Finite Group Databasebeta

← #288732

Subgroups — #288732

Order 512.

642 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28531C211yesmaximal, characteristic
2#28265C211yesmaximal, characteristic
3#25471C212yesmaximal
4C2×#1676C211yesmaximal, characteristic
5C2×C8×Q16C211yesmaximal, characteristic
6#10414C212yesmaximal
7#10237C212yesmaximal
8#9823C211yesmaximal, characteristic
9#9614C211yesmaximal, characteristic
10#9216C212yesmaximal
11C8×#103C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#477C2212yes—
14C2×#477C2212yes—
15C2×C8×Q8C2211yescharacteristic
16C2×C8×Q8C2212yes—
17C2×C4×Q16C2211yescharacteristic
18#2582C2212yes—
19#2567C2212yes—
20#2095—22no—
21#2087—22no—
22#2068—22no—
23#2068—22no—
24#1990C2212yes—
25#1969—21no—
26#1969—21no—
27#1956C2212yes—
28#1947C2212yes—
29#1942—21no—
30#1942—22no—
31#1933—22no—
32#1845—21no—
33#1845C2211yescharacteristic
34#1803C2212yes—
35C8×C4⋊C4C2211yescharacteristic
36C4×#103C2211yescharacteristic
37#1676—24no—
38C8×Q16—24no—
39C2×#335C2211yescharacteristic
40C2×#333C2212yes—
41C2×Q8⋊C8C2211yescharacteristic
42C2×Q8⋊C8C2212yes—
43C2×C82C2211yescharacteristic
44#1296C2212yes—
45#1279C2211yescharacteristic
46#1277C2211yescharacteristic
47C22×Q16D411yescharacteristic
48C2×C4⋊Q8C2312yes—
49C2×C4×Q8C2311yescharacteristic
50C2×C4×Q8C2312yes—
51#500—22no—
52#500—22no—
53#477—24no—
54#477—24no—
55#477—42no—
56#477—42no—
57C8×Q8—24no—
58C8×Q8—22no—
59C8×Q8—22no—
60C8×Q8—42no—
61C4×Q16—22no—
62C4×Q16—22no—
63C2×#107—21no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107—22no—
67C2×#107—22no—
68C2×#107—22no—
69C2×#107—21no—
70C2×#107D411yescharacteristic
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8C2311yescharacteristic
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2311yescharacteristic
76C2×#103—22no—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—42no—
80C2×#103D411yescharacteristic
81C2×#103D412yes—
82C2×#103D411yescharacteristic
83C2×C4×C8—21no—
84C2×C4×C8—22no—
85C2×C4×C8—21no—
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#398—42no—
90#395—42no—
91#391—22no—
92#391—22no—
93#391—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#384—22no—
99#384C2312yes—
100C4×C4⋊C4C2311yescharacteristic
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340D411yescharacteristic
106#340D411yescharacteristic
107#336—21no—
108#335D414yes—
109#333—24no—
110Q8⋊C8—24no—
111Q8⋊C8—22no—
112C82—22no—
113C22×Q8—22no—
114C22×Q8C2×D412yes—
115C22×Q8C2×D411yescharacteristic
116C2×Q16—22no—
117C2×Q16—22no—
118C2×Q16Q1614yes—
119C22×C8—21no—
120C22×C8—22no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8C2×D411yescharacteristic
124C22×C8C2×D411yescharacteristic
125C22×C8C2×D411yescharacteristic
126C22×C8C2×D411yescharacteristic
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4×Q8—22no—
132C4×Q8—21no—
133C4×Q8—21no—
134C4×Q8—22no—
135C4×Q8—21no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4C2×D412yes—
150C2×C4⋊C4C2×D411yescharacteristic
151C2×C4⋊C4C2×D411yescharacteristic
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C42—22no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42C2411yescharacteristic
157#107—22no—
158#107—44no—
159#107—44no—
160#107—42no—
161#107—42no—
162#107—42no—
163#107—44no—
164#107—81no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—22no—
168C4⋊C8—21no—
169C4⋊C8—44no—
170C4⋊C8—42no—
171C4⋊C8C2×D414yes—
172C4⋊C8#4112yes—
173#103—24no—
174#103—22no—
175#103—44no—
176#103—44no—
177#103—82no—
178#103—84no—
179#103Q1614yescomplemented
180C4×C8—21no—
181C4×C8—21no—
182C4×C8—21no—
183C4×C8—21no—
184C4×C8—44no—
185C4×C8—42no—
186C4×C8—42no—
187C4×C8C2×D412yes—
188C4×C8#4112yes—
189#95—21no—
190#95—42no—
191#95—42no—
192#95—42no—
193#95—42no—
194#95—42no—
195C2×Q8—21no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—21no—
199C2×Q8—21no—
200C2×Q8—22no—
201C2×Q8—21no—
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—21no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—82no—
208C2×Q8—82no—
209C2×Q8C2×Q1612yes—
210C2×Q8C2×Q1614yes—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4C22×D411yescharacteristic
220C22×C4C22×D411yescharacteristic
221C22×C4H(5,𝔽2)11yescharacteristic
222Q16—28no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—22no—
228C2×C8—21no—
229C2×C8—24no—
230C2×C8—22no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8#12112yes—
243C2×C8#12112yes—
244C2×C8C2×Q1612yes—
245C2×C8C2×Q1612yes—
246C2×C8#13512yes—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—22no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—22no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—82no—
266C4⋊C4—162no—
267C4⋊C4—162no—
268C4⋊C4—162no—
269C4⋊C4C2×Q1612yes—
270C4⋊C4#13512yes—
271C42—41no—
272C42—42no—
273C42—42no—
274C42—41no—
275C42—41no—
276C42—41no—
277C42C22×D411yescharacteristic
278C42C2×#4111yescharacteristic
279C42C2×#4111yescharacteristic
280C42H(5,𝔽2)11yescharacteristic
281C23D4211yescharacteristic
282Q8—22no—
283Q8—22no—
284Q8—24no—
285Q8—24no—
286Q8—41no—
287Q8—41no—
288Q8—42no—
289Q8—42no—
290Q8—84no—
291Q8—162no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4C2×#12111yescharacteristic
319C2×C4#53211yescharacteristic
320C2×C4#54011yescharacteristic
321C2×C4D4211yescharacteristic
322C2×C4#54611yescharacteristic
323C2×C4#55011yescharacteristic
324C2×C4C22×Q1611yescharacteristic
325C2×C4C22×Q1611yescharacteristic
326C2×C4C2×#13511yescharacteristic
327C2×C4C2×#13511yescharacteristic
328C2×C4#57711yescharacteristic
329C2×C4#57811yescharacteristic
330C2×C4#57811yescharacteristic
331C2×C4#57811yescharacteristic
332C8—22no—
333C8—22no—
334C8—42no—
335C8—41no—
336C8—44no—
337C8—42no—
338C8—81no—
339C8#46714yes—
340C22#242811yescharacteristic
341C22#328411yescharacteristic
342C22#328511yescharacteristic
343C22D4×Q1611yescharacteristic
344C22D4×Q1611yescharacteristic
345C22#329011yescharacteristic
346C22#329211yescharacteristic
347C4—21no—
348C4—21no—
349C4—42no—
350C4—44no—
351C4—42no—
352C4—42no—
353C4—82no—
354C4—81no—
355C4—162no—
356C4—162no—
357C4C2×#46711yescharacteristic
358C4#306311yescharacteristic
359C4#317011yescharacteristic
360C4#317711yescharacteristic
361C4#328311yescharacteristic
362C4D4×Q1611yescharacteristic
363C4#329911yescharacteristic
364C4#330211yescharacteristic
365C2#2000711yescharacteristic
366C2#2000811yescharacteristic
367C2Q16211yescharacteristic
368C2#2001211yescharacteristic
369C2#2538611yescharacteristic
370C2#2541511yescharacteristic
371C2#2544111yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
179. #103Q16semidirect

Maximal subgroups

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

subgroupindex
#285312
#282652
#254712
C2×#16762
C2×C8×Q162
#104142
#102372
#98232
#96142
#92162
C8×#1032