Finite Group Databasebeta

← #414038

Subgroups — #414038

Order 512.

1220 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1991C211yesmaximal, characteristic, complemented
2C2×C8×C22⋊C4C211yesmaximal, characteristic
3#15223C211yesmaximal, characteristic
4#15216C211yesmaximal, characteristic
5#11385C211yesmaximal, characteristic, complemented
6#11358C211yesmaximal, characteristic, complemented
7#11220C212yesmaximal, complemented
8#11095C212yesmaximal
9#10880C212yesmaximal, complemented
10#10861C211yesmaximal, characteristic, complemented
11#10721C212yesmaximal
12C24×C8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#106C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#389C2212yes—
19C2×#382C2211yescharacteristic
20C2×#380C2212yes—
21C2×#380C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23#1991—24no—
24C8×C22⋊C4—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34#1445—22no—
35#1442—22no—
36#1439—22no—
37#1439—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C24×C4C2311yescharacteristic
42C23×C8—22no—
43C23×C8—22no—
44C23×C8—42no—
45C23×C8—42no—
46C23×C8Q814yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×#125—24no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—42no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4D412yes—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×#106—24no—
67C2×#106—24no—
68C2×#106—42no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8Q814yes—
72C2×C4×C8—22no—
73C2×C4×C8—42no—
74C2×C4×C8—42no—
75C2×C4×C8D414yes—
76#389—24no—
77#389—42no—
78#382—22no—
79#382—22no—
80#380—24no—
81#380—24no—
82#380—42no—
83#380—42no—
84C4×C22⋊C4—22no—
85C4×C22⋊C4—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95D412yes—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—42no—
100#340—42no—
101#340—42no—
102#336—22no—
103#336—22no—
104C25C2×D411yescharacteristic
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D412yes—
120C23×C4C2×Q812yes—
121C23×C4C2×Q812yes—
122C23×C4C2411yescharacteristic
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—28no—
136C22×C8—48no—
137C22×C8—44no—
138C22×C8—44no—
139C22×C8—48no—
140C22×C8C2×D412yes—
141C22×C8C2×D412yes—
142C22×C8C2×Q812yes—
143C22×C8C2×Q812yes—
144C22×C8C2×Q812yes—
145C22×C8C2×Q812yes—
146C22×C8#4112yes—
147C22×C8#4112yes—
148#125—28no—
149#125—48no—
150#125—44no—
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—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4#4114yes—
180C2×C4⋊C4#4114yes—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—82no—
189C2×C22⋊C4—82no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42C2×D412yes—
197C2×C42#4112yes—
198#106—48no—
199C22⋊C8—28no—
200C4×C8—44no—
201C4×C8—44no—
202#95—24no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—82no—
209#95—82no—
210#95—82no—
211C24—21no—
212C24—21no—
213C24—22no—
214C24—22no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24C4⋊Q812yes—
219C24#13514yes—
220C24C22×D411yescharacteristic
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
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#12114yes—
284C22×C4C4⋊Q812yes—
285C22×C4#13514yes—
286C22×C4C22×D411yescharacteristic
287C22×C4C22×Q811yescharacteristic
288C22×C4C22×Q811yescharacteristic
289C22×C4C2×#4111yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C22×C4C2×#4112yes—
292C22×C4C2×#4112yes—
293C22×C4C2×#4111yescharacteristic
294C22×C4H(5,𝔽2)12yes—
295C22×C4H(5,𝔽2)12yes—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—416no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8C22⋊Q814yes—
316C2×C8C22⋊Q814yes—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—416no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—88no—
335C22⋊C4—216no—
336C22⋊C4—88no—
337C22⋊C4—88no—
338C42—22no—
339C42—22no—
340C42—24no—
341C42—41no—
342C42—41no—
343C42—42no—
344C23—22no—
345C23—22no—
346C23—24no—
347C23—24no—
348C23—24no—
349C23—24no—
350C23—24no—
351C23—24no—
352C23—24no—
353C23—24no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—41no—
359C23—42no—
360C23—42no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23#49914yes—
365C23C2×#12111yescharacteristic
366C23C2×C4⋊Q811yescharacteristic
367C23#53211yescharacteristic
368C23#54612yes—
369C23#54612yes—
370C23C2×#13512yes—
371C23C2×#13512yes—
372C23#57712yes—
373C23#57712yes—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4C2×C22⋊Q811yescharacteristic
428C2×C4C2×C22⋊Q811yescharacteristic
429C2×C4#52511yescharacteristic
430C2×C4#52912yes—
431C2×C4#53211yescharacteristic
432C2×C4#53311yescharacteristic
433C2×C4#53311yescharacteristic
434C2×C4#53312yes—
435C2×C4#53411yescharacteristic
436C2×C4#53511yescharacteristic
437C2×C4#53812yes—
438C2×C4#54312yes—
439C2×C4#55112yes—
440C2×C4#55112yes—
441C2×C4#55412yes—
442C2×C4#55412yes—
443C2×C4#55912yes—
444C2×C4#55912yes—
445C8—24no—
446C8—24no—
447C8—42no—
448C8—42no—
449C22—28no—
450C22—24no—
451C22—44no—
452C22—44no—
453C22—42no—
454C22—44no—
455C22—44no—
456C22—42no—
457C22—44no—
458C22#253411yescharacteristic
459C22#254411yescharacteristic
460C22#254912yes—
461C22#255712yes—
462C22#285711yescharacteristic
463C22#305012yes—
464C22#306012yes—
465C22C2×#49911yescharacteristic
466C22#316011yescharacteristic
467C22#316312yes—
468C22#316812yes—
469C22#317411yescharacteristic
470C22#317411yescharacteristic
471C22#329612yes—
472C22#329612yes—
473C22#329612yes—
474C22#330112yes—
475C22#330112yes—
476C22#330112yes—
477C22#330312yes—
478C22#330412yes—
479C4—42no—
480C4—44no—
481C4—44no—
482C4—84no—
483C4—84no—
484C4—162no—
485C4—162no—
486C4#239611yescharacteristic
487C4#256411yescharacteristic
488C4#258011yescharacteristic
489C4#258912yes—
490C4#260712yes—
491C4#262611yescharacteristic
492C2—44no—
493C2#1516111yescharacteristic
494C2#2809912yes—
495C2#2860212yes—
496C2#2877012yes—
497C2#2877412yes—
498C2#2898012yes—
499C2#2898212yes—
500C2#3320811yescharacteristic
501C2#3321411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1991C2semidirect
5. #11385C2semidirect
6. #11358C2semidirect
7. #11220C2semidirect
9. #10880C2semidirect
10. #10861C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19912
C2×C8×C22⋊C42
#152232
#152162
#113852
#113582
#112202
#110952
#108802
#108612
#107212