Finite Group Databasebeta

← #410669

Subgroups — #410669

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1968C211yesmaximal, characteristic
2C2×#1934C212yesmaximal
3C2×#1933C212yesmaximal, complemented
4C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
5#15193C211yesmaximal, characteristic, complemented
6#15148C211yesmaximal, characteristic
7#11327C212yesmaximal, complemented
8#11077C211yesmaximal, characteristic, complemented
9#11076C211yesmaximal, characteristic
10#11023C212yesmaximal
11#10820C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19C22×C4×C8C2212yes—
20C22×C4×C8C2211yescharacteristic
21C2×#399—21no—
22C2×#391C2211yescharacteristic
23C2×#391C2212yes—
24C2×#390C2211yescharacteristic
25C2×#390C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1968—24no—
30#1934—28no—
31#1933—28no—
32C8×C4⋊C4—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38#1445—22no—
39#1445—21no—
40#1444—21no—
41#1439—22no—
42#1439—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C23×Q8C2311yescharacteristic
46C23×D4C2311yescharacteristic
47C22×SD16—22no—
48C22×SD16—22no—
49C22×SD16—22no—
50C22×SD16—24no—
51C22×SD16—42no—
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—21no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69C2×#106—22no—
70C2×#106—42no—
71C2×C4⋊C8—24no—
72C2×C4⋊C8—22no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—22no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—22no—
83C2×C4×C8—24no—
84C2×C4×C8—24no—
85C2×C4×C8—24no—
86C2×C4×C8—24no—
87C2×C4×C8D414yes—
88#399—44no—
89#391—24no—
90#391—22no—
91#391—22no—
92#391—42no—
93#390—24no—
94#390—22no—
95#390—22no—
96#390—42no—
97#389—24no—
98#389—24no—
99#389—42no—
100#389—42no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95D412yes—
114#340—24no—
115#340—24no—
116#340—24no—
117#340D418yes—
118#336—24no—
119C25—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8#4112yes—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4#4112yes—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
150C2×SD16—28no—
151C2×SD16—48no—
152C2×SD16—48no—
153C2×SD16—44no—
154C2×SD16—44no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8C2×D412yes—
168C22×C8C2×D414yes—
169C22×C8C2×D412yes—
170C22×C8#4114yes—
171C22×C8#4114yes—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
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—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—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4SD1614yescomplemented
201C2×C4⋊C4C2×D414yes—
202C2×C4⋊C4C2×D414yes—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—82no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4—82no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42C2×D412yes—
223C2×C42#4112yes—
224#106—44no—
225C4⋊C8—28no—
226#103—216no—
227#103—44no—
228#102—216no—
229#102—44no—
230C4×C8—28no—
231C4×C8—48no—
232#95—24no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—42no—
238#95—42no—
239#95—42no—
240#95—42no—
241#95—81no—
242#95—81no—
243C24—22no—
244C24—41no—
245C24—42no—
246C24—42no—
247C24—41no—
248C24—41no—
249C24—42no—
250C24—41no—
251C24—41no—
252C24—42no—
253C24—41no—
254C24C22×D411yescharacteristic
255C2×Q8—24no—
256C2×Q8—22no—
257C2×Q8—24no—
258C2×Q8—22no—
259C2×Q8—41no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4—24no—
276C2×D4—22no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—41no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4#12112yes—
339C22×C4#12112yes—
340C22×C4C2×SD1614yes—
341C22×C4C2×SD1612yes—
342C22×C4#13514yes—
343C22×C4#13512yes—
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D412yes—
346C22×C4C2×#4112yes—
347C22×C4C2×#4112yes—
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4H(5,𝔽2)12yes—
352C22×C42−1+411yescharacteristic
353C22×C42−1+411yescharacteristic
354SD16—416no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8#12314yes—
371C2×C8#12714yes—
372C4⋊C4—28no—
373C4⋊C4—28no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—88no—
389C4⋊C4—88no—
390C22⋊C4—88no—
391C22⋊C4—88no—
392C22⋊C4—84no—
393C22⋊C4—84no—
394C42—22no—
395C42—22no—
396C42—24no—
397C42—42no—
398C42—42no—
399C42—42no—
400C42—42no—
401C23—42no—
402C23—44no—
403C23—42no—
404C23—44no—
405C23—42no—
406C23—81no—
407C23—81no—
408C23—82no—
409C23—81no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23C2×#12111yescharacteristic
425C23#53212yes—
426C23D4212yes—
427C23#54811yescharacteristic
428C23#54811yescharacteristic
429C23C22×SD1611yescharacteristic
430C23C2×#13512yes—
431C23C2×#13511yescharacteristic
432C23#57611yescharacteristic
433C23#57712yes—
434C23#57811yescharacteristic
435Q8—42no—
436Q8—44no—
437Q8—42no—
438Q8—81no—
439Q8—82no—
440Q8—81no—
441D4—42no—
442D4—44no—
443D4—42no—
444D4—81no—
445D4—82no—
446D4—81no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4#46514yes—
500C2×C4C2×#12311yescharacteristic
501C2×C4#52511yescharacteristic
502C2×C4#52512yes—
503C2×C4#52512yes—
504C2×C4#52511yescharacteristic
505C2×C4#52912yes—
506C2×C4C2×#12711yescharacteristic
507C2×C4#53511yescharacteristic
508C2×C4#53611yescharacteristic
509C2×C4#54212yes—
510C2×C4#54212yes—
511C2×C4#54712yes—
512C2×C4#54712yes—
513C2×C4#55012yes—
514C2×C4#55112yes—
515C2×C4#55112yes—
516C2×C4#55211yescharacteristic
517C2×C4#55211yescharacteristic
518C8—24no—
519C8—24no—
520C8—44no—
521C22—82no—
522C22—82no—
523C22—84no—
524C22—84no—
525C22—82no—
526C22—161no—
527C22—161no—
528C22—161no—
529C22—161no—
530C22—162no—
531C22—162no—
532C22#246112yes—
533C22#246112yes—
534C22#253611yescharacteristic
535C22#254111yescharacteristic
536C22#285411yescharacteristic
537C22C2×#46511yescharacteristic
538C22#305011yescharacteristic
539C22#305711yescharacteristic
540C22#306111yescharacteristic
541C22#316712yes—
542C22#316812yes—
543C22#317812yes—
544C22#317912yes—
545C22#328012yes—
546C22D4×SD1612yes—
547C22#328412yes—
548C22#328512yes—
549C22#335911yescharacteristic
550C22#336012yes—
551C22#336111yescharacteristic
552C22#336211yescharacteristic
553C22#336411yescharacteristic
554C22#336512yes—
555C4—24no—
556C4—44no—
557C4—44no—
558C4—42no—
559C4—84no—
560C4—84no—
561C4—82no—
562C4—161no—
563C4—161no—
564C4#239711yescharacteristic
565C4#247912yes—
566C4#247912yes—
567C4#248211yescharacteristic
568C4#250111yescharacteristic
569C4#263211yescharacteristic
570C2—162no—
571C2#1508411yescharacteristic
572C2#2681612yes—
573C2#2681912yes—
574C2#2682012yes—
575C2#2682312yes—
576C2#2818611yescharacteristic
577C2#2818911yescharacteristic
578C2#2848211yescharacteristic
579C2#2848411yescharacteristic
580C2#3312811yescharacteristic
581C2#3313111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×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
3. C2×#1933C2semidirect
4. C2×C8×C4⋊C4C2semidirect
5. #15193C2semidirect
7. #11327C2semidirect
8. #11077C2semidirect
11. #10820C2semidirect
200. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#19682
C2×#19342
C2×#19332
C2×C8×C4⋊C42
#151932
#151482
#113272
#110772
#110762
#110232
#108202