Finite Group Databasebeta

← #477713

Subgroups — #477713

Order 512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2073C212yesmaximal, complemented
2#15222C212yesmaximal, complemented
3#11346C211yesmaximal, characteristic, complemented
4#11328C211yesmaximal, characteristic, complemented
5#11268C212yesmaximal, complemented
6#11123C212yesmaximal, complemented
7#11066C212yesmaximal, complemented
8#11022C212yesmaximal
9#11022C211yesmaximal, characteristic
10C23×D8C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#102C2212yes—
13C22×#102C2212yes—
14C22×#102C2212yes—
15C2×#390C2212yes—
16C2×#390C2212yes—
17C2×#390C2212yes—
18C2×#389C2212yes—
19C2×#382C2212yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21#2073—28no—
22C2×#340C2212yes—
23C2×#340C2211yescharacteristic
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31#1439—22no—
32#1439—22no—
33#1439—22no—
34#1439—22no—
35C4⋊#95C2212yes—
36C23×D4C2312yes—
37C22×D8—24no—
38C22×D8—22no—
39C22×D8—41no—
40C22×D8—42no—
41C22×D8—41no—
42C23×C8C2312yes—
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C22×C4⋊C4—22no—
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C42C2312yes—
58C22×C42C2311yescharacteristic
59C2×#107—42no—
60C2×#107—41no—
61C2×#102—24no—
62C2×#102—24no—
63C2×#102—24no—
64C2×#102—24no—
65C2×#102—42no—
66C2×#102—42no—
67C2×#102—42no—
68C2×#102—42no—
69C2×#102—42no—
70#390—24no—
71#390—42no—
72#390—42no—
73#390—42no—
74#390—42no—
75#390—42no—
76#389—42no—
77#389—42no—
78#382—24no—
79#382—42no—
80C4×C4⋊C4—41no—
81C4×C4⋊C4—41no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95D412yes—
88#340—24no—
89#340—42no—
90#340—42no—
91#340—41no—
92#340D414yes—
93#336—24no—
94#336—24no—
95#336—41no—
96#336—41no—
97C25—22no—
98C22×D4—22no—
99C22×D4—22no—
100C22×D4—22no—
101C22×D4—22no—
102C22×D4—22no—
103C22×D4—22no—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4—22no—
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4#4114yes—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2411yescharacteristic
120C2×D8—48no—
121C2×D8—44no—
122C2×D8—44no—
123C2×D8—48no—
124C2×D8—44no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—21no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8C2×D412yes—
141C22×C8#4112yes—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4C2×D414yes—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—82no—
179C2×C22⋊C4—82no—
180C2×C22⋊C4—82no—
181C2×C22⋊C4—82no—
182C2×C22⋊C4—82no—
183C2×C22⋊C4—82no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196#107—44no—
197#102—48no—
198#102—48no—
199#102—48no—
200#95—42no—
201#95—44no—
202#95—44no—
203#95—44no—
204#95—82no—
205#95—82no—
206C24—24no—
207C24—42no—
208C24—42no—
209C24—42no—
210C24—42no—
211C24—42no—
212C24—42no—
213C24—42no—
214C24—42no—
215C24—42no—
216C24—42no—
217C24—42no—
218C24—42no—
219C24—42no—
220C24—42no—
221C24C22×D411yescharacteristic
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4#13512yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C2×#4112yes—
297C22×C4C2×#4111yescharacteristic
298C22×C4H(5,𝔽2)11yescharacteristic
299C22×C4H(5,𝔽2)12yes—
300C22×C4H(5,𝔽2)11yescharacteristic
301C22×C4H(5,𝔽2)11yescharacteristic
302C22×C4H(5,𝔽2)12yes—
303C22×C42−1+411yescharacteristic
304C22×C42−1+412yes—
305D8—416no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—88no—
338C22⋊C4—88no—
339C22⋊C4—88no—
340C22⋊C4—88no—
341C22⋊C4—88no—
342C22⋊C4—88no—
343C22⋊C4—88no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42—41no—
348C42—42no—
349C42—41no—
350C42—42no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—44no—
355C23—44no—
356C23—44no—
357C23—44no—
358C23—82no—
359C23—82no—
360C23—82no—
361C23—82no—
362C23—82no—
363C23—82no—
364C23—82no—
365C23—82no—
366C23—82no—
367C23—82no—
368C23—82no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23—82no—
373C23—82no—
374C23—82no—
375C23—82no—
376C23—82no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23—82no—
386C23#53411yescharacteristic
387C23#53512yes—
388C23D4211yescharacteristic
389C23#54711yescharacteristic
390C23#54812yes—
391C23C2×#13511yescharacteristic
392C23#57512yes—
393C23#57511yescharacteristic
394C23#57611yescharacteristic
395C23#57612yes—
396C23#57611yescharacteristic
397D4—44no—
398D4—44no—
399D4—44no—
400D4—44no—
401D4—82no—
402D4—82no—
403D4—82no—
404D4—82no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4#53711yescharacteristic
451C2×C4#53712yes—
452C2×C4#53812yes—
453C2×C4#53811yescharacteristic
454C2×C4#53912yes—
455C2×C4#53912yes—
456C2×C4#54212yes—
457C2×C4#54612yes—
458C2×C4#55011yescharacteristic
459C2×C4#55111yescharacteristic
460C2×C4#55112yes—
461C2×C4#56011yescharacteristic
462C2×C4#56012yes—
463C2×C4#56011yescharacteristic
464C2×C4#56211yescharacteristic
465C2×C4#56212yes—
466C2×C4#56312yes—
467C2×C4#56311yescharacteristic
468C8—42no—
469C8—42no—
470C8—82no—
471C22—84no—
472C22—84no—
473C22—84no—
474C22—84no—
475C22—84no—
476C22—84no—
477C22—84no—
478C22—162no—
479C22—162no—
480C22—162no—
481C22—162no—
482C22—162no—
483C22—162no—
484C22—162no—
485C22—162no—
486C22#267612yes—
487C22#268211yescharacteristic
488C22#272312yes—
489C22#272811yescharacteristic
490C22#281511yescharacteristic
491C22#319111yescharacteristic
492C22#319711yescharacteristic
493C22#319911yescharacteristic
494C22#319911yescharacteristic
495C22#321112yes—
496C22#321912yes—
497C22#322012yes—
498C22#322012yes—
499C22#326812yes—
500C22#327311yescharacteristic
501C22#328711yescharacteristic
502C22#334111yescharacteristic
503C22#334712yes—
504C22#335511yescharacteristic
505C22#336112yes—
506C22#337012yes—
507C22#337012yes—
508C22#337012yes—
509C4—42no—
510C4—84no—
511C4—82no—
512C4—82no—
513C4—81no—
514C4—162no—
515C4—162no—
516C4#269311yescharacteristic
517C4#270512yes—
518C4#273511yescharacteristic
519C4#282311yescharacteristic
520C4#282712yes—
521C4#283011yescharacteristic
522C2—164no—
523C2#1712111yescharacteristic
524C2#3088712yes—
525C2#3089112yes—
526C2#3119211yescharacteristic
527C2#3120711yescharacteristic
528C2#3243712yes—
529C2#3244312yes—
530C2#3254511yescharacteristic
531C2#3255811yescharacteristic
532C2#3276311yescharacteristic
533C2#3276411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
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
1. C2×#2073C2semidirect
2. #15222C2semidirect
3. #11346C2semidirect
4. #11328C2semidirect
5. #11268C2semidirect
6. #11123C2semidirect
7. #11066C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20732
#152222
#113462
#113282
#112682
#111232
#110662
#110222