Finite Group Databasebeta

← #352537

Subgroups — #352537

Order 512.

615 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33200C211yesmaximal, characteristic
2#33196C211yesmaximal, characteristic
3#30304C211yesmaximal, characteristic
4#20569C211yesmaximal, characteristic
5#9900C211yesmaximal, characteristic
6#9896C211yesmaximal, characteristic
7#9616C212yesmaximal
8#9615C212yesmaximal
9#8810C211yesmaximal, characteristic
10#8649C212yesmaximal
11#8645C212yesmaximal
12C2×#479—21no—
13C2×#475—21no—
14C2×C8×Q8C2212yes—
15#2868C2211yescharacteristic
16#2859C2211yescharacteristic
17C42×Q8C2211yescharacteristic
18#2083—22no—
19#2057—21no—
20#2057—21no—
21#1989—21no—
22#1988—21no—
23#1949—21no—
24#1948—21no—
25#1948C2211yescharacteristic
26#1948C2212yes—
27#1947C2211yescharacteristic
28#1845C2212yes—
29#1842C2211yescharacteristic
30#1842C2211yescharacteristic
31#1834C2212yes—
32#1805—21no—
33#1803—21no—
34#1803C2212yes—
35#1803C2211yescharacteristic
36#1803C2211yescharacteristic
37C8×C4⋊C4—22no—
38C8×C4⋊C4C2212yes—
39C4×C4⋊C8—21no—
40C4×C4⋊C8C2211yescharacteristic
41#1744—21no—
42#1744C2211yescharacteristic
43C2×#335C2212yes—
44C2×#335C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×#125—21no—
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8C2311yescharacteristic
56#479—44no—
57#475—44no—
58C8×Q8—24no—
59C8×Q8Q818yescomplemented
60C2×#107—21no—
61C2×#107—22no—
62C2×#107—22no—
63C2×#106—22no—
64C2×#106—21no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2312yes—
74C2×#103—22no—
75C2×#103—21no—
76C2×#103—21no—
77C2×C4×C8—22no—
78C2×C4×C8—22no—
79C2×C4×C8—21no—
80C2×C4×C8C2312yes—
81#389—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384C2311yescharacteristic
89#384C2312yes—
90#384C2311yescharacteristic
91#382—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4D411yescharacteristic
96C4×C4⋊C4D411yescharacteristic
97C4×C4⋊C4C2311yescharacteristic
98C43—21no—
99C43C2311yescharacteristic
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—21no—
105#340—21no—
106#340—21no—
107#340—21no—
108#336—22no—
109#336—21no—
110#336D411yescharacteristic
111#336D411yescharacteristic
112#336D412yes—
113#335—24no—
114#335—24no—
115C22×Q8—41no—
116C22×Q8C2×D411yescharacteristic
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8C2×D412yes—
122C22×C8C2×D412yes—
123C4⋊Q8—42no—
124C4⋊Q8—42no—
125C4⋊Q8—82no—
126C4⋊Q8—82no—
127#125—42no—
128#125—42no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8D814yes—
133C4×Q8Q1614yes—
134C4×Q8C2×Q814yes—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C4⋊C4C2×D411yescharacteristic
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
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—42no—
169#107—82no—
170#107—82no—
171#106—44no—
172#106—42no—
173C4⋊C8—28no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—42no—
180C4⋊C8—42no—
181C4⋊C8—42no—
182C4⋊C8—42no—
183C4⋊C8C2×Q814yes—
184#103—44no—
185#103—42no—
186#103—42no—
187C4×C8—28no—
188C4×C8—81no—
189C4×C8—82no—
190C4×C8C2×Q814yes—
191C4×C8#4114yes—
192#95—21no—
193#95—21no—
194#95—21no—
195#95—41no—
196#95—41no—
197C2×Q8—21no—
198C2×Q8—21no—
199C2×Q8—21no—
200C2×Q8—82no—
201C2×Q8—82no—
202C2×Q8—82no—
203C2×Q8C4⋊Q812yes—
204C2×Q8C2×D812yes—
205C2×Q8C2×Q1612yes—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4C22×D411yescharacteristic
218C22×C4C22×D411yescharacteristic
219C22×C4C2×#4111yescharacteristic
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—24no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—42no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—42no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8C22⋊Q814yes—
237C2×C8C22⋊Q814yes—
238C4⋊C4—21no—
239C4⋊C4—24no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—81no—
256C4⋊C4—81no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4C4⋊Q812yes—
264C4⋊C4C2×D812yes—
265C4⋊C4C2×Q1612yes—
266C42—24no—
267C42—21no—
268C42—22no—
269C42—21no—
270C42—24no—
271C42—24no—
272C42—41no—
273C42—41no—
274C42—41no—
275C42—41no—
276C42—81no—
277C42—81no—
278C42C2×D812yes—
279C42C2×Q1612yes—
280C42#13512yes—
281C42#13512yes—
282C42C22×Q812yes—
283C42C2×#4112yes—
284C23C2×#12111yescharacteristic
285Q8—22no—
286Q8—22no—
287Q8—22no—
288Q8—164no—
289Q8#50014yes—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
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—82no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4#46612yes—
322C2×C4#46612yes—
323C2×C4#46712yes—
324C2×C4#46712yes—
325C2×C4C2×C22⋊Q812yes—
326C2×C4C2×C4⋊Q811yescharacteristic
327C2×C4#53211yescharacteristic
328C2×C4#53312yes—
329C2×C4C22×D811yescharacteristic
330C2×C4C22×Q1611yescharacteristic
331C2×C4C2×#13511yescharacteristic
332C2×C4C2×#13511yescharacteristic
333C2×C4#57712yes—
334C2×C4#57711yescharacteristic
335C2×C4#57811yescharacteristic
336C8—24no—
337C8—24no—
338C8—44no—
339C8—82no—
340C8—81no—
341C8—81no—
342C22#285711yescharacteristic
343C22#304311yescharacteristic
344C22#304611yescharacteristic
345C22C2×#46611yescharacteristic
346C22C2×#46711yescharacteristic
347C22#306811yescharacteristic
348C22#306911yescharacteristic
349C4—24no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—42no—
356C4—42no—
357C4—42no—
358C4—161no—
359C4—162no—
360C4—161no—
361C4#308212yes—
362C4#308412yes—
363C4C2×#50011yescharacteristic
364C4#316011yescharacteristic
365C4#316911yescharacteristic
366C4#317011yescharacteristic
367C2#1939811yescharacteristic
368C2#1939911yescharacteristic
369C2#1941611yescharacteristic
370C2#1941711yescharacteristic
371C2#3320411yescharacteristic
372C2#3320911yescharacteristic
373C2#3322311yescharacteristic
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. C8×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#332002
#331962
#303042
#205692
#99002
#98962
#96162
#96152
#88102
#86492
#86452