Finite Group Databasebeta

← #452453

Subgroups — #452453

Order 512.

1024 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14021C212yesmaximal
2#11379C211yesmaximal, characteristic
3#11346C211yesmaximal, characteristic
4#11190C212yesmaximal
5#11124C212yesmaximal
6#11024C211yesmaximal, characteristic
7#11024C212yesmaximal
8#10932C212yesmaximal
9#10806C212yesmaximal
10C23×Q16C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#103C2212yes—
13C22×#103C2212yes—
14C22×#103C2212yes—
15C2×#391C2212yes—
16C2×#391C2212yes—
17C2×#391C2212yes—
18C2×C4×C4⋊C4C2212yes—
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2211yescharacteristic
23C2×#340C2212yes—
24C2×#340C2211yescharacteristic
25C2×#336C2212yes—
26C2×#336C2211yescharacteristic
27C2×#336C2212yes—
28C2×#336C2211yescharacteristic
29#1443—22no—
30#1442—22no—
31#1436—22no—
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34#1434—22no—
35C23×Q8C2312yes—
36C22×Q16—24no—
37C22×Q16—22no—
38C22×Q16—42no—
39C22×Q16—41no—
40C22×Q16—41no—
41C23×C8C2312yes—
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C42—22no—
50C22×C42C2311yescharacteristic
51C22×C42C2312yes—
52C2×#107—22no—
53C2×#107—41no—
54C2×#107—41no—
55C2×#103—24no—
56C2×#103—24no—
57C2×#103—24no—
58C2×#103—24no—
59C2×#103—42no—
60C2×#103—42no—
61C2×#103—42no—
62C2×#103—42no—
63C2×#103—42no—
64#391—24no—
65#391—24no—
66#391—24no—
67#391—42no—
68#391—42no—
69#391—42no—
70C4×C4⋊C4—24no—
71C4×C4⋊C4—41no—
72C4×C4⋊C4—41no—
73C4×C4⋊C4—42no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95D412yes—
86#340—24no—
87#340—24no—
88#340—41no—
89#340—42no—
90#340—41no—
91#336—41no—
92#336—42no—
93#336—41no—
94#336—42no—
95C22×Q8—22no—
96C22×Q8—22no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8—22no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8#4114yes—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2411yescharacteristic
118C2×Q16—48no—
119C2×Q16—44no—
120C2×Q16—44no—
121C2×Q16—48no—
122C2×Q16—44no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8#4112yes—
139C22×C8#4112yes—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
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—24no—
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—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4#4114yes—
162C2×C4⋊C4#4114yes—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—44no—
174C2×C42—44no—
175C2×C42—82no—
176#107—44no—
177#103—48no—
178#103—48no—
179#103—48no—
180#95—28no—
181#95—28no—
182#95—42no—
183#95—44no—
184#95—44no—
185#95—44no—
186#95—44no—
187#95—44no—
188#95—44no—
189#95—44no—
190#95—82no—
191#95—82no—
192C24C22×D411yescharacteristic
193C2×Q8—24no—
194C2×Q8—24no—
195C2×Q8—24no—
196C2×Q8—24no—
197C2×Q8—24no—
198C2×Q8—24no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
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#13512yes—
272C22×C4#13714yes—
273C22×C4C2×#4111yescharacteristic
274C22×C4C2×#4111yescharacteristic
275C22×C4C2×#4112yes—
276C22×C4C2×#4112yes—
277C22×C4C2×#4112yes—
278C22×C4H(5,𝔽2)11yescharacteristic
279C22×C4H(5,𝔽2)11yescharacteristic
280C22×C4H(5,𝔽2)12yes—
281C22×C4H(5,𝔽2)11yescharacteristic
282C22×C4H(5,𝔽2)11yescharacteristic
283Q16—416no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C42—42no—
316C42—42no—
317C42—41no—
318C42—42no—
319C42—42no—
320C42—41no—
321C42—42no—
322C42—88no—
323C23#52512yes—
324C23#53411yescharacteristic
325C23#54612yes—
326C23#55011yescharacteristic
327C23#55011yescharacteristic
328C23C2×#13511yescharacteristic
329C23C2×#13712yes—
330C23#57511yescharacteristic
331C23#57611yescharacteristic
332C23#57812yes—
333C23#57811yescharacteristic
334Q8—44no—
335Q8—44no—
336Q8—44no—
337Q8—44no—
338Q8—82no—
339Q8—82no—
340Q8—82no—
341Q8—82no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4#52912yes—
395C2×C4#52912yes—
396C2×C4#53811yescharacteristic
397C2×C4#53811yescharacteristic
398C2×C4#53812yes—
399C2×C4#54012yes—
400C2×C4#54012yes—
401C2×C4#55111yescharacteristic
402C2×C4#55112yes—
403C2×C4#55112yes—
404C2×C4#55112yes—
405C2×C4#55312yes—
406C2×C4#55312yes—
407C2×C4#55511yescharacteristic
408C2×C4#55511yescharacteristic
409C2×C4#55911yescharacteristic
410C2×C4#56011yescharacteristic
411C2×C4H(3,𝔽4)11yescharacteristic
412C8—42no—
413C8—42no—
414C8—82no—
415C22#246712yes—
416C22#247112yes—
417C22#260011yescharacteristic
418C22#269911yescharacteristic
419C22#273311yescharacteristic
420C22#310812yes—
421C22#311612yes—
422C22#312412yes—
423C22#312712yes—
424C22#319111yescharacteristic
425C22#319511yescharacteristic
426C22#320211yescharacteristic
427C22#320311yescharacteristic
428C22#330212yes—
429C22#331212yes—
430C22#331412yes—
431C22#332712yes—
432C22#339311yescharacteristic
433C22#339611yescharacteristic
434C22#339812yes—
435C22#339812yes—
436C22#340211yescharacteristic
437C22#340311yescharacteristic
438C4—42no—
439C4—44no—
440C4—84no—
441C4—84no—
442C4—84no—
443C4—82no—
444C4—81no—
445C4—162no—
446C4#253112yes—
447C4#257211yescharacteristic
448C4#259311yescharacteristic
449C4#273511yescharacteristic
450C4#275211yescharacteristic
451C4#275312yes—
452C2#1604711yescharacteristic
453C2#2706712yes—
454C2#2709212yes—
455C2#2719112yes—
456C2#2721612yes—
457C2#2941611yescharacteristic
458C2#2942711yescharacteristic
459C2#3172311yescharacteristic
460C2#3173211yescharacteristic
461C2#3267211yescharacteristic
462C2#3267911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#140212
#113792
#113462
#111902
#111242
#110242
#109322
#108062