Finite Group Databasebeta

← #471722

Subgroups — #471722

Order 512.

1122 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2073C211yesmaximal, characteristic
2C2×#2025C211yesmaximal, characteristic
3#14709C211yesmaximal, characteristic, complemented
4#14700C211yesmaximal, characteristic
5#11385C211yesmaximal, characteristic, complemented
6#11358C211yesmaximal, characteristic, complemented
7#11315C212yesmaximal, complemented
8#11228C212yesmaximal, complemented
9#11189C212yesmaximal
10#11076C212yesmaximal
11#10832C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16C2×#390C2211yescharacteristic
17C2×#390C2211yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#388—22no—
21C2×#382C2211yescharacteristic
22C2×C43C2211yescharacteristic
23#2073—24no—
24#2025—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1437—22no—
35#1436—22no—
36#1436—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C23×D4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×#125—24no—
45C2×#125—22no—
46C2×#125—22no—
47C2×#125—42no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—21no—
59C22×C42D412yes—
60C22×C42D412yes—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×C4⋊C8—24no—
65C2×C4⋊C8—42no—
66C2×C4⋊C8—42no—
67C2×#102—24no—
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—42no—
71C2×#102—42no—
72C2×#102—42no—
73#390—22no—
74#390—24no—
75#390—42no—
76#390D414yes—
77#389—24no—
78#388—48no—
79#382—22no—
80#382—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C43—42no—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—42no—
94#340—42no—
95#340—42no—
96#336—22no—
97#336—22no—
98#336—42no—
99C25—21no—
100C22×D4—22no—
101C22×D4—22no—
102C22×D4—21no—
103C22×D4—22no—
104C22×D4—21no—
105C22×D4—21no—
106C22×D4—21no—
107C22×D4—21no—
108C22×D4—22no—
109C22×D4C2×D412yes—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2411yescharacteristic
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134#125—48no—
135#125—48no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4C2×D412yes—
164C2×C4⋊C4#4114yes—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—44no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—24no—
181C2×C42—24no—
182C2×C42—21no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42—42no—
190C2×C42C2×D412yes—
191C2×C42#4112yes—
192C2×C42#4114yes—
193C4⋊C8—48no—
194#102—48no—
195#102—48no—
196#95—44no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—42no—
201#95—44no—
202#95—44no—
203#95—44no—
204C24—22no—
205C24—41no—
206C24—41no—
207C24—41no—
208C24—41no—
209C24—41no—
210C24—41no—
211C24—42no—
212C24—42no—
213C24—42no—
214C24—42no—
215C24C22×D411yescharacteristic
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—41no—
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#12114yes—
281C22×C4#12412yes—
282C22×C4#12412yes—
283C22×C4#13514yes—
284C22×C4AGL(1,ℤ/8)14yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C2×#4112yes—
287C22×C4C2×#4112yes—
288C22×C4C2×#4111yescharacteristic
289C22×C4H(5,𝔽2)11yescharacteristic
290C22×C4H(5,𝔽2)11yescharacteristic
291C22×C4H(5,𝔽2)12yes—
292C22×C42−1+411yescharacteristic
293C22×C42−1+411yescharacteristic
294C22×C42−1+412yes—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—416no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—84no—
325C22⋊C4—416no—
326C22⋊C4—84no—
327C22⋊C4—88no—
328C22⋊C4—84no—
329C42—24no—
330C42—24no—
331C42—24no—
332C42—24no—
333C42—48no—
334C42—48no—
335C42—42no—
336C42—42no—
337C23—42no—
338C23—44no—
339C23—44no—
340C23—42no—
341C23—42no—
342C23—81no—
343C23—82no—
344C23—82no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—82no—
352C23—82no—
353C23—82no—
354C23—82no—
355C23—82no—
356C23—82no—
357C23—82no—
358C23—82no—
359C23—81no—
360C23C2×#12111yescharacteristic
361C23#52511yescharacteristic
362C23C2×#12411yescharacteristic
363C23#53411yescharacteristic
364C23#53611yescharacteristic
365C23#53711yescharacteristic
366C23#53711yescharacteristic
367C23C2×#13512yes—
368C23C2×AGL(1,ℤ/8)12yes—
369C23#57512yes—
370C23#57512yes—
371D4—44no—
372D4—44no—
373D4—82no—
374D4—82no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—81no—
411C2×C4—82no—
412C2×C4—81no—
413C2×C4—82no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
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×C4#48814yes—
428C2×C4#53511yescharacteristic
429C2×C4#53611yescharacteristic
430C2×C4#53812yes—
431C2×C4#53912yes—
432C2×C4#54112yes—
433C2×C4#54212yes—
434C2×C4#54211yescharacteristic
435C2×C4#54211yescharacteristic
436C2×C4#54612yes—
437C2×C4#54712yes—
438C2×C4#55112yes—
439C2×C4#55612yes—
440C2×C4#55612yes—
441C2×C4#55912yes—
442C2×C4#56212yes—
443C2×C4#56312yes—
444C8—82no—
445C8—82no—
446C22—82no—
447C22—82no—
448C22—84no—
449C22—82no—
450C22—84no—
451C22—161no—
452C22—161no—
453C22—162no—
454C22—161no—
455C22—162no—
456C22—161no—
457C22#253411yescharacteristic
458C22#254711yescharacteristic
459C22#259211yescharacteristic
460C22#262011yescharacteristic
461C22#263711yescharacteristic
462C22#263911yescharacteristic
463C22#281411yescharacteristic
464C22#303512yes—
465C22#303612yes—
466C22#310612yes—
467C22#311112yes—
468C22C2×#48811yescharacteristic
469C22#313511yescharacteristic
470C22#313811yescharacteristic
471C22#314211yescharacteristic
472C22#319112yes—
473C22#319412yes—
474C22#323312yes—
475C22#323412yes—
476C22#324712yes—
477C22#325012yes—
478C22#325112yes—
479C22#325612yes—
480C4—24no—
481C4—44no—
482C4—44no—
483C4—42no—
484C4—82no—
485C4—84no—
486C4—162no—
487C4—161no—
488C4—161no—
489C4#268912yes—
490C4#271012yes—
491C4#275912yes—
492C4#279812yes—
493C2—162no—
494C2#1680511yescharacteristic
495C2#2811612yes—
496C2#2872111yescharacteristic
497C2#2872211yescharacteristic
498C2#2934811yescharacteristic
499C2#2935311yescharacteristic
500C2#2983212yes—
501C2#3015412yes—
502C2#3018611yescharacteristic
503C2#3020111yescharacteristic
504C2#3274712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14709C2semidirect
5. #11385C2semidirect
6. #11358C2semidirect
7. #11315C2semidirect
8. #11228C2semidirect
11. #10832C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20732
C2×#20252
#147092
#147002
#113852
#113582
#113152
#112282
#111892
#110762
#108322