Finite Group Databasebeta

← #352477

Subgroups — #352477

Order 512.

570 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic
2#29984C211yesmaximal, characteristic
3#29983C211yesmaximal, characteristic
4#20516C211yesmaximal, characteristic
5#9905C211yesmaximal, characteristic
6#9903C211yesmaximal, characteristic
7#9619C212yesmaximal
8#9613C212yesmaximal
9#8807C211yesmaximal, characteristic
10#8655C212yesmaximal
11#8647C212yesmaximal
12C2×#446C2211yescharacteristic
13C2×C8×Q8C2211yescharacteristic
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16#2295C2211yescharacteristic
17#2100—21no—
18#2099—21no—
19#2096—21no—
20#2095—21no—
21#2058—21no—
22#2057—21no—
23#2024—21no—
24#2024—21no—
25#1990—21no—
26#1990—21no—
27#1990—21no—
28#1990—21no—
29#1948C2212yes—
30#1947C2212yes—
31#1834C2212yes—
32#1834C2212yes—
33#1831C2211yescharacteristic
34#1831C2211yescharacteristic
35#1803C2212yes—
36#1803C2212yes—
37#1771—21no—
38#1770C2211yescharacteristic
39C8×C4⋊C4—21no—
40C8×C4⋊C4C2211yescharacteristic
41#1768—21no—
42#1767C2211yescharacteristic
43#1726—21no—
44C42×C8C2211yescharacteristic
45C2×#335C2212yes—
46C2×#334C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1296C2212yes—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4×Q8C2311yescharacteristic
58#446—22no—
59#446Q814yescomplemented
60C8×Q8—22no—
61C8×Q8Q814yescomplemented
62C2×#107—22no—
63C2×#106—22no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2311yescharacteristic
70C2×#103—22no—
71C2×#103—22no—
72C2×#97—21no—
73C2×#97—21no—
74C2×#97—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8D411yescharacteristic
78C2×C4×C8D411yescharacteristic
79C2×C4×C8D411yescharacteristic
80C2×C4×C8D411yescharacteristic
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2311yescharacteristic
83#391D411yescharacteristic
84#391D411yescharacteristic
85#389D411yescharacteristic
86#389D411yescharacteristic
87#387—21no—
88#387—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384—21no—
93#384—21no—
94#384—21no—
95#384C2312yes—
96#384C2312yes—
97C4×C4⋊C4—21no—
98C4×C4⋊C4C2311yescharacteristic
99#376—21no—
100C43C2311yescharacteristic
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#335—24no—
117#334—24no—
118C22×Q8—41no—
119C22×Q8—41no—
120C22×Q8C2×D411yescharacteristic
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8C2×D411yescharacteristic
126C22×C8C2×D411yescharacteristic
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C4⋊Q8—42no—
130C4⋊Q8—42no—
131C4⋊Q8—42no—
132C4⋊Q8—42no—
133C4⋊Q8—82no—
134C4⋊Q8—82no—
135C4⋊Q8—82no—
136C4⋊Q8—82no—
137C4×Q8—21no—
138C4×Q8—21no—
139C4×Q8C2×Q812yes—
140C4×Q8C2×Q812yes—
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—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4C2×D411yescharacteristic
157C2×C4⋊C4C2×D411yescharacteristic
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C42—21no—
160C2×C42—21no—
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42C2411yescharacteristic
168#107—82no—
169#106—82no—
170C4⋊C8—24no—
171C4⋊C8—24no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8C2×Q812yes—
179C4⋊C8C2×Q812yes—
180#103—82no—
181#103—82no—
182#97—24no—
183#97—42no—
184#97—42no—
185C4×C8—24no—
186C4×C8—22no—
187C4×C8—22no—
188C4×C8—41no—
189C4×C8—41no—
190C4×C8—42no—
191C4×C8C2×Q812yes—
192C4×C8C2×Q812yes—
193C4×C8#4112yes—
194C4×C8#4112yes—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202C2×Q8—21no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—82no—
206C2×Q8—82no—
207C2×Q8—82no—
208C2×Q8—82no—
209C2×Q8—82no—
210C2×Q8—82no—
211C2×Q8C4⋊Q812yes—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4C22×D411yescharacteristic
223C22×C4C22×D411yescharacteristic
224C22×C4C2×#4111yescharacteristic
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8C22⋊Q812yes—
246C2×C8C22⋊Q812yes—
247C2×C8C22⋊Q812yes—
248C2×C8C22⋊Q812yes—
249C4⋊C4—21no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C4⋊C4C4⋊Q812yes—
273C42—22no—
274C42—22no—
275C42—21no—
276C42—21no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—41no—
282C42—41no—
283C42#13512yes—
284C42#13712yes—
285C42C22×Q811yescharacteristic
286C42C22×Q811yescharacteristic
287C42C2×#4111yescharacteristic
288C42C2×#4111yescharacteristic
289C23C2×#12111yescharacteristic
290Q8—24no—
291Q8—42no—
292Q8—164no—
293Q8—164no—
294C2×C4—21no—
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—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4#46412yes—
323C2×C4#46412yes—
324C2×C4#47212yes—
325C2×C4#47312yes—
326C2×C4C2×C22⋊Q811yescharacteristic
327C2×C4C2×C22⋊Q811yescharacteristic
328C2×C4C2×C4⋊Q811yescharacteristic
329C2×C4#53211yescharacteristic
330C2×C4#53311yescharacteristic
331C2×C4#53311yescharacteristic
332C2×C4C2×#13511yescharacteristic
333C2×C4C2×#13711yescharacteristic
334C2×C4#57511yescharacteristic
335C2×C4#57511yescharacteristic
336C2×C4#57712yes—
337C2×C4#57712yes—
338C8—22no—
339C8—22no—
340C8—22no—
341C8—22no—
342C8—42no—
343C8—42no—
344C8—82no—
345C8—82no—
346C22#285711yescharacteristic
347C22C2×#46411yescharacteristic
348C22#304011yescharacteristic
349C22#305711yescharacteristic
350C22#305711yescharacteristic
351C22C2×#47211yescharacteristic
352C22C2×#47311yescharacteristic
353C4—22no—
354C4—22no—
355C4—22no—
356C4—22no—
357C4—21no—
358C4—21no—
359C4—42no—
360C4—42no—
361C4—81no—
362C4—81no—
363C4—162no—
364C4—162no—
365C4#307611yescharacteristic
366C4#307611yescharacteristic
367C4#307811yescharacteristic
368C4#307811yescharacteristic
369C4#316312yes—
370C4#317312yes—
371C2#1921211yescharacteristic
372C2#1921311yescharacteristic
373C2#1936211yescharacteristic
374C2#1936311yescharacteristic
375C2#3320211yescharacteristic
376C2#3321311yescharacteristic
377C2#3322211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
59. #446Q8semidirect
61. C8×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#332892
#299842
#299832
#205162
#99052
#99032
#96192
#96132
#88072
#86552
#86472