205C2×C4⋊C4C2×Q812yes—
206C2×C4⋊C4#4112yes—
207C2×C4⋊C4#4112yes—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42C22×C412yes—
221C2×C42C2×D412yes—
222C2×C42#4112yes—
223C2×C42#4112yes—
224C2×C42#4112yes—
225C2×C42#4112yes—
226C4⋊C8—28no—
227C4×C8—28no—
228C4×C8—48no—
229#95—24no—
230#95—44no—
231#95—42no—
232#95—42no—
233#95—44no—
234#95—82no—
235#95—82no—
236#95—82no—
237#95—82no—
238#95—82no—
239C24C22×D411yescharacteristic
240C2×Q8—48no—
241C2×Q8—84no—
242C2×Q8—84no—
243C2×Q8—84no—
244C2×Q8—84no—
245C2×Q8—84no—
246C2×Q8—84no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
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—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
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—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4C4×D412yes—
306C22×C4C4×D412yes—
307C22×C4#12112yes—
308C22×C4#12112yes—
309C22×C4#12412yes—
310C22×C4#12412yes—
311C22×C4C2×SD1614yes—
312C22×C4C2×SD1612yes—
313C22×C4C2×Q1614yes—
314C22×C4C2×Q1612yes—
315C22×C4#13512yes—
316C22×C4#13512yes—
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×Q811yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4H(5,𝔽2)11yescharacteristic
328C22×C42−1+411yescharacteristic
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—22no—
339C2×C8—22no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8C4×Q814yes—
352C4⋊C4—28no—
353C4⋊C4—28no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—41no—
377C4⋊C4—42no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—84no—
381C4⋊C4—88no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—42no—
386C42—84no—
387C42—84no—
388C42#11714yes—
389C42C4×D414yes—
390C42#12314yes—
391C42#12614yes—
392C23C2×C4×D411yescharacteristic
393C23C2×#12111yescharacteristic
394C23C2×#12411yescharacteristic
395C23#54611yescharacteristic
396C23#54711yescharacteristic
397C23#54811yescharacteristic
398C23D4×Q811yescharacteristic
399C23C22×SD1611yescharacteristic
400C23C22×Q1611yescharacteristic
401C23C2×#13511yescharacteristic
402C23C2×#13511yescharacteristic
403C23#57611yescharacteristic
404C23#57611yescharacteristic
405C23#57711yescharacteristic
406C23#57711yescharacteristic
407Q8—816no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
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—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4#38814yes—
471C2×C4#38814yes—
472C2×C4C4×SD1614yes—
473C2×C4C4×Q1614yes—
474C2×C4#46514yes—
475C2×C4#46714yes—
476C2×C4#48714yes—
477C2×C4#48814yes—
478C2×C4C2×#11711yescharacteristic
479C2×C4C2×C4×D411yescharacteristic
480C2×C4C2×C4×Q811yescharacteristic
481C2×C4C4×#4111yescharacteristic
482C2×C4#51911yescharacteristic
483C2×C4#52011yescharacteristic
484C2×C4C2×#12311yescharacteristic
485C2×C4C2×#12611yescharacteristic
486C2×C4#52911yescharacteristic
487C2×C4#52911yescharacteristic
488C2×C4#53311yescharacteristic
489C2×C4#53311yescharacteristic
490C2×C4#54111yescharacteristic
491C2×C4#54111yescharacteristic
492C2×C4#54211yescharacteristic
493C2×C4#54211yescharacteristic
494C2×C4#54611yescharacteristic
495C2×C4#54711yescharacteristic
496C2×C4#54811yescharacteristic
497C2×C4D4×Q811yescharacteristic
498C2×C4#55211yescharacteristic
499C2×C4#55211yescharacteristic
500C2×C4#55411yescharacteristic
501C2×C4#55611yescharacteristic
502C2×C4#55611yescharacteristic
503C2×C4#55611yescharacteristic
504C2×C4#55711yescharacteristic
505C2×C4#55911yescharacteristic
506C8—24no—
507C8—42no—
508C8—44no—
509C22C2×#38811yescharacteristic
510C22#236511yescharacteristic
511C22#236711yescharacteristic
512C22#242711yescharacteristic
513C22#242911yescharacteristic
514C22#248911yescharacteristic
515C22#249211yescharacteristic
516C22C2×C4×SD1611yescharacteristic
517C22C2×C4×Q1611yescharacteristic
518C22#294811yescharacteristic
519C22#294911yescharacteristic
520C22C2×#46511yescharacteristic
521C22C2×#46711yescharacteristic
522C22#306211yescharacteristic
523C22#306211yescharacteristic
524C22C2×#48711yescharacteristic
525C22C2×#48811yescharacteristic
526C22#313611yescharacteristic
527C22#313611yescharacteristic
528C22#329511yescharacteristic
529C22#329911yescharacteristic
530C22#330311yescharacteristic
531C22#330511yescharacteristic
532C22#333511yescharacteristic
533C22#333811yescharacteristic
534C22#334011yescharacteristic
535C22#334111yescharacteristic
536C22#336011yescharacteristic
537C22#336111yescharacteristic
538C22#336211yescharacteristic
539C22#336311yescharacteristic
540C22Q8×SD1611yescharacteristic
541C22#338011yescharacteristic
542C22#338111yescharacteristic
543C22Q8×Q1611yescharacteristic
544C4—24no—
545C4—44no—
546C4—44no—
547C4—82no—
548C4—84no—
549C4—82no—
550C4—82no—
551C4—84no—
552C4—82no—
553C4#193318yes—
554C4#229511yescharacteristic
555C4#232011yescharacteristic
556C4#236011yescharacteristic
557C4#236211yescharacteristic
558C4#244611yescharacteristic
559C4#244711yescharacteristic
560C4#251111yescharacteristic
561C4#251411yescharacteristic
562C2#1414211yescharacteristic
563C2C2×#193311yescharacteristic
564C2#2108911yescharacteristic
565C2#2315611yescharacteristic
566C2#2315811yescharacteristic
567C2#2321511yescharacteristic
568C2#2321711yescharacteristic
569C2#2533911yescharacteristic
570C2#2534011yescharacteristic
571C2#2547011yescharacteristic
572C2#2547111yescharacteristic
573C2#2772811yescharacteristic
574C2#2772911yescharacteristic
575C2#2781411yescharacteristic
576C2#2781511yescharacteristic
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
22. #1990C4semidirect
109. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19902
C2×#19872
C2×C8×C4⋊C42
#141612
#141512
#113162
#112932
#108802
#108492
#108202
C4×#3402