Finite Group Databasebeta

← #444465

Subgroups — #444465

Order 512.

1106 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2056C212yesmaximal
2C2×#1928C212yesmaximal
3#14003C212yesmaximal
4#11322C211yesmaximal, characteristic
5#11256C211yesmaximal, characteristic
6#11077C212yesmaximal
7#11052C212yesmaximal
8#11024C211yesmaximal, characteristic
9#10850C212yesmaximal
10C23×Q16C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×C4⋊C8C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C2×#391C2212yes—
18C2×#391C2212yes—
19C2×#391C2212yes—
20C2×#382C2212yes—
21C2×C4×C4⋊C4C2212yes—
22#2056—28no—
23#1928—28no—
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31#1443—22no—
32#1443—22no—
33C4⋊#95C2212yes—
34#1434—22no—
35#1434—22no—
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⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C42—22no—
50C22×C42C2311yescharacteristic
51C22×C42C2312yes—
52C2×#107—24no—
53C2×#107—41no—
54C2×C4⋊C8—24no—
55C2×C4⋊C8—24no—
56C2×C4⋊C8—42no—
57C2×C4⋊C8—41no—
58C2×C4⋊C8—42no—
59C2×#103—24no—
60C2×#103—24no—
61C2×#103—24no—
62C2×#103—24no—
63C2×#103—24no—
64C2×#103—24no—
65C2×#103—42no—
66C2×#103—42no—
67C2×#103—42no—
68#391—24no—
69#391—24no—
70#391—24no—
71#391—24no—
72#391—42no—
73#391D418yes—
74#382—24no—
75#382—42no—
76C4×C4⋊C4—24no—
77C4×C4⋊C4—24no—
78C2×#95—22no—
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×#95D411yescharacteristic
90C2×#95D411yescharacteristic
91C2×#95D412yes—
92#340—22no—
93#340—24no—
94#340—24no—
95#340—41no—
96#340—42no—
97#336—24no—
98#336—41no—
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×D412yes—
120C23×C4C2×D411yescharacteristic
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—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
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⋊C4C2×D414yes—
161C2×C4⋊C4#4114yes—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—21no—
168C2×C42—22no—
169C2×C42—21no—
170C2×C42—44no—
171C2×C42—44no—
172C2×C42—44no—
173C2×C42C2×D414yes—
174C2×C42#4114yes—
175#107—44no—
176C4⋊C8—48no—
177C4⋊C8—44no—
178#103—216no—
179#103—48no—
180#103—48no—
181#95—28no—
182#95—28no—
183#95—22no—
184#95—41no—
185#95—42no—
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—44no—
195C24C22×D411yescharacteristic
196C2×Q8—24no—
197C2×Q8—24no—
198C2×Q8—24no—
199C2×Q8—24no—
200C2×Q8—24no—
201C2×Q8—24no—
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—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C22×C4—21no—
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—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
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—42no—
272C22×C4—42no—
273C22×C4#12114yes—
274C22×C4#12114yes—
275C22×C4#13512yes—
276C22×C4C22×D412yes—
277C22×C4C2×#4111yescharacteristic
278C22×C4C2×#4112yes—
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4112yes—
281C22×C4H(5,𝔽2)11yescharacteristic
282C22×C4H(5,𝔽2)12yes—
283C22×C4H(5,𝔽2)11yescharacteristic
284C22×C42−1+411yescharacteristic
285C22×C42−1+411yescharacteristic
286Q16—416no—
287C2×C8—24no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—24no—
291C2×C8—42no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C42—24no—
317C42—24no—
318C42—24no—
319C42—24no—
320C42—42no—
321C42—41no—
322C42—41no—
323C42—88no—
324C23C2×#12112yes—
325C23#53711yescharacteristic
326C23#54612yes—
327C23#54711yescharacteristic
328C23#55011yescharacteristic
329C23C2×#13511yescharacteristic
330C23#57512yes—
331C23#57511yescharacteristic
332C23#57811yescharacteristic
333C23#57812yes—
334C23#57811yescharacteristic
335Q8—44no—
336Q8—44no—
337Q8—44no—
338Q8—44no—
339Q8—82no—
340Q8—82no—
341Q8—82no—
342Q8—82no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—42no—
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#52512yes—
396C2×C4#52912yes—
397C2×C4#53512yes—
398C2×C4#53811yescharacteristic
399C2×C4#53911yescharacteristic
400C2×C4#53912yes—
401C2×C4#54212yes—
402C2×C4D4211yescharacteristic
403C2×C4#54612yes—
404C2×C4#54712yes—
405C2×C4#54712yes—
406C2×C4#55312yes—
407C2×C4#55511yescharacteristic
408C2×C4#55512yes—
409C2×C4#55611yescharacteristic
410C2×C4#55911yescharacteristic
411C2×C4H(3,𝔽4)11yescharacteristic
412C2×C4#56311yescharacteristic
413C8—42no—
414C8—42no—
415C8—82no—
416C22#241412yes—
417C22#242512yes—
418C22#253711yescharacteristic
419C22#271211yescharacteristic
420C22#272711yescharacteristic
421C22#303512yes—
422C22#304012yes—
423C22#304412yes—
424C22#304412yes—
425C22#325011yescharacteristic
426C22#325611yescharacteristic
427C22#326111yescharacteristic
428C22#326111yescharacteristic
429C22#330012yes—
430C22#331512yes—
431C22#331512yes—
432C22#332712yes—
433C22#334211yescharacteristic
434C22#334912yes—
435C22#335211yescharacteristic
436C22#339311yescharacteristic
437C22#340211yescharacteristic
438C22#340212yes—
439C4—44no—
440C4—42no—
441C4—44no—
442C4—84no—
443C4—84no—
444C4—81no—
445C4—84no—
446C4—162no—
447C4#245511yescharacteristic
448C4#247412yes—
449C4#259311yescharacteristic
450C4#267011yescharacteristic
451C4#271812yes—
452C4#278511yescharacteristic
453C2#1578011yescharacteristic
454C2#2463712yes—
455C2#2464512yes—
456C2#2520112yes—
457C2#2522912yes—
458C2#2825611yescharacteristic
459C2#2826111yescharacteristic
460C2#3211511yescharacteristic
461C2#3212511yescharacteristic
462C2#3252811yescharacteristic
463C2#3253911yescharacteristic
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×#20562
C2×#19282
#140032
#113222
#112562
#110772
#110522
#110242
#108502