Finite Group Databasebeta

← #288062

Subgroups — #288062

Order 512.

630 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28737C211yesmaximal, characteristic
2#25454C212yesmaximal
3#23158C211yesmaximal, characteristic
4C2×#1707C211yesmaximal, characteristic
5C2×#1676C211yesmaximal, characteristic
6#10544C212yesmaximal
7#10061C212yesmaximal
8#9733C211yesmaximal, characteristic
9#9594C211yesmaximal, characteristic
10#9060C212yesmaximal
11C8×#103C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#498C2212yes—
14C2×#494C2211yescharacteristic
15C2×#487C2212yes—
16C2×#477C2212yes—
17C2×C8×Q8C2211yescharacteristic
18#2568C2212yes—
19#2369C2212yes—
20#2085—22no—
21#2066—22no—
22#2056—22no—
23#1988C2212yes—
24#1987C2212yes—
25#1981C2212yes—
26#1969—21no—
27#1969—21no—
28#1949—22no—
29#1942—21no—
30#1939C2212yes—
31#1927—22no—
32#1921—22no—
33C4×#107—21no—
34C4×#107C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C4×#103C2211yescharacteristic
37#1707—24no—
38#1676—24no—
39C2×#335C2211yescharacteristic
40C2×#333C2212yes—
41C2×#332C2212yes—
42C2×Q8⋊C8C2211yescharacteristic
43C2×C82C2211yescharacteristic
44#1279C2211yescharacteristic
45#1279C2212yes—
46#1277C2211yescharacteristic
47C22×Q16—21no—
48C22×Q16—21no—
49C2×C4⋊Q8C2312yes—
50C2×C4⋊Q8C2312yes—
51C2×C4×Q8C2311yescharacteristic
52#500—22no—
53#500D414yes—
54#498—24no—
55#498—42no—
56#494—22no—
57#494—22no—
58#487—24no—
59#487—24no—
60#477—24no—
61#477—42no—
62C8×Q8—22no—
63C8×Q8—22no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107—22no—
67C2×#107D411yescharacteristic
68C2×#107D411yescharacteristic
69C2×#106—22no—
70C2×#106—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×#103—22no—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—42no—
80C2×#103D411yescharacteristic
81C2×#103D411yescharacteristic
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#400—42no—
88#391—22no—
89#391—22no—
90#391—22no—
91#391—42no—
92#389—22no—
93#389—22no—
94#382—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4C2312yes—
98C4×C4⋊C4C2311yescharacteristic
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340D412yes—
104#340D411yescharacteristic
105#340D411yescharacteristic
106#336—22no—
107#336—21no—
108#335—22no—
109#333—24no—
110#332—42no—
111Q8⋊C8—22no—
112C82—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8C2×D411yescharacteristic
116C2×Q16—42no—
117C2×Q16—42no—
118C2×Q16—42no—
119C2×Q16—42no—
120C2×Q16—81no—
121C2×Q16—81no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8C2×D411yescharacteristic
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C22×C8C2×D411yescharacteristic
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8C2×D414yes—
138C4×Q8—21no—
139C4×Q8—21no—
140C4×Q8—21no—
141C4×Q8—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4C2×D411yescharacteristic
149C2×C4⋊C4C2×D411yescharacteristic
150C2×C4⋊C4C2×D412yes—
151C2×C4⋊C4C2×D412yes—
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—22no—
156C2×C42—42no—
157C2×C42C2411yescharacteristic
158#107—24no—
159#107—24no—
160#107—22no—
161#107—22no—
162#107—44no—
163#106—44no—
164#106—44no—
165C4⋊C8—21no—
166C4⋊C8—22no—
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—22no—
171C4⋊C8—42no—
172C4⋊C8—44no—
173#103—22no—
174#103—44no—
175#103—48no—
176#103—82no—
177#103—82no—
178#103Q1614yescomplemented
179C4×C8—21no—
180C4×C8—21no—
181C4×C8—21no—
182C4×C8—42no—
183C4×C8—42no—
184C4×C8C2×D412yes—
185C4×C8#4112yes—
186C4×C8#4112yes—
187#95—21no—
188#95—22no—
189#95—42no—
190#95—42no—
191#95—42no—
192#95—42no—
193#95—42no—
194#95—42no—
195C2×Q8—21no—
196C2×Q8—21no—
197C2×Q8—21no—
198C2×Q8—24no—
199C2×Q8—21no—
200C2×Q8—21no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—82no—
209C2×Q8—82no—
210C2×Q8C2×Q1612yes—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4C22×D411yescharacteristic
220C22×C4C22×D411yescharacteristic
221C22×C4H(5,𝔽2)11yescharacteristic
222Q16—84no—
223Q16—84no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—22no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—42no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—42no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—42no—
242C2×C8#12112yes—
243C2×C8#12112yes—
244C2×C8C2×Q1612yes—
245C2×C8#13512yes—
246C2×C8#13512yes—
247C2×C8#13712yes—
248C4⋊C4—21no—
249C4⋊C4—24no—
250C4⋊C4—21no—
251C4⋊C4—24no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—21no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—162no—
268C4⋊C4C2×Q1612yes—
269C42—41no—
270C42—41no—
271C42—41no—
272C42—42no—
273C42—41no—
274C42—42no—
275C42—162no—
276C42C22×D411yescharacteristic
277C42C2×#4111yescharacteristic
278C42C2×#4111yescharacteristic
279C42H(5,𝔽2)11yescharacteristic
280C23D4211yescharacteristic
281Q8—22no—
282Q8—22no—
283Q8—41no—
284Q8—41no—
285Q8—84no—
286Q8—82no—
287Q8—82no—
288Q8—82no—
289Q8—82no—
290Q8—162no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4C2×#12111yescharacteristic
318C2×C4#53211yescharacteristic
319C2×C4#54011yescharacteristic
320C2×C4D4211yescharacteristic
321C2×C4#54611yescharacteristic
322C2×C4#55011yescharacteristic
323C2×C4C22×Q1611yescharacteristic
324C2×C4C2×#13511yescharacteristic
325C2×C4C2×#13511yescharacteristic
326C2×C4C2×#13711yescharacteristic
327C2×C4#57711yescharacteristic
328C2×C4#57711yescharacteristic
329C2×C4#57811yescharacteristic
330C2×C4#57811yescharacteristic
331C8—22no—
332C8—22no—
333C8—22no—
334C8—42no—
335C8—81no—
336C8—82no—
337C8—81no—
338C8#46414yes—
339C22#242811yescharacteristic
340C22#327811yescharacteristic
341C22#328311yescharacteristic
342C22#328411yescharacteristic
343C22D4×Q1611yescharacteristic
344C22#329011yescharacteristic
345C22#329011yescharacteristic
346C4—21no—
347C4—21no—
348C4—42no—
349C4—42no—
350C4—82no—
351C4—81no—
352C4—81no—
353C4—84no—
354C4—82no—
355C4—162no—
356C4C2×#46411yescharacteristic
357C4#304311yescharacteristic
358C4#317011yescharacteristic
359C4#317711yescharacteristic
360C4#328511yescharacteristic
361C4D4×Q1611yescharacteristic
362C4#339411yescharacteristic
363C4#340011yescharacteristic
364C2#1993211yescharacteristic
365C2#1995911yescharacteristic
366C2#2000711yescharacteristic
367C2#2001611yescharacteristic
368C2#2536911yescharacteristic
369C2#2541511yescharacteristic
370C2#2543311yescharacteristic
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
178. #103Q16semidirect

Maximal subgroups

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

subgroupindex
#287372
#254542
#231582
C2×#17072
C2×#16762
#105442
#100612
#97332
#95942
#90602
C8×#1032