Finite Group Databasebeta

← #477318

Subgroups — #477318

Order 512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2066C212yesmaximal
2C2×#2056C212yesmaximal
3#15186C212yesmaximal
4#11328C211yesmaximal, characteristic
5#11322C211yesmaximal, characteristic
6#11231C212yesmaximal
7#11124C212yesmaximal
8#11122C212yesmaximal
9#11024C211yesmaximal, characteristic
10C23×Q16C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#106C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C2×#391C2212yes—
19C2×#391C2212yes—
20C2×#391C2212yes—
21C2×#389C2212yes—
22#2066—28no—
23#2056—28no—
24C2×#340C2212yes—
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#336C2211yescharacteristic
29C2×#336C2212yes—
30#1445—22no—
31#1443—22no—
32#1443—22no—
33#1442—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C4×#95C2211yescharacteristic
37C23×Q8C2312yes—
38C22×Q16—24no—
39C22×Q16—22no—
40C22×Q16—41no—
41C22×Q16—41no—
42C22×Q16—42no—
43C23×C8C2312yes—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C22×C4⋊C4—22no—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C42C2312yes—
51C22×C42C2311yescharacteristic
52C2×#107—42no—
53C2×#107—41no—
54C2×#106—24no—
55C2×#106—42no—
56C2×#106—42no—
57C2×C4⋊C8—24no—
58C2×C4⋊C8—24no—
59C2×C4⋊C8—24no—
60C2×C4⋊C8—42no—
61C2×C4⋊C8—41no—
62C2×#103—24no—
63C2×#103—24no—
64C2×#103—24no—
65C2×#103—42no—
66C2×#103—42no—
67C2×#103—42no—
68C2×#103—42no—
69C2×#103—42no—
70C2×#103D418yes—
71#391—24no—
72#391—24no—
73#391—24no—
74#391—42no—
75#391—42no—
76#391—42no—
77#389—24no—
78#389—42no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95D411yescharacteristic
91C2×#95D412yes—
92C2×#95D411yescharacteristic
93#340—24no—
94#340—22no—
95#340—24no—
96#340—41no—
97#336—41no—
98#336—42no—
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—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8C2×D414yes—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4C2411yescharacteristic
122C2×Q16—48no—
123C2×Q16—44no—
124C2×Q16—44no—
125C2×Q16—48no—
126C2×Q16—44no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8C2×D414yes—
142C22×C8#4112yes—
143C22×C8#4112yes—
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—22no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4C2×D414yes—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—21no—
175#107—44no—
176#106—48no—
177C4⋊C8—44no—
178C4⋊C8—48no—
179#103—48no—
180#103—48no—
181#103—48no—
182#95—22no—
183#95—24no—
184#95—41no—
185#95—44no—
186#95—44no—
187#95—44no—
188#95—44no—
189#95—44no—
190#95—44no—
191#95—44no—
192#95—44no—
193#95—44no—
194#95—82no—
195#95—82no—
196C24C22×D411yescharacteristic
197C2×Q8—24no—
198C2×Q8—24no—
199C2×Q8—24no—
200C2×Q8—24no—
201C2×Q8—24no—
202C2×Q8—24no—
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—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
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—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4#13512yes—
276C22×C4#13714yes—
277C22×C4C22×D412yes—
278C22×C4C2×#4111yescharacteristic
279C22×C4C2×#4111yescharacteristic
280C22×C4H(5,𝔽2)11yescharacteristic
281C22×C4H(5,𝔽2)12yes—
282C22×C4H(5,𝔽2)12yes—
283C22×C4H(5,𝔽2)11yescharacteristic
284C22×C42−1+411yescharacteristic
285C22×C42−1+412yes—
286C22×C42−1+411yescharacteristic
287Q16—416no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—42no—
303C2×C8—41no—
304C2×C8—42no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—88no—
318C42—42no—
319C42—42no—
320C42—42no—
321C42—41no—
322C42—42no—
323C42—41no—
324C42—42no—
325C23#53712yes—
326C23#53711yescharacteristic
327C23D4212yes—
328C23#54711yescharacteristic
329C23#55011yescharacteristic
330C23C2×#13511yescharacteristic
331C23C2×#13712yes—
332C23#57511yescharacteristic
333C23#57712yes—
334C23#57811yescharacteristic
335C23#57811yescharacteristic
336Q8—44no—
337Q8—44no—
338Q8—44no—
339Q8—44no—
340Q8—82no—
341Q8—82no—
342Q8—82no—
343Q8—82no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—41no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—82no—
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—82no—
395C2×C4—82no—
396C2×C4#53412yes—
397C2×C4#53612yes—
398C2×C4#53712yes—
399C2×C4#53812yes—
400C2×C4#53811yescharacteristic
401C2×C4#53911yescharacteristic
402C2×C4#54212yes—
403C2×C4D4211yescharacteristic
404C2×C4#54612yes—
405C2×C4#54712yes—
406C2×C4#55611yescharacteristic
407C2×C4#56011yescharacteristic
408C2×C4#56011yescharacteristic
409C2×C4#56212yes—
410C2×C4#56212yes—
411C2×C4#56311yescharacteristic
412C2×C4#56312yes—
413C2×C4#56311yescharacteristic
414C8—42no—
415C8—44no—
416C8—42no—
417C22#267212yes—
418C22#267511yescharacteristic
419C22#268212yes—
420C22#272711yescharacteristic
421C22#282311yescharacteristic
422C22#324812yes—
423C22#325011yescharacteristic
424C22#325212yes—
425C22#325511yescharacteristic
426C22#325812yes—
427C22#325812yes—
428C22#326111yescharacteristic
429C22#326311yescharacteristic
430C22#327012yes—
431C22#327112yes—
432C22#327612yes—
433C22#328212yes—
434C22#334211yescharacteristic
435C22#334412yes—
436C22#335211yescharacteristic
437C22#339311yescharacteristic
438C22#340012yes—
439C22#340211yescharacteristic
440C4—42no—
441C4—44no—
442C4—81no—
443C4—84no—
444C4—82no—
445C4—84no—
446C4—162no—
447C4—162no—
448C4#266811yescharacteristic
449C4#268011yescharacteristic
450C4#271512yes—
451C4#281412yes—
452C4#282611yescharacteristic
453C4#283011yescharacteristic
454C2#1710911yescharacteristic
455C2#3066812yes—
456C2#3068612yes—
457C2#3085211yescharacteristic
458C2#3086411yescharacteristic
459C2#3121512yes—
460C2#3122612yes—
461C2#3252811yescharacteristic
462C2#3253511yescharacteristic
463C2#3286311yescharacteristic
464C2#3286611yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ 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
C2×#20662
C2×#20562
#151862
#113282
#113222
#112312
#111242
#111222
#110242