Finite Group Databasebeta

← #314764

Subgroups — #314764

Order 512.

638 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27648C211yesmaximal, characteristic
2#27647C211yesmaximal, characteristic
3#21597C211yesmaximal, characteristic
4C2×#1834C211yesmaximal, characteristic
5#10656C212yesmaximal, complemented
6#10624C212yesmaximal, complemented
7#9622C212yesmaximal, complemented
8#8724C211yesmaximal, characteristic
9#8723C211yesmaximal, characteristic
10#8657C212yesmaximal, complemented
11#8595C211yesmaximal, characteristic
12C2×#484—21no—
13C2×#483—21no—
14C2×#446C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C4×C22⋊Q8C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19#2084—21no—
20#2083—21no—
21#2072—21no—
22#2072—21no—
23#1984C2211yescharacteristic
24#1948C2212yes—
25#1947C2212yes—
26#1939C2211yescharacteristic
27#1916—21no—
28#1850—21no—
29#1834—21no—
30#1834—21no—
31#1834C2212yes—
32#1834C2212yes—
33#1824—21no—
34#1823—21no—
35#1803C2212yes—
36#1803C2212yes—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40#1771—21no—
41#1770C2211yescharacteristic
42#1752—21no—
43C2×#340—21no—
44C2×#340—21no—
45C2×#334C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1277C2212yes—
51C23×C8D411yescharacteristic
52C23×C8D411yescharacteristic
53C2×C22⋊Q8D411yescharacteristic
54C2×C4×Q8C2311yescharacteristic
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C42C2311yescharacteristic
62#484—44no—
63#483—44no—
64#446—24no—
65C2×#107—22no—
66C2×#106—22no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2311yescharacteristic
72C2×C4⋊C8C2311yescharacteristic
73C2×#103—22no—
74C2×#103—22no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×#97—21no—
79C2×#97—21no—
80C2×C4×C8—24no—
81C2×C4×C8D411yescharacteristic
82C2×C4×C8D411yescharacteristic
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85#391D411yescharacteristic
86#389D411yescharacteristic
87#387—21no—
88#385—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384—21no—
93#384—21no—
94#384—21no—
95#384C2312yes—
96#384C2312yes—
97#380D411yescharacteristic
98C4×C4⋊C4—21no—
99C4×C4⋊C4C2311yescharacteristic
100C4×C22⋊C4—21no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#336—21no—
110#336—21no—
111#336—21no—
112#336—21no—
113#336—21no—
114#336—21no—
115#336—21no—
116#334—24no—
117C22×Q8C2×D411yescharacteristic
118C23×C4—41no—
119C23×C4—41no—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4#4111yescharacteristic
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—41no—
132C22×C8—41no—
133C22×C8C2×D411yescharacteristic
134C22×C8C2×D411yescharacteristic
135C22×C8C2×D411yescharacteristic
136C22×C8C2×D411yescharacteristic
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
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—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4C2×D411yescharacteristic
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42C2411yescharacteristic
179#107—44no—
180#106—44no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—42no—
188C4⋊C8—42no—
189#103—44no—
190#103—44no—
191C22⋊C8—42no—
192C22⋊C8—42no—
193C22⋊C8—42no—
194#97—42no—
195#97—42no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8#4114yes—
200#95—21no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—21no—
205C24#12111yescharacteristic
206C2×Q8—22no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—22no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#13512yes—
235C22×C4AGL(1,ℤ/8)12yes—
236C22×C4AGL(1,ℤ/8)12yes—
237C22×C4#13712yes—
238C22×C4C22×D411yescharacteristic
239C22×C4C22×D411yescharacteristic
240C22×C4C2×#4111yescharacteristic
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—44no—
252C2×C8—44no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8#12312yes—
266C2×C8#12312yes—
267C4⋊C4—22no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—22no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C42—21no—
293C42—21no—
294C42—22no—
295C42—22no—
296C42—41no—
297C42—41no—
298C42—41no—
299C42—41no—
300C42C2×#4112yes—
301C422−1+412yes—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23#46412yes—
307C23#46912yes—
308C23#47112yes—
309C23C2×#12111yescharacteristic
310Q8—42no—
311Q8—42no—
312C2×C4—28no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
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—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—164no—
346C2×C4—164no—
347C2×C4#46412yes—
348C2×C4#46912yes—
349C2×C4#47112yes—
350C2×C4C2×#12311yescharacteristic
351C2×C4#53611yescharacteristic
352C2×C4#54712yes—
353C2×C4#55212yes—
354C2×C4C2×#13511yescharacteristic
355C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
356C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
357C2×C4C2×#13711yescharacteristic
358C2×C4#57511yescharacteristic
359C2×C4#57511yescharacteristic
360C2×C4#57812yes—
361C8—22no—
362C8—22no—
363C8—41no—
364C8—41no—
365C8—82no—
366C8—82no—
367C8—81no—
368C8—81no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22#170214yes—
376C22#248611yescharacteristic
377C22C2×#46411yescharacteristic
378C22#304011yescharacteristic
379C22C2×#46911yescharacteristic
380C22#305911yescharacteristic
381C22C2×#47111yescharacteristic
382C22C2×#47111yescharacteristic
383C4—22no—
384C4—22no—
385C4—22no—
386C4—22no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4—162no—
392C4—162no—
393C4#309111yescharacteristic
394C4#309511yescharacteristic
395C4#323211yescharacteristic
396C4#323611yescharacteristic
397C4#334312yes—
398C4#334312yes—
399C2—24no—
400C2C2×#170211yescharacteristic
401C2#1921511yescharacteristic
402C2#1930111yescharacteristic
403C2#1936511yescharacteristic
404C2#2765311yescharacteristic
405C2#2766711yescharacteristic
406C2#2767211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10656C2semidirect
6. #10624C2semidirect
7. #9622C2semidirect
10. #8657C2semidirect

Maximal subgroups

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

subgroupindex
#276482
#276472
#215972
C2×#18342
#106562
#106242
#96222
#87242
#87232
#86572
#85952