Finite Group Databasebeta

← #475794

Subgroups — #475794

Order 512.

1196 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2055C212yesmaximal
2C2×#1968C211yesmaximal, characteristic
3C2×#1939C211yesmaximal, characteristic, complemented
4#15192C211yesmaximal, characteristic, complemented
5#15147C211yesmaximal, characteristic
6#11231C212yesmaximal, complemented
7#11122C212yesmaximal, complemented
8#11121C212yesmaximal
9#11077C211yesmaximal, characteristic, complemented
10#11076C211yesmaximal, characteristic
11#10861C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×#106C2212yes—
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C22×#102C2211yescharacteristic
20C22×#102C2212yes—
21C22×C4×C8C2211yescharacteristic
22C2×#397—21no—
23C2×#391C2211yescharacteristic
24C2×#391C2212yes—
25C2×#390C2211yescharacteristic
26C2×#390C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28#2055—28no—
29#1968—24no—
30#1939—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35#1445—21no—
36#1443—22no—
37#1442—21no—
38#1440—22no—
39#1439—21no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C23×Q8C2311yescharacteristic
45C23×D4C2311yescharacteristic
46C22×SD16—22no—
47C22×SD16—22no—
48C22×SD16—42no—
49C22×SD16—42no—
50C22×SD16D414yes—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—21no—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66C2×#106—22no—
67C2×#106—24no—
68C2×#106—42no—
69C2×#106—42no—
70C2×#106—42no—
71C2×C4⋊C8—24no—
72C2×C4⋊C8—24no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—42no—
76C2×#103—24no—
77C2×#103—22no—
78C2×#103—24no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#102—24no—
82C2×#102—22no—
83C2×#102—24no—
84C2×#102—42no—
85C2×#102—42no—
86C2×C4×C8—42no—
87C2×C4×C8D414yes—
88#397—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—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95D412yes—
112#340—24no—
113#340—42no—
114#340—42no—
115#336—24no—
116C25—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—21no—
126C22×Q8C2×D412yes—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4C2×D412yes—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C2×SD16—28no—
148C2×SD16—48no—
149C2×SD16—44no—
150C2×SD16—44no—
151C2×SD16—48no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8C2×D412yes—
165C22×C8C2×D412yes—
166C22×C8C2×D412yes—
167C22×C8#4112yes—
168C22×C8#4114yes—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
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—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4SD1614yescomplemented
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4—82no—
204C2×C22⋊C4—82no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42C2×D412yes—
214C2×C42#4112yes—
215#106—44no—
216#106—48no—
217C4⋊C8—44no—
218C4⋊C8—48no—
219#103—48no—
220#103—44no—
221#102—48no—
222#102—44no—
223C4×C8—44no—
224#95—24no—
225#95—44no—
226#95—44no—
227#95—44no—
228#95—42no—
229#95—42no—
230#95—44no—
231#95—42no—
232#95—42no—
233#95—81no—
234#95—82no—
235#95—81no—
236#95—82no—
237C24—22no—
238C24—41no—
239C24—42no—
240C24—41no—
241C24—42no—
242C24—41no—
243C24—41no—
244C24—42no—
245C24—42no—
246C24—41no—
247C24—41no—
248C24C22×D411yescharacteristic
249C2×Q8—22no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—22no—
253C2×Q8—41no—
254C2×Q8—42no—
255C2×Q8—41no—
256C2×Q8—42no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×D4—22no—
268C2×D4—24no—
269C2×D4—22no—
270C2×D4—24no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
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—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4#12112yes—
334C22×C4#12112yes—
335C22×C4C2×SD1614yes—
336C22×C4C2×SD1612yes—
337C22×C4#13512yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4112yes—
342C22×C4C2×#4111yescharacteristic
343C22×C4H(5,𝔽2)12yes—
344C22×C4H(5,𝔽2)12yes—
345C22×C4H(5,𝔽2)11yescharacteristic
346C22×C42−1+412yes—
347C22×C42−1+411yescharacteristic
348SD16—416no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8#12114yes—
365C2×C8#12414yes—
366C4⋊C4—28no—
367C4⋊C4—44no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C22⋊C4—88no—
383C22⋊C4—84no—
384C22⋊C4—88no—
385C22⋊C4—84no—
386C42—22no—
387C42—22no—
388C42—24no—
389C42—42no—
390C42—42no—
391C42—42no—
392C42—42no—
393C23—44no—
394C23—44no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—81no—
399C23—81no—
400C23—82no—
401C23—81no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23—82no—
410C23—82no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23C2×#12111yescharacteristic
417C23#53712yes—
418C23D4211yescharacteristic
419C23#54711yescharacteristic
420C23#55012yes—
421C23C22×SD1611yescharacteristic
422C23C2×#13511yescharacteristic
423C23#57512yes—
424C23#57711yescharacteristic
425C23#57711yescharacteristic
426C23#57712yes—
427Q8—42no—
428Q8—44no—
429Q8—42no—
430Q8—81no—
431Q8—81no—
432Q8—82no—
433D4—42no—
434D4—44no—
435D4—42no—
436D4—81no—
437D4—81no—
438D4—82no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—81no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4#46514yes—
492C2×C4C2×#12111yescharacteristic
493C2×C4C2×#12411yescharacteristic
494C2×C4#53411yescharacteristic
495C2×C4#53412yes—
496C2×C4#53511yescharacteristic
497C2×C4#53611yescharacteristic
498C2×C4#53711yescharacteristic
499C2×C4#53812yes—
500C2×C4#53812yes—
501C2×C4#53912yes—
502C2×C4#53912yes—
503C2×C4#54711yescharacteristic
504C2×C4#54712yes—
505C2×C4#55511yescharacteristic
506C2×C4#55612yes—
507C2×C4#55912yes—
508C2×C4#56212yes—
509C2×C4#56312yes—
510C8—24no—
511C8—24no—
512C8—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—84no—
517C22—84no—
518C22—162no—
519C22—162no—
520C22—161no—
521C22—161no—
522C22—161no—
523C22—161no—
524C22#253711yescharacteristic
525C22#254811yescharacteristic
526C22#261811yescharacteristic
527C22#268212yes—
528C22#272712yes—
529C22C2×#46511yescharacteristic
530C22#305411yescharacteristic
531C22#305711yescharacteristic
532C22#306011yescharacteristic
533C22#324612yes—
534C22#325012yes—
535C22#326012yes—
536C22#326312yes—
537C22#327512yes—
538C22D4×SD1611yescharacteristic
539C22#328411yescharacteristic
540C22#333511yescharacteristic
541C22#334011yescharacteristic
542C22#335112yes—
543C22#339212yes—
544C22#339412yes—
545C22#340112yes—
546C22#340112yes—
547C4—24no—
548C4—44no—
549C4—42no—
550C4—82no—
551C4—82no—
552C4—161no—
553C4—161no—
554C4—162no—
555C4—162no—
556C4#254711yescharacteristic
557C4#261711yescharacteristic
558C4#263911yescharacteristic
559C4#272412yes—
560C4#280812yes—
561C4#282011yescharacteristic
562C2—162no—
563C2#1706911yescharacteristic
564C2#2822311yescharacteristic
565C2#2822911yescharacteristic
566C2#2873811yescharacteristic
567C2#2874211yescharacteristic
568C2#2977411yescharacteristic
569C2#2978111yescharacteristic
570C2#3121412yes—
571C2#3121712yes—
572C2#3252512yes—
573C2#3252712yes—
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×#1939C2semidirect
4. #15192C2semidirect
6. #11231C2semidirect
7. #11122C2semidirect
9. #11077C2semidirect
11. #10861C2semidirect
193. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#20552
C2×#19682
C2×#19392
#151922
#151472
#112312
#111222
#111212
#110772
#110762
#108612