Finite Group Databasebeta

← #325154

Subgroups — #325154

Order 512.

548 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic
2#29213C211yesmaximal, characteristic
3#28694C211yesmaximal, characteristic
4C8×#125C211yesmaximal, characteristic
5#10658C212yesmaximal
6#10657C212yesmaximal
7#9905C211yesmaximal, characteristic
8#9903C211yesmaximal, characteristic
9#9613C212yesmaximal
10#9613C212yesmaximal
11#8273C211yesmaximal, characteristic
12C2×#501—21no—
13#2868C2211yescharacteristic
14#2868C2211yescharacteristic
15C4×#125C2211yescharacteristic
16#2084—21no—
17#2083—21no—
18#2058—21no—
19#2057—21no—
20#1993—21no—
21#1992—21no—
22#1990—21no—
23#1990—21no—
24#1990—21no—
25#1990—21no—
26#1948C2212yes—
27#1948C2212yes—
28#1947C2212yes—
29#1947C2212yes—
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1916—21no—
33#1916—21no—
34#1847—21no—
35#1834C2212yes—
36#1834C2212yes—
37#1831C2211yescharacteristic
38C8×C4⋊C4—21no—
39C8×C4⋊C4—21no—
40C8×C4⋊C4C2211yescharacteristic
41C8×C4⋊C4C2211yescharacteristic
42#1767C2211yescharacteristic
43#1767C2211yescharacteristic
44C42×C8C2211yescharacteristic
45C2×#334C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1277C2212yes—
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×#125D411yescharacteristic
58#501—44no—
59C2×#107—22no—
60C2×#107—22no—
61C2×#106—22no—
62C2×#106—22no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2311yescharacteristic
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×#97—21no—
70C2×#97—21no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8D411yescharacteristic
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79#389D411yescharacteristic
80#389D411yescharacteristic
81#387—21no—
82#387—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384C2312yes—
90#384C2312yes—
91#382D411yescharacteristic
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4C2311yescharacteristic
95C4×C4⋊C4C2311yescharacteristic
96C43C2311yescharacteristic
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#336—21no—
106#336—21no—
107#336—21no—
108#336—21no—
109#336—21no—
110#336—21no—
111#336—21no—
112#336—21no—
113#334Q818yescomplemented
114C22×Q8—41no—
115C22×Q8—41no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D411yescharacteristic
124C4⋊Q8—42no—
125C4⋊Q8—42no—
126C4⋊Q8—42no—
127C4⋊Q8—42no—
128C4⋊Q8—42no—
129C4⋊Q8—42no—
130C4⋊Q8—82no—
131C4⋊Q8—82no—
132C4⋊Q8—82no—
133#125—22no—
134#125SD1614yes—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
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—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4C2×D411yescharacteristic
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C4⋊C4C2×D411yescharacteristic
155C2×C42—21no—
156C2×C42—21no—
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42C2411yescharacteristic
164#107—44no—
165#107—82no—
166#106—44no—
167#106—82no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—42no—
173C4⋊C8—42no—
174C4⋊C8C2×Q814yes—
175C4⋊C8C2×Q814yes—
176#97—42no—
177#97—42no—
178C4×C8—22no—
179C4×C8—41no—
180C4×C8—41no—
181C4×C8—41no—
182C4×C8—41no—
183C4×C8—42no—
184C4×C8—42no—
185C4×C8C2×Q814yes—
186#95—21no—
187#95—21no—
188#95—21no—
189#95—21no—
190#95—21no—
191#95—21no—
192C2×Q8—82no—
193C2×Q8—82no—
194C2×Q8—82no—
195C2×Q8—82no—
196C2×Q8—82no—
197C2×Q8—82no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4#12111yescharacteristic
205C22×C4#12111yescharacteristic
206C22×C4#12111yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4C22×D411yescharacteristic
209C22×C4C22×D411yescharacteristic
210C22×C4C2×#4111yescharacteristic
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—24no—
218C2×C8—24no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8C4⋊Q812yes—
234C2×C8C4⋊Q812yes—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4C2×SD1612yes—
260C4⋊C4C2×SD1612yes—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—41no—
269C42—41no—
270C42—41no—
271C42—41no—
272C42C2×SD1612yes—
273C42#13512yes—
274C42C22×Q812yes—
275C422−1+412yes—
276C23C2×#12111yescharacteristic
277Q8—164no—
278Q8—164no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4#46512yes—
308C2×C4#46512yes—
309C2×C4#47212yes—
310C2×C4#47312yes—
311C2×C4C2×C4⋊Q811yescharacteristic
312C2×C4#53611yescharacteristic
313C2×C4D4×Q812yes—
314C2×C4#55712yes—
315C2×C4C22×SD1611yescharacteristic
316C2×C4C2×#13511yescharacteristic
317C2×C4#57511yescharacteristic
318C2×C4#57511yescharacteristic
319C2×C4#57511yescharacteristic
320C2×C4#57511yescharacteristic
321C2×C4#57712yes—
322C8—22no—
323C8—22no—
324C8—44no—
325C8—44no—
326C8—41no—
327C8—41no—
328C8—81no—
329C8—81no—
330C22#254611yescharacteristic
331C22#304011yescharacteristic
332C22#304011yescharacteristic
333C22C2×#46511yescharacteristic
334C22#305711yescharacteristic
335C22C2×#47211yescharacteristic
336C22C2×#47311yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—42no—
343C4—42no—
344C4—81no—
345C4—81no—
346C4—162no—
347C4—162no—
348C4#316211yescharacteristic
349C4#316611yescharacteristic
350C4#324011yescharacteristic
351C4#324011yescharacteristic
352C4Q8×SD1612yes—
353C4#338112yes—
354C2#1927411yescharacteristic
355C2#1927511yescharacteristic
356C2#1936211yescharacteristic
357C2#1936311yescharacteristic
358C2#2867711yescharacteristic
359C2#2869511yescharacteristic
360C2#2869911yescharacteristic
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
113. #334Q8semidirect

Maximal subgroups

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

subgroupindex
#332892
#292132
#286942
C8×#1252
#106582
#106572
#99052
#99032
#96132
#82732