Finite Group Databasebeta

← #403528

Subgroups — #403528

Order 512.

1420 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×Q8C211yesmaximal, characteristic, complemented
2C2×#2065C212yesmaximal
3C2×#1934C211yesmaximal, characteristic
4#15202C211yesmaximal, characteristic, complemented
5#15157C211yesmaximal, characteristic
6#11231C212yesmaximal, complemented
7#11228C211yesmaximal, characteristic, complemented
8#11076C211yesmaximal, characteristic
9#10925C212yesmaximal
10#10880C212yesmaximal, complemented
11#10861C211yesmaximal, characteristic, complemented
12C22×C4×Q8C2211yescharacteristic
13C2×C8×Q8—28no—
14C2×C8×Q8—24no—
15C22×#106C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×#102C2211yescharacteristic
19C22×#102C2212yes—
20C22×#102C2211yescharacteristic
21C22×C4×C8C2212yes—
22C22×C4×C8C2211yescharacteristic
23C2×#393—21no—
24C2×#390C2212yes—
25C2×#390C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2211yescharacteristic
28#2065—28no—
29#1934—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36#1446—21no—
37#1444—21no—
38#1444—22no—
39#1439—22no—
40#1439—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8D411yescharacteristic
46C23×D4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4×Q8—24no—
51C2×C4×Q8—22no—
52C2×C4×Q8—24no—
53C2×C4×Q8—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D412yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—21no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69C8×Q8—232no—
70C2×#106—24no—
71C2×#106—24no—
72C2×#106—42no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—24no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8D418yes—
78C2×#102—22no—
79C2×#102—22no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—22no—
83C2×#102—24no—
84C2×#102—42no—
85C2×#102Q814yescomplemented
86C2×C4×C8—24no—
87C2×C4×C8—24no—
88C2×C4×C8—24no—
89C2×C4×C8—24no—
90C2×C4×C8D414yes—
91#393—44no—
92#390—24no—
93#390—22no—
94#390—24no—
95#390—22no—
96#389—24no—
97#389—22no—
98#389—24no—
99#389—22no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—22no—
110#340—24no—
111#340—22no—
112#340—22no—
113#340—24no—
114#340—24no—
115#340—42no—
116C25—21no—
117C22×Q8—24no—
118C22×Q8—22no—
119C22×Q8—24no—
120C22×Q8—22no—
121C22×Q8SD1614yes—
122C22×D4—21no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4C2×Q812yes—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2411yescharacteristic
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8C2×D414yes—
154C22×C8C2×D412yes—
155C22×C8C2×D414yes—
156C22×C8C2×D412yes—
157C22×C8C2×Q812yes—
158C22×C8#4112yes—
159C4×Q8—28no—
160C4×Q8—216no—
161C4×Q8—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4C2×Q812yes—
197C2×C4⋊C4#4114yes—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4—82no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42C2×D414yes—
216C2×C42C2×D412yes—
217C2×C42#4114yes—
218C2×C42#4112yes—
219#106—48no—
220C4⋊C8—28no—
221C4⋊C8—216no—
222#102—28no—
223#102—48no—
224#102—44no—
225C4×C8—216no—
226C4×C8—28no—
227#95—44no—
228#95—42no—
229#95—44no—
230#95—42no—
231#95—44no—
232#95—42no—
233#95—82no—
234#95—82no—
235#95—81no—
236#95—81no—
237C24—22no—
238C24—42no—
239C24—41no—
240C24—42no—
241C24—41no—
242C24—41no—
243C24—42no—
244C24—41no—
245C24—42no—
246C24—41no—
247C24—41no—
248C24C22×D411yescharacteristic
249C2×Q8—28no—
250C2×Q8—28no—
251C2×Q8—216no—
252C2×Q8—28no—
253C2×Q8—216no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—22no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—41no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4#12112yes—
318C22×C4#12114yes—
319C22×C4#12114yes—
320C22×C4#12112yes—
321C22×C4C2×SD1614yes—
322C22×C4C2×SD1612yes—
323C22×C4#13514yes—
324C22×C4#13512yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D412yes—
327C22×C4C22×Q811yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4112yes—
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4112yes—
332C22×C4H(5,𝔽2)11yescharacteristic
333C22×C42−1+412yes—
334C22×C42−1+411yescharacteristic
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8#12014yes—
351C2×C8C22⋊Q814yes—
352C4⋊C4—216no—
353C4⋊C4—28no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—84no—
373C4⋊C4—88no—
374C22⋊C4—84no—
375C22⋊C4—88no—
376C22⋊C4—88no—
377C22⋊C4—84no—
378C42—24no—
379C42—22no—
380C42—22no—
381C42—24no—
382C42—24no—
383C42—24no—
384C42—24no—
385C23—42no—
386C23—44no—
387C23—42no—
388C23—44no—
389C23—42no—
390C23—81no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23—82no—
395C23—81no—
396C23—81no—
397C23—82no—
398C23—82no—
399C23—81no—
400C23—82no—
401C23—81no—
402C23—82no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23—82no—
407C23—82no—
408C23C2×#12112yes—
409C23C2×#12111yescharacteristic
410C23#54812yes—
411C23D4×Q811yescharacteristic
412C23#55011yescharacteristic
413C23C22×SD1611yescharacteristic
414C23C2×#13512yes—
415C23C2×#13511yescharacteristic
416C23#57612yes—
417C23#57711yescharacteristic
418C23#57711yescharacteristic
419Q8—232no—
420D4—42no—
421D4—42no—
422D4—44no—
423D4—81no—
424D4—81no—
425D4—82no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—82no—
458C2×C4—81no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4#46514yes—
478C2×C4#46518yes—
479C2×C4C2×#12011yescharacteristic
480C2×C4C2×C22⋊Q811yescharacteristic
481C2×C4#52511yescharacteristic
482C2×C4#52512yes—
483C2×C4#52912yes—
484C2×C4#53211yescharacteristic
485C2×C4#53312yes—
486C2×C4#53611yescharacteristic
487C2×C4#53711yescharacteristic
488C2×C4#54112yes—
489C2×C4#54212yes—
490C2×C4#54612yes—
491C2×C4#54712yes—
492C2×C4#54812yes—
493C2×C4D4×Q812yes—
494C2×C4#55411yescharacteristic
495C2×C4#55612yes—
496C2×C4#55611yescharacteristic
497C8—24no—
498C8—24no—
499C8—44no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—84no—
504C22—84no—
505C22—161no—
506C22—161no—
507C22—162no—
508C22—162no—
509C22—161no—
510C22—161no—
511C22#242612yes—
512C22#242912yes—
513C22#253811yescharacteristic
514C22#254511yescharacteristic
515C22#284711yescharacteristic
516C22C2×#46511yescharacteristic
517C22C2×#46512yes—
518C22#305011yescharacteristic
519C22#305012yes—
520C22#305811yescharacteristic
521C22#306011yescharacteristic
522C22#306212yes—
523C22#306212yes—
524C22#336012yes—
525C22#336112yes—
526C22#336212yes—
527C22#336512yes—
528C22Q8×SD1611yescharacteristic
529C22#338012yes—
530C22#338111yescharacteristic
531C22#339211yescharacteristic
532C22#339411yescharacteristic
533C22#339612yes—
534C4—28no—
535C4—24no—
536C4—42no—
537C4—44no—
538C4—82no—
539C4—84no—
540C4—161no—
541C4—162no—
542C4—161no—
543C4#239611yescharacteristic
544C4#240911yescharacteristic
545C4#243011yescharacteristic
546C4#246512yes—
547C4#246912yes—
548C4#263711yescharacteristic
549C2—162no—
550C2#1487711yescharacteristic
551C2#2529512yes—
552C2#2529612yes—
553C2#2547012yes—
554C2#2547312yes—
555C2#2826411yescharacteristic
556C2#2826711yescharacteristic
557C2#2864011yescharacteristic
558C2#2864211yescharacteristic
559C2#3292911yescharacteristic
560C2#3293911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C8×Q8C2semidirect
4. #15202C2semidirect
6. #11231C2semidirect
7. #11228C2semidirect
10. #10880C2semidirect
11. #10861C2semidirect
85. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C8×Q82
C2×#20652
C2×#19342
#152022
#151572
#112312
#112282
#110762
#109252
#108802
#108612