Finite Group Databasebeta

← #507954

Subgroups — #507954

Order 512.

638 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33241C211yesmaximal, characteristic, complemented
2#21275C211yesmaximal, characteristic, complemented
3#13452C211yesmaximal, characteristic, complemented
4#13407C211yesmaximal, characteristic, complemented
5#13363C211yesmaximal, characteristic, complemented
6#13325C211yesmaximal, characteristic
7#13239C212yesmaximal, complemented
8#13149C212yesmaximal, complemented
9#13131C212yesmaximal, complemented
10#13012C212yesmaximal, complemented
11#12591C211yesmaximal, characteristic, complemented
12C22×D16C2212yes—
13C2×#500C2211yescharacteristic
14C2×#493C2211yescharacteristic
15C2×#486—21no—
16C2×C4×D8C2211yescharacteristic
17#2858—21no—
18#2293—21no—
19C2×#367C2212yes—
20C2×#357C2212yes—
21C2×#357C2212yes—
22C2×#357C2212yes—
23C2×#357C2212yes—
24C2×#346C2212yes—
25#1970—21no—
26#1970—21no—
27#1970C2211yescharacteristic
28#1970C2211yescharacteristic
29#1970C2212yes—
30#1970C2212yes—
31#1942C2211yescharacteristic
32#1942C2211yescharacteristic
33#1934—21no—
34#1934—21no—
35C4×#107C2211yescharacteristic
36C4×#102—21no—
37C42×C8C2211yescharacteristic
38#1380C2212yes—
39#1380C2212yes—
40#1374C2211yescharacteristic
41#1374C2211yescharacteristic
42#1374C2212yes—
43C22×D8—21no—
44C22×D8C2312yes—
45C22×D8C2311yescharacteristic
46C2×C4⋊Q8—21no—
47C2×#127—21no—
48C2×#124—41no—
49C2×#124—41no—
50C2×C4×D4—21no—
51C2×#117—41no—
52C2×D16—24no—
53C2×D16—24no—
54C2×D16—42no—
55C22×C16C2312yes—
56C22×C16C2312yes—
57#500—42no—
58#493—42no—
59#486—44no—
60C4×D8—22no—
61C4×D8—22no—
62C2×#107—21no—
63C2×#107C2312yes—
64C2×#107C2311yescharacteristic
65C2×#107C2311yescharacteristic
66C2×#107C2311yescharacteristic
67C2×#102—21no—
68C2×#102—22no—
69C2×#102—22no—
70C2×#102—21no—
71C2×#102—21no—
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#390—22no—
79#390—22no—
80#390—21no—
81#390—21no—
82#390—41no—
83#390—41no—
84#389—21no—
85#389—21no—
86#388—41no—
87#388—41no—
88C4×C4⋊C4—21no—
89C4×C22⋊C4—41no—
90C43D411yescharacteristic
91#367—42no—
92#357—24no—
93#357—24no—
94#357—42no—
95#357—42no—
96#346—24no—
97#340—21no—
98#340—21no—
99#340—21no—
100C22×Q8—41no—
101C22×D4—21no—
102C22×D4—22no—
103C22×D4—41no—
104C22×D4—41no—
105C23×C4—41no—
106C2×D8—22no—
107C2×D8—21no—
108C2×D8—22no—
109C2×D8—22no—
110C2×D8—22no—
111C2×D8—22no—
112C2×D8—22no—
113C2×D8—22no—
114C2×D8—22no—
115C2×D8—42no—
116C2×D8—42no—
117C2×D8—42no—
118C2×D8#4112yes—
119C22×C8C2×D411yescharacteristic
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C22×C8C2411yescharacteristic
123C4⋊Q8—42no—
124C4⋊Q8—42no—
125#127—42no—
126#127—42no—
127#124—84no—
128#124—84no—
129C4×D4—24no—
130C4×D4—42no—
131#117—84no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—41no—
148C2×C42—41no—
149C2×C42D811yescharacteristic
150C2×C42D811yescharacteristic
151C2×C42D811yescharacteristic
152C2×C42D811yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156D16—48no—
157C2×C16—22no—
158C2×C16—22no—
159C2×C16—22no—
160C2×C16—22no—
161C2×C16—22no—
162C2×C16#4114yes—
163#107—21no—
164#107—22no—
165#107—22no—
166#107—22no—
167#107—22no—
168#107—42no—
169#107#4112yes—
170#102—24no—
171#102—44no—
172#102—42no—
173#102—42no—
174#102—44no—
175C4×C8—22no—
176C4×C8—22no—
177C4×C8—22no—
178C4×C8—21no—
179C4×C8—21no—
180C4×C8—22no—
181C4×C8—41no—
182#95—41no—
183#95—41no—
184#95—41no—
185C24—41no—
186C24—42no—
187C2×Q8—82no—
188C2×Q8—82no—
189C2×Q8—82no—
190C2×D4—22no—
191C2×D4—42no—
192C2×D4—41no—
193C2×D4—42no—
194C2×D4—41no—
195C2×D4—42no—
196C2×D4—41no—
197C2×D4—42no—
198C2×D4—41no—
199C2×D4—42no—
200C2×D4—41no—
201C2×D4—42no—
202C2×D4—82no—
203C2×D4—82no—
204C2×D4—82no—
205C2×D4—82no—
206C2×D4—82no—
207C2×D4—82no—
208C22×C4—44no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4C2×D811yescharacteristic
216C22×C4C2×D811yescharacteristic
217C22×C4C2×D811yescharacteristic
218C22×C4C2×D811yescharacteristic
219C22×C4C2×D811yescharacteristic
220C22×C4C2×D811yescharacteristic
221C22×C4C22×D411yescharacteristic
222D8—24no—
223D8—42no—
224D8—42no—
225D8—42no—
226D8—42no—
227D8—42no—
228D8—84no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8#13512yes—
237C2×C8AGL(1,ℤ/8)12yes—
238C2×C8C2×#4111yescharacteristic
239C2×C8C2×#4112yes—
240C2×C8C2×#4111yescharacteristic
241C2×C8H(5,𝔽2)12yes—
242C4⋊C4—22no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—41no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C22⋊C4—44no—
254C22⋊C4—82no—
255C22⋊C4—82no—
256C22⋊C4—84no—
257C22⋊C4—84no—
258C22⋊C4—84no—
259C22⋊C4—82no—
260C22⋊C4—82no—
261C22⋊C4—84no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—41no—
272C42—82no—
273C16—24no—
274C16—42no—
275C23—42no—
276C23—81no—
277C23—81no—
278C23—81no—
279C23—81no—
280C23—81no—
281C23—81no—
282C23—82no—
283C23—82no—
284C23—82no—
285C23—82no—
286C23—82no—
287C23—82no—
288C23—82no—
289C23C22×D811yescharacteristic
290Q8—164no—
291D4—42no—
292D4—42no—
293D4—82no—
294D4—82no—
295D4—81no—
296D4—82no—
297D4—81no—
298D4—82no—
299D4—164no—
300D4—164no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—42no—
317C2×C4—88no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—82no—
328C2×C4—81no—
329C2×C4#50912yes—
330C2×C4#50912yes—
331C2×C4#50912yes—
332C2×C4#52511yescharacteristic
333C2×C4#53211yescharacteristic
334C2×C4#53511yescharacteristic
335C2×C4C2×#13511yescharacteristic
336C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
337C2×C4#57511yescharacteristic
338C8—22no—
339C8—22no—
340C8—41no—
341C8#55112yes—
342C8#55512yes—
343C22—82no—
344C22—82no—
345C22—82no—
346C22—161no—
347C22—161no—
348C22—161no—
349C22—161no—
350C22—162no—
351C22—162no—
352C22—162no—
353C22—162no—
354C22—162no—
355C22—162no—
356C22—162no—
357C22#310011yescharacteristic
358C22#316911yescharacteristic
359C22#320511yescharacteristic
360C22C2×#50911yescharacteristic
361C22C2×#50911yescharacteristic
362C22C2×#50911yescharacteristic
363C22#341411yescharacteristic
364C4—22no—
365C4—22no—
366C4—22no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—82no—
371C4—161no—
372C4—161no—
373C4—162no—
374C4#261011yescharacteristic
375C4#310611yescharacteristic
376C4#318011yescharacteristic
377C4#321111yescharacteristic
378C2—162no—
379C2—322no—
380C2#2955211yescharacteristic
381C2#3366911yescharacteristic
382C2#3367511yescharacteristic
383C2#3373711yescharacteristic
384C2#3374811yescharacteristic
385C2#3377911yescharacteristic
386C2#3378311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#13325C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #33241C2semidirect
2. #21275C2semidirect
3. #13452C2semidirect
4. #13407C2semidirect
5. #13363C2semidirect
7. #13239C2semidirect
8. #13149C2semidirect
9. #13131C2semidirect
10. #13012C2semidirect
11. #12591C2semidirect

Maximal subgroups

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

subgroupindex
#332412
#212752
#134522
#134072
#133632
#133252
#132392
#131492
#131312
#130122
#125912