Finite Group Databasebeta

← #478365

Subgroups — #478365

Order 512.

904 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15152C211yesmaximal, characteristic
2#15128C211yesmaximal, characteristic
3#11384C212yesmaximal
4#11346C211yesmaximal, characteristic
5#11316C212yesmaximal
6#11316C212yesmaximal
7#11316C212yesmaximal
8#11293C212yesmaximal
9#11207C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C2×#400—21no—
12C2×#395—21no—
13C2×#389C2212yes—
14C2×#384C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16C2×#340C2212yes—
17C2×#340C2212yes—
18C2×#340C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#336C2212yes—
25C2×#336C2211yescharacteristic
26C2×#336C2211yescharacteristic
27C2×#336C2212yes—
28#1446—21no—
29#1445—21no—
30#1444—21no—
31#1442—21no—
32#1441—21no—
33#1441—21no—
34C4⋊#95C2212yes—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C23×Q8—21no—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—42no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59#400—44no—
60#395—44no—
61#389—24no—
62#389—42no—
63#384—41no—
64#384—41no—
65C4×C4⋊C4—41no—
66C4×C4⋊C4—41no—
67C2×#95—22no—
68C2×#95—21no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95D411yescharacteristic
79C2×#95D411yescharacteristic
80#340—24no—
81#340—24no—
82#340—24no—
83#340—24no—
84#340—42no—
85#340—42no—
86#340—42no—
87#340—42no—
88#336—42no—
89#336—41no—
90#336—41no—
91#336—42no—
92C22×Q8—44no—
93C22×Q8—42no—
94C22×Q8—42no—
95C22×Q8—42no—
96C22×Q8—82no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2411yescharacteristic
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C4⋊Q8—44no—
123C4⋊Q8—48no—
124C4⋊Q8—44no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
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—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4Q1614yes—
163C2×C4⋊C4#4114yes—
164C2×C4⋊C4#4114yes—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—22no—
177#95—41no—
178#95—41no—
179#95—44no—
180#95—42no—
181#95—42no—
182#95—44no—
183#95—42no—
184#95—44no—
185#95—42no—
186#95—44no—
187#95—82no—
188#95—82no—
189#95—82no—
190C24C22×D411yescharacteristic
191C2×Q8—48no—
192C2×Q8—88no—
193C2×Q8—84no—
194C2×Q8—88no—
195C2×Q8—84no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4C2×Q1612yes—
252C22×C4C2×Q1612yes—
253C22×C4C2×Q1612yes—
254C22×C4AGL(1,ℤ/8)12yes—
255C22×C4AGL(1,ℤ/8)12yes—
256C22×C4C2×#4112yes—
257C22×C4C2×#4112yes—
258C22×C4H(5,𝔽2)11yescharacteristic
259C22×C4H(5,𝔽2)12yes—
260C22×C4H(5,𝔽2)11yescharacteristic
261C22×C42−1+412yes—
262C22×C42−1+412yes—
263C22×C42−1+412yes—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—48no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—88no—
298C4⋊C4—84no—
299C4⋊C4—84no—
300C42—42no—
301C42—42no—
302C42—41no—
303C42—41no—
304C42—41no—
305C42—41no—
306C42—42no—
307C42—42no—
308C23#53411yescharacteristic
309C23#53611yescharacteristic
310C23#53611yescharacteristic
311C23#54612yes—
312C23#54812yes—
313C23C22×Q1611yescharacteristic
314C23C2×AGL(1,ℤ/8)11yescharacteristic
315C23C2×AGL(1,ℤ/8)11yescharacteristic
316C23#57511yescharacteristic
317C23#57712yes—
318C23#57712yes—
319Q8—816no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4#53812yes—
381C2×C4#53912yes—
382C2×C4#54112yes—
383C2×C4#54112yes—
384C2×C4#54212yes—
385C2×C4#54212yes—
386C2×C4#55111yescharacteristic
387C2×C4#55212yes—
388C2×C4#55511yescharacteristic
389C2×C4#55612yes—
390C2×C4#56011yescharacteristic
391C2×C4#56311yescharacteristic
392C2×C4#56312yes—
393C2×C4#56312yes—
394C2×C4#56312yes—
395C2×C4#56312yes—
396C8—82no—
397C8—82no—
398C22#269012yes—
399C22#269711yescharacteristic
400C22#271612yes—
401C22#273111yescharacteristic
402C22#281711yescharacteristic
403C22#318911yescharacteristic
404C22#319211yescharacteristic
405C22#320011yescharacteristic
406C22#320011yescharacteristic
407C22#323011yescharacteristic
408C22#323011yescharacteristic
409C22#323511yescharacteristic
410C22#323511yescharacteristic
411C22#324412yes—
412C22#324412yes—
413C22#329912yes—
414C22#330612yes—
415C22#331012yes—
416C22#332612yes—
417C22#336312yes—
418C22#336812yes—
419C22#337312yes—
420C22#337612yes—
421C4—42no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—42no—
426C4—81no—
427C4—84no—
428C4—84no—
429C4—162no—
430C4—162no—
431C4#275812yes—
432C4#277512yes—
433C4#283312yes—
434C4#283812yes—
435C2#1714411yescharacteristic
436C2#3146012yes—
437C2#3148212yes—
438C2#3166911yescharacteristic
439C2#3168111yescharacteristic
440C2#3223212yes—
441C2#3225212yes—
442C2#3263111yescharacteristic
443C2#3264611yescharacteristic
444C2#3279011yescharacteristic
445C2#3279411yescharacteristic
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
#151522
#151282
#113842
#113462
#113162
#112932
#112072