Finite Group Databasebeta

← #404755

Subgroups — #404755

Order 512.

1192 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1988C211yesmaximal, characteristic, complemented
2C2×#1926C211yesmaximal, characteristic
3#15144C211yesmaximal, characteristic
4#13812C211yesmaximal, characteristic, complemented
5#11384C211yesmaximal, characteristic, complemented
6#11316C212yesmaximal, complemented
7#11255C211yesmaximal, characteristic, complemented
8#11189C212yesmaximal
9#11076C211yesmaximal, characteristic
10#10931C212yesmaximal
11#10908C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2212yes—
18C2×#393—21no—
19C2×#393—21no—
20C2×#390C2211yescharacteristic
21C2×#390C2212yes—
22C2×C4×C4⋊C4C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24#1988—24no—
25#1926—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1439—22no—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39#1434—21no—
40#1434—22no—
41C4×#95—21no—
42C4×#95C2211yescharacteristic
43C23×Q8—21no—
44C23×D4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—42no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C42—22no—
63C22×C42—21no—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66C2×#106—22no—
67C2×#106—24no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—42no—
70C2×#102—22no—
71C2×#102—22no—
72C2×#102—22no—
73C2×#102—24no—
74C2×#102—22no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—42no—
78#393—44no—
79#393—44no—
80#390—24no—
81#390—22no—
82#390—22no—
83#390—42no—
84C4×C4⋊C4—24no—
85C4×C4⋊C4—24no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4D414yes—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95D411yescharacteristic
97C2×#95D412yes—
98C2×#95D411yescharacteristic
99#340—22no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—42no—
105#336—24no—
106#336—22no—
107#336—42no—
108C25—21no—
109C22×Q8—44no—
110C22×Q8—42no—
111C22×Q8—42no—
112C22×Q8—42no—
113C22×Q8—82no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—21no—
117C22×D4—21no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4#4112yes—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8#4112yes—
151C4⋊Q8—28no—
152C4⋊Q8—48no—
153C4⋊Q8—44no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4#4114yes—
180C2×C4⋊C4#4114yes—
181C2×C4⋊C4#4114yes—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4—82no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—28no—
202C2×C42—24no—
203C2×C42—82no—
204C2×C42—82no—
205C2×C42—82no—
206C2×C42C2×D412yes—
207C2×C42#4112yes—
208#106—44no—
209C4⋊C8—44no—
210#102—28no—
211#102—48no—
212#102—44no—
213#95—28no—
214#95—24no—
215#95—22no—
216#95—42no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—42no—
221#95—81no—
222#95—82no—
223#95SD1614yes—
224C24—22no—
225C24—41no—
226C24—42no—
227C24—42no—
228C24—41no—
229C24—41no—
230C24—41no—
231C24—42no—
232C24—42no—
233C24—41no—
234C24—41no—
235C24C22×D411yescharacteristic
236C2×Q8—48no—
237C2×Q8—88no—
238C2×Q8—88no—
239C2×Q8—84no—
240C2×Q8—84no—
241C2×D4—22no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—41no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4#12112yes—
308C22×C4#12112yes—
309C22×C4C2×SD1614yes—
310C22×C4C2×SD1612yes—
311C22×C4AGL(1,ℤ/8)12yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4112yes—
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4112yes—
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4112yes—
319C22×C4H(5,𝔽2)12yes—
320C22×C42−1+411yescharacteristic
321C22×C42−1+411yescharacteristic
322C2×C8—22no—
323C2×C8—24no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—48no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4#12414yes—
358C22⋊C4—88no—
359C22⋊C4—84no—
360C22⋊C4—88no—
361C22⋊C4—84no—
362C42—22no—
363C42—22no—
364C42—24no—
365C42—42no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42—84no—
370C42—88no—
371C23—42no—
372C23—44no—
373C23—42no—
374C23—42no—
375C23—44no—
376C23—81no—
377C23—82no—
378C23—81no—
379C23—82no—
380C23—81no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—81no—
385C23—82no—
386C23—81no—
387C23—82no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23C2×#12111yescharacteristic
395C23#52512yes—
396C23#54612yes—
397C23#54711yescharacteristic
398C23#54811yescharacteristic
399C23C22×SD1611yescharacteristic
400C23C2×AGL(1,ℤ/8)11yescharacteristic
401C23#57512yes—
402C23#57612yes—
403C23#57711yescharacteristic
404C23#57811yescharacteristic
405Q8—816no—
406D4—42no—
407D4—42no—
408D4—44no—
409D4—81no—
410D4—81no—
411D4—82no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4#46014yes—
465C2×C4#52511yescharacteristic
466C2×C4#52511yescharacteristic
467C2×C4C2×#12411yescharacteristic
468C2×C4#52912yes—
469C2×C4#52912yes—
470C2×C4#52912yes—
471C2×C4#53211yescharacteristic
472C2×C4#53511yescharacteristic
473C2×C4#53611yescharacteristic
474C2×C4#54212yes—
475C2×C4#54212yes—
476C2×C4#55112yes—
477C2×C4#55112yes—
478C2×C4#55212yes—
479C2×C4#55211yescharacteristic
480C2×C4#55512yes—
481C2×C4#55612yes—
482C2×C4#55611yescharacteristic
483C8—42no—
484C8—82no—
485C8—81no—
486C22—84no—
487C22—82no—
488C22—84no—
489C22—82no—
490C22—82no—
491C22—161no—
492C22—161no—
493C22—161no—
494C22—161no—
495C22—162no—
496C22—162no—
497C22#241111yescharacteristic
498C22#241511yescharacteristic
499C22#247412yes—
500C22#247712yes—
501C22#285011yescharacteristic
502C22C2×#46011yescharacteristic
503C22#304011yescharacteristic
504C22#304111yescharacteristic
505C22#304611yescharacteristic
506C22#310312yes—
507C22#311512yes—
508C22#311812yes—
509C22#312412yes—
510C22#329512yes—
511C22#331312yes—
512C22#331812yes—
513C22#332612yes—
514C22#333511yescharacteristic
515C22#334712yes—
516C22#335311yescharacteristic
517C22#335911yescharacteristic
518C22#337012yes—
519C22#337311yescharacteristic
520C4—24no—
521C4—44no—
522C4—42no—
523C4—44no—
524C4—84no—
525C4—82no—
526C4—84no—
527C4—82no—
528C4—161no—
529C4#239811yescharacteristic
530C4#252712yes—
531C4#253212yes—
532C4#260411yescharacteristic
533C4#261011yescharacteristic
534C4#264011yescharacteristic
535C2—162no—
536C2#1490211yescharacteristic
537C2#2447811yescharacteristic
538C2#2449711yescharacteristic
539C2#2467811yescharacteristic
540C2#2469311yescharacteristic
541C2#2733012yes—
542C2#2735412yes—
543C2#2742612yes—
544C2#2745012yes—
545C2#3302511yescharacteristic
546C2#3303611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1988C2semidirect
4. #13812C2semidirect
5. #11384C2semidirect
6. #11316C2semidirect
7. #11255C2semidirect
11. #10908C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19882
C2×#19262
#151442
#138122
#113842
#113162
#112552
#111892
#110762
#109312
#109082