Finite Group Databasebeta

← #450777

Subgroups — #450777

Order 512.

1080 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×#1928C212yesmaximal
2#14142C212yesmaximal
3#11379C211yesmaximal, characteristic
4#11255C211yesmaximal, characteristic
5#11077C212yesmaximal
6#11067C212yesmaximal
7#11024C211yesmaximal, characteristic
8#11024C212yesmaximal
9#10884C212yesmaximal
10C23×Q16C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C2×#391C2212yes—
17C2×#391C2212yes—
18C2×#391C2212yes—
19C2×#389C2212yes—
20C2×C4×C4⋊C4C2212yes—
21#1928—28no—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2211yescharacteristic
26C2×#340C2211yescharacteristic
27C2×#336C2212yes—
28C2×#336C2211yescharacteristic
29C2×#336C2212yes—
30#1443—22no—
31#1442—22no—
32#1436—22no—
33C4⋊#95C2212yes—
34#1434—22no—
35C4×#95C2211yescharacteristic
36C23×Q8C2312yes—
37C22×Q16—24no—
38C22×Q16—22no—
39C22×Q16—42no—
40C22×Q16—41no—
41C22×Q16—41no—
42C23×C8C2312yes—
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C22×C4⋊C4—22no—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C42—22no—
50C22×C42C2311yescharacteristic
51C22×C42C2312yes—
52C2×#107—41no—
53C2×#107—41no—
54C2×#107D414yes—
55C2×C4⋊C8—41no—
56C2×C4⋊C8—42no—
57C2×#103—24no—
58C2×#103—24no—
59C2×#103—24no—
60C2×#103—24no—
61C2×#103—24no—
62C2×#103—42no—
63C2×#103—42no—
64C2×#103—42no—
65C2×#103—42no—
66#391—24no—
67#391—24no—
68#391—24no—
69#391—24no—
70#391—42no—
71#391—42no—
72#389—24no—
73#389—42no—
74C4×C4⋊C4—24no—
75C4×C4⋊C4—42no—
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×#95—22no—
86C2×#95D412yes—
87C2×#95D411yescharacteristic
88C2×#95D411yescharacteristic
89#340—24no—
90#340—24no—
91#340—24no—
92#340—41no—
93#340—41no—
94#336—24no—
95#336—42no—
96#336—41no—
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—22no—
109C22×Q8—22no—
110C22×Q8#4114yes—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D412yes—
119C23×C4C2411yescharacteristic
120C2×Q16—48no—
121C2×Q16—44no—
122C2×Q16—44no—
123C2×Q16—48no—
124C2×Q16—44no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8C2×D412yes—
141C22×C8#4112yes—
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—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—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4C2×D414yes—
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—22no—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42—44no—
174C2×C42—44no—
175C2×C42—82no—
176#107—44no—
177C4⋊C8—44no—
178#103—216no—
179#103—48no—
180#103—48no—
181#95—28no—
182#95—28no—
183#95—22no—
184#95—24no—
185#95—41no—
186#95—44no—
187#95—44no—
188#95—44no—
189#95—44no—
190#95—44no—
191#95—44no—
192#95—44no—
193#95—82no—
194C24C22×D411yescharacteristic
195C2×Q8—24no—
196C2×Q8—24no—
197C2×Q8—24no—
198C2×Q8—24no—
199C2×Q8—24no—
200C2×Q8—24no—
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—
221C2×Q8—42no—
222C2×Q8—42no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
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#13512yes—
274C22×C4#13714yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C2×#4112yes—
277C22×C4C2×#4112yes—
278C22×C4C2×#4111yescharacteristic
279C22×C4C2×#4112yes—
280C22×C4H(5,𝔽2)11yescharacteristic
281C22×C4H(5,𝔽2)11yescharacteristic
282C22×C4H(5,𝔽2)11yescharacteristic
283C22×C4H(5,𝔽2)12yes—
284C22×C42−1+411yescharacteristic
285Q16—416no—
286C2×C8—24no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—24no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—88no—
316C42—42no—
317C42—42no—
318C42—42no—
319C42—42no—
320C42—42no—
321C42—41no—
322C42—41no—
323C42—88no—
324C23#52512yes—
325C23#53711yescharacteristic
326C23D4211yescharacteristic
327C23#54612yes—
328C23#55011yescharacteristic
329C23C2×#13511yescharacteristic
330C23C2×#13712yes—
331C23#57511yescharacteristic
332C23#57611yescharacteristic
333C23#57811yescharacteristic
334C23#57812yes—
335Q8—44no—
336Q8—44no—
337Q8—44no—
338Q8—44no—
339Q8—82no—
340Q8—82no—
341Q8—82no—
342Q8—82no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—41no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—44no—
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#53411yescharacteristic
398C2×C4#53811yescharacteristic
399C2×C4#53812yes—
400C2×C4#53812yes—
401C2×C4#54212yes—
402C2×C4#54711yescharacteristic
403C2×C4#55012yes—
404C2×C4#55111yescharacteristic
405C2×C4#55112yes—
406C2×C4#55212yes—
407C2×C4#55312yes—
408C2×C4#55312yes—
409C2×C4#55511yescharacteristic
410C2×C4#55911yescharacteristic
411C2×C4H(3,𝔽4)11yescharacteristic
412C2×C4#56211yescharacteristic
413C8—42no—
414C8—42no—
415C8—82no—
416C22#245612yes—
417C22#246712yes—
418C22#260911yescharacteristic
419C22#268211yescharacteristic
420C22#271211yescharacteristic
421C22#310712yes—
422C22#311712yes—
423C22#312412yes—
424C22#312712yes—
425C22#325011yescharacteristic
426C22#325511yescharacteristic
427C22#326111yescharacteristic
428C22#326211yescharacteristic
429C22#327012yes—
430C22#327311yescharacteristic
431C22#328311yescharacteristic
432C22#330012yes—
433C22#331412yes—
434C22#331512yes—
435C22#332812yes—
436C22#339611yescharacteristic
437C22#339812yes—
438C22#340211yescharacteristic
439C4—44no—
440C4—42no—
441C4—84no—
442C4—84no—
443C4—81no—
444C4—84no—
445C4—84no—
446C4—82no—
447C4#247912yes—
448C4#259311yescharacteristic
449C4#260011yescharacteristic
450C4#273311yescharacteristic
451C4#273611yescharacteristic
452C4#277912yes—
453C2#1597811yescharacteristic
454C2#2662212yes—
455C2#2664512yes—
456C2#2707312yes—
457C2#2708212yes—
458C2#2954011yescharacteristic
459C2#2955111yescharacteristic
460C2#3121011yescharacteristic
461C2#3122511yescharacteristic
462C2#3211511yescharacteristic
463C2#3212411yescharacteristic
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×#19282
#141422
#113792
#112552
#110772
#110672
#110242
#108842