Finite Group Databasebeta

← #508782

Subgroups — #508782

Order 512.

643 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33266C211yesmaximal, characteristic, complemented
2C4×#493C211yesmaximal, characteristic, complemented
3#13538C211yesmaximal, characteristic, complemented
4#13527C211yesmaximal, characteristic, complemented
5#13476C212yesmaximal, complemented
6#13448C212yesmaximal, complemented
7#13407C211yesmaximal, characteristic, complemented
8#13325C211yesmaximal, characteristic
9#13012C212yesmaximal, complemented
10#12937C212yesmaximal, complemented
11#12879C211yesmaximal, characteristic, complemented
12C22×D16C2212yes—
13C2×#493C2211yescharacteristic
14C2×C4×D8—21no—
15C2×C4×D8C2212yes—
16#2860—21no—
17C4×#127—21no—
18C2×#367C2212yes—
19C2×C4⋊C16C2212yes—
20C2×#357C2212yes—
21C2×#357C2212yes—
22C2×#357C2212yes—
23C2×#357C2212yes—
24#1970—21no—
25#1970C2211yescharacteristic
26#1970C2211yescharacteristic
27#1970C2212yes—
28#1970C2211yescharacteristic
29#1957—21no—
30#1942C2211yescharacteristic
31#1942C2211yescharacteristic
32#1942C2211yescharacteristic
33#1934—21no—
34#1934—21no—
35#1934—21no—
36#1845C2211yescharacteristic
37C42×C8C2211yescharacteristic
38#1380C2212yes—
39#1380C2212yes—
40#1374C2211yescharacteristic
41#1374C2211yescharacteristic
42#1374C2212yes—
43C22×D8—21no—
44C22×D8C2311yescharacteristic
45C22×D8C2312yes—
46C2×#127—21no—
47C2×#126—41no—
48C2×C4×D4—22no—
49C2×C4×D4—41no—
50C2×C4×D4—41no—
51C2×D16—24no—
52C2×D16—42no—
53C2×D16—42no—
54C22×C16C2312yes—
55C22×C16C2312yes—
56#493—24no—
57C4×D8—42no—
58C4×D8—42no—
59C4×D8—44no—
60C2×#107—21no—
61C2×#107C2312yes—
62C2×#107C2311yescharacteristic
63C2×#107C2311yescharacteristic
64C2×#107C2311yescharacteristic
65C2×#102—22no—
66C2×#102—22no—
67C2×#102—21no—
68C2×#102—21no—
69C2×#102—21no—
70C2×C4×C8D411yescharacteristic
71C2×C4×C8D411yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#390—22no—
77#390—21no—
78#390—21no—
79#390—21no—
80#390—41no—
81#390—41no—
82#389—21no—
83#389—21no—
84#389—21no—
85#388—41no—
86#388—41no—
87#388—41no—
88#384—21no—
89C43D411yescharacteristic
90#367—42no—
91C4⋊C16—42no—
92#357—24no—
93#357—42no—
94#357—42no—
95#357—42no—
96#340—21no—
97#340—21no—
98#340—21no—
99C22×D4—22no—
100C22×D4—21no—
101C22×D4—41no—
102C22×D4—41no—
103C23×C4—42no—
104C2×D8—24no—
105C2×D8—22no—
106C2×D8—22no—
107C2×D8—22no—
108C2×D8—22no—
109C2×D8—22no—
110C2×D8—22no—
111C2×D8—21no—
112C2×D8—22no—
113C2×D8—21no—
114C2×D8—42no—
115C2×D8—42no—
116C2×D8#4114yes—
117C22×C8C2×D411yescharacteristic
118C22×C8C2×D411yescharacteristic
119C22×C8C2×D411yescharacteristic
120C22×C8C2411yescharacteristic
121#127—24no—
122#127—42no—
123#126—84no—
124C4×D4—44no—
125C4×D4—44no—
126C4×D4—84no—
127C4×D4—84no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C42D811yescharacteristic
147C2×C42D811yescharacteristic
148C2×C42D811yescharacteristic
149C2×C42D811yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153D16—48no—
154C2×C16—22no—
155C2×C16—22no—
156C2×C16—22no—
157C2×C16—22no—
158C2×C16—22no—
159C2×C16—22no—
160#107—22no—
161#107—21no—
162#107—22no—
163#107—22no—
164#107—22no—
165#107—21no—
166#107—42no—
167#102—42no—
168#102—42no—
169#102—44no—
170#102—44no—
171#102—42no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—21no—
175C4×C8—22no—
176C4×C8—21no—
177C4×C8—41no—
178C4×C8—41no—
179#95—41no—
180#95—41no—
181#95—41no—
182C24—42no—
183C24—41no—
184C2×D4—24no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4—41no—
188C2×D4—42no—
189C2×D4—41no—
190C2×D4—42no—
191C2×D4—41no—
192C2×D4—42no—
193C2×D4—41no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—82no—
199C2×D4—82no—
200C2×D4—82no—
201C2×D4—82no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—84no—
207C22×C4—84no—
208C22×C4—84no—
209C22×C4C2×D811yescharacteristic
210C22×C4C2×D811yescharacteristic
211C22×C4C2×D811yescharacteristic
212C22×C4C2×D811yescharacteristic
213C22×C4C2×D811yescharacteristic
214C22×C4C2×D811yescharacteristic
215C22×C4C22×D411yescharacteristic
216D8—24no—
217D8—24no—
218D8—48no—
219D8—42no—
220D8—42no—
221D8—42no—
222D8—42no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8AGL(1,ℤ/8)12yes—
232C2×C8C2×#4112yes—
233C2×C8H(5,𝔽2)11yescharacteristic
234C2×C8H(5,𝔽2)12yes—
235C2×C82−1+411yescharacteristic
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—82no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C22⋊C4—84no—
250C22⋊C4—84no—
251C22⋊C4—82no—
252C22⋊C4—82no—
253C22⋊C4—82no—
254C22⋊C4—82no—
255C22⋊C4—82no—
256C22⋊C4—84no—
257C22⋊C4—82no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—22no—
268C16—42no—
269C16—42no—
270C23—44no—
271C23—82no—
272C23—82no—
273C23—82no—
274C23—82no—
275C23—82no—
276C23—82no—
277C23—81no—
278C23—81no—
279C23—81no—
280C23—81no—
281C23—81no—
282C23—81no—
283C23—81no—
284C23C22×D811yescharacteristic
285D4—44no—
286D4—44no—
287D4—88no—
288D4—88no—
289D4—81no—
290D4—81no—
291D4—82no—
292D4—81no—
293D4—82no—
294D4—81no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
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—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—82no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—168no—
322C2×C4#50812yes—
323C2×C4#50912yes—
324C2×C4#50912yes—
325C2×C4#50912yes—
326C2×C4#52511yescharacteristic
327C2×C4#53511yescharacteristic
328C2×C4#53711yescharacteristic
329C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
330C2×C4#57511yescharacteristic
331C2×C4#57511yescharacteristic
332C8—22no—
333C8—41no—
334C8—41no—
335C8#53812yes—
336C8#53912yes—
337C22—84no—
338C22—84no—
339C22—84no—
340C22—162no—
341C22—162no—
342C22—162no—
343C22—162no—
344C22—161no—
345C22—161no—
346C22—161no—
347C22—161no—
348C22—161no—
349C22—161no—
350C22—161no—
351C22#310011yescharacteristic
352C22#320511yescharacteristic
353C22#324511yescharacteristic
354C22C2×#50811yescharacteristic
355C22C2×#50911yescharacteristic
356C22C2×#50911yescharacteristic
357C22C2×#50911yescharacteristic
358C4—22no—
359C4—22no—
360C4—22no—
361C4—22no—
362C4—41no—
363C4—41no—
364C4—161no—
365C4—161no—
366C4—161no—
367C4—162no—
368C4#263611yescharacteristic
369C4#311411yescharacteristic
370C4#321911yescharacteristic
371C4#325111yescharacteristic
372C2—164no—
373C2—321no—
374C2#3012511yescharacteristic
375C2#3366511yescharacteristic
376C2#3366911yescharacteristic
377C2#3377311yescharacteristic
378C2#3378311yescharacteristic
379C2#3381311yescharacteristic
380C2#3381911yescharacteristic
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. #33266C2semidirect
2. C4×#493C2semidirect
3. #13538C2semidirect
4. #13527C2semidirect
5. #13476C2semidirect
6. #13448C2semidirect
7. #13407C2semidirect
9. #13012C2semidirect
10. #12937C2semidirect
11. #12879C2semidirect

Maximal subgroups

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

subgroupindex
#332662
C4×#4932
#135382
#135272
#134762
#134482
#134072
#133252
#130122
#129372
#128792