Finite Group Databasebeta

← #414503

Subgroups — #414503

Order 512.

1066 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2084C211yesmaximal, characteristic, complemented
2C2×#2071C211yesmaximal, characteristic
3#14260C211yesmaximal, characteristic
4#13779C211yesmaximal, characteristic, complemented
5#11315C212yesmaximal, complemented
6#11208C211yesmaximal, characteristic, complemented
7#11121C212yesmaximal
8#11051C211yesmaximal, characteristic
9#10931C212yesmaximal
10#10908C212yesmaximal, complemented
11#10796C211yesmaximal, characteristic, complemented
12C22×#106C2212yes—
13C22×#102C2212yes—
14C22×#102C2212yes—
15C2×#390C2211yescharacteristic
16C2×#390C2211yescharacteristic
17C2×#390C2211yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2212yes—
22C2×#376C2211yescharacteristic
23#2084—24no—
24#2071—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1440—22no—
35#1439—22no—
36#1436—22no—
37#1436—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C23×D4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C22⋊C4—22no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—21no—
56C22×C42—22no—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×#106—24no—
61C2×#106—24no—
62C2×#106—42no—
63C2×#102—24no—
64C2×#102—24no—
65C2×#102—24no—
66C2×#102—24no—
67C2×#102—42no—
68C2×#102—42no—
69#390—22no—
70#390—22no—
71#390—22no—
72#390—22no—
73#390—24no—
74#389—22no—
75#389—22no—
76#384—22no—
77#384D414yes—
78#382—24no—
79C4×C4⋊C4—42no—
80C4×C4⋊C4—42no—
81#376—24no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95D412yes—
91C2×#95D412yes—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—42no—
97#336—24no—
98#336—22no—
99#336—22no—
100#336—24no—
101C25—21no—
102C22×D4—21no—
103C22×D4—22no—
104C22×D4—21no—
105C22×D4—22no—
106C22×D4—22no—
107C22×D4—21no—
108C22×D4—21no—
109C22×D4—21no—
110C22×D4—22no—
111C22×D4#4112yes—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4C2411yescharacteristic
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—
134C22×C8—22no—
135C22×C8#4114yes—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4C2×D414yes—
166C2×C4⋊C4#4114yes—
167C2×C4⋊C4#4112yes—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—82no—
173C2×C22⋊C4—82no—
174C2×C22⋊C4—82no—
175C2×C22⋊C4—82no—
176C2×C22⋊C4—82no—
177C2×C22⋊C4—82no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—44no—
188C2×C42—82no—
189C2×C42—82no—
190C2×C42C2×D412yes—
191C2×C42#4112yes—
192C2×C42#4114yes—
193#106—48no—
194#102—48no—
195#102—48no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—44no—
201#95—42no—
202#95—82no—
203#95—82no—
204#95—82no—
205#95—82no—
206C24—22no—
207C24—41no—
208C24—41no—
209C24—41no—
210C24—41no—
211C24—41no—
212C24—41no—
213C24—42no—
214C24—42no—
215C24—42no—
216C24—42no—
217C24C22×D411yescharacteristic
218C2×D4—24no—
219C2×D4—22no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—41no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—41no—
232C2×D4—42no—
233C2×D4—42no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
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—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4#12112yes—
285C22×C4#12112yes—
286C22×C4#12314yes—
287C22×C4C22×D411yescharacteristic
288C22×C4C2×#4112yes—
289C22×C4C2×#4111yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C22×C4C2×#4112yes—
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4112yes—
294C22×C4H(5,𝔽2)11yescharacteristic
295C22×C4H(5,𝔽2)11yescharacteristic
296C22×C42−1+412yes—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—24no—
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—
309C2×C8—42no—
310C2×C8—42no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—84no—
328C4⋊C4—84no—
329C4⋊C4—84no—
330C22⋊C4—88no—
331C22⋊C4—84no—
332C22⋊C4—84no—
333C22⋊C4—88no—
334C42—24no—
335C42—24no—
336C42—24no—
337C42—24no—
338C42—42no—
339C42—42no—
340C42—88no—
341C23—42no—
342C23—44no—
343C23—44no—
344C23—42no—
345C23—42no—
346C23—82no—
347C23—82no—
348C23—81no—
349C23—81no—
350C23—82no—
351C23—82no—
352C23—81no—
353C23—82no—
354C23—82no—
355C23—82no—
356C23—82no—
357C23—82no—
358C23—82no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23C2×#12111yescharacteristic
365C23C2×#12311yescharacteristic
366C23#52511yescharacteristic
367C23#52511yescharacteristic
368C23#53211yescharacteristic
369C23#53511yescharacteristic
370C23#53611yescharacteristic
371C23#57612yes—
372C23#57712yes—
373C23#57712yes—
374C23#57712yes—
375D4—44no—
376D4—44no—
377D4—82no—
378D4—82no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
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—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—44no—
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—81no—
421C2×C4—81no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4#52912yes—
433C2×C4#52912yes—
434C2×C4#52912yes—
435C2×C4#53212yes—
436C2×C4#53912yes—
437C2×C4#53912yes—
438C2×C4#54611yescharacteristic
439C2×C4#54611yescharacteristic
440C2×C4#54812yes—
441C2×C4#55112yes—
442C2×C4#55212yes—
443C2×C4#55311yescharacteristic
444C2×C4#55311yescharacteristic
445C2×C4#55312yes—
446C2×C4#55612yes—
447C2×C4#55612yes—
448C8—44no—
449C8—82no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22—84no—
454C22—84no—
455C22—162no—
456C22—161no—
457C22—161no—
458C22—162no—
459C22—161no—
460C22—161no—
461C22#239511yescharacteristic
462C22#253511yescharacteristic
463C22#254511yescharacteristic
464C22#259911yescharacteristic
465C22#260311yescharacteristic
466C22#262311yescharacteristic
467C22#285411yescharacteristic
468C22#304111yescharacteristic
469C22#304211yescharacteristic
470C22#304311yescharacteristic
471C22#304311yescharacteristic
472C22#309612yes—
473C22#309912yes—
474C22#311912yes—
475C22#311912yes—
476C22#312312yes—
477C22#312512yes—
478C22#317312yes—
479C22#317912yes—
480C22#322112yes—
481C22#322412yes—
482C22#324112yes—
483C22#324412yes—
484C4—42no—
485C4—42no—
486C4—44no—
487C4—82no—
488C4—84no—
489C4—84no—
490C4—82no—
491C4—161no—
492C4—161no—
493C4#255812yes—
494C4#256012yes—
495C4#257712yes—
496C4#259412yes—
497C2—162no—
498C2#1517411yescharacteristic
499C2#2377112yes—
500C2#2816611yescharacteristic
501C2#2816711yescharacteristic
502C2#2866211yescharacteristic
503C2#2866711yescharacteristic
504C2#2940612yes—
505C2#2951012yes—
506C2#2991612yes—
507C2#3311811yescharacteristic
508C2#3312111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2084C2semidirect
4. #13779C2semidirect
5. #11315C2semidirect
6. #11208C2semidirect
10. #10908C2semidirect
11. #10796C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20842
C2×#20712
#142602
#137792
#113152
#112082
#111212
#110512
#109312
#109082
#107962