Finite Group Databasebeta

← #288065

Subgroups — #288065

Order 512.

632 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28675C211yesmaximal, characteristic
2#25278C212yesmaximal
3#23158C211yesmaximal, characteristic
4C2×#1710C211yesmaximal, characteristic
5C2×#1676C211yesmaximal, characteristic
6#10541C212yesmaximal
7#10059C212yesmaximal
8#9731C211yesmaximal, characteristic
9#9594C211yesmaximal, characteristic
10#9060C212yesmaximal
11C8×#103C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#500C2211yescharacteristic
14C2×#487C2212yes—
15C2×#487C2212yes—
16C2×#477C2212yes—
17C2×C8×Q8C2211yescharacteristic
18#2546C2212yes—
19#2369C2212yes—
20#2086—22no—
21#2068—22no—
22#2043—22no—
23#1988C2212yes—
24#1987C2212yes—
25#1981—22no—
26#1981C2212yes—
27#1942—21no—
28#1942—21no—
29#1942—21no—
30#1939C2212yes—
31#1928—22no—
32#1921—22no—
33C4×#107—21no—
34C4×#107C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C4×#103C2211yescharacteristic
37#1710—24no—
38#1676—24no—
39C2×#335C2211yescharacteristic
40C2×#333C2212yes—
41C2×#332C2212yes—
42C2×Q8⋊C8C2211yescharacteristic
43C2×C82C2211yescharacteristic
44#1279C2211yescharacteristic
45#1279C2212yes—
46#1277C2211yescharacteristic
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8C2312yes—
49C2×C4⋊Q8C2312yes—
50C2×C4×Q8C2311yescharacteristic
51#500—22no—
52#500—22no—
53#500—22no—
54#500D414yes—
55#487—24no—
56#487—24no—
57#487—24no—
58#487—42no—
59#477—24no—
60#477—42no—
61C8×Q8—22no—
62C8×Q8—22no—
63C2×#107—21no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107—21no—
67C2×#107—22no—
68C2×#107D411yescharacteristic
69C2×#107D411yescharacteristic
70C2×C4⋊C8—21no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2311yescharacteristic
74C2×C4⋊C8C2311yescharacteristic
75C2×#103—22no—
76C2×#103—22no—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—42no—
81C2×#103D411yescharacteristic
82C2×#103D411yescharacteristic
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88#398—42no—
89#391—22no—
90#391—42no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4C2311yescharacteristic
98C4×C4⋊C4C2312yes—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340D411yescharacteristic
104#340D412yes—
105#340D411yescharacteristic
106#336—21no—
107#336—22no—
108#335—22no—
109#333—24no—
110#332—42no—
111Q8⋊C8—22no—
112C82—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8C2×D411yescharacteristic
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D411yescharacteristic
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—44no—
132C4⋊Q8—82no—
133C4⋊Q8C2×D414yes—
134C4×Q8—21no—
135C4×Q8—21no—
136C4×Q8—21no—
137C4×Q8—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4C2×D411yescharacteristic
147C2×C4⋊C4C2×D412yes—
148C2×C4⋊C4C2×D411yescharacteristic
149C2×C4⋊C4C2×D412yes—
150C2×C4⋊C4C2×D411yescharacteristic
151C2×C42—21no—
152C2×C42—22no—
153C2×C42—21no—
154C2×C42—42no—
155C2×C42C2411yescharacteristic
156#107—24no—
157#107—24no—
158#107—22no—
159#107—44no—
160#107—42no—
161#107—42no—
162#107Q1614yes—
163C4⋊C8—21no—
164C4⋊C8—22no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—21no—
168C4⋊C8—22no—
169C4⋊C8—42no—
170C4⋊C8—82no—
171#103—22no—
172#103—22no—
173#103—44no—
174#103—44no—
175#103—44no—
176#103—48no—
177#103—44no—
178#103—82no—
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—
193C2×Q8—21no—
194C2×Q8—21no—
195C2×Q8—21no—
196C2×Q8—24no—
197C2×Q8—21no—
198C2×Q8—21no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—82no—
207C2×Q8—82no—
208C2×Q8#13512yes—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4C22×D411yescharacteristic
218C22×C4C22×D411yescharacteristic
219C22×C4H(5,𝔽2)11yescharacteristic
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—22no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—42no—
233C2×C8—41no—
234C2×C8—42no—
235C2×C8—41no—
236C2×C8—42no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8#12112yes—
241C2×C8#12112yes—
242C2×C8C2×Q1612yes—
243C2×C8#13712yes—
244C4⋊C4—21no—
245C4⋊C4—22no—
246C4⋊C4—24no—
247C4⋊C4—21no—
248C4⋊C4—24no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—22no—
252C4⋊C4—21no—
253C4⋊C4—22no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—84no—
263C4⋊C4—84no—
264C4⋊C4—162no—
265C4⋊C4C2×Q1614yes—
266C4⋊C4#13512yes—
267C42—41no—
268C42—41no—
269C42—42no—
270C42—41no—
271C42—42no—
272C42—41no—
273C42—162no—
274C42C22×D411yescharacteristic
275C42C2×#4111yescharacteristic
276C42C2×#4111yescharacteristic
277C42H(5,𝔽2)11yescharacteristic
278C23D4211yescharacteristic
279Q8—22no—
280Q8—22no—
281Q8—41no—
282Q8—41no—
283Q8—84no—
284Q8—82no—
285Q8—82no—
286Q8—82no—
287Q8—82no—
288Q8—162no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—44no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4C2×#12111yescharacteristic
316C2×C4#53211yescharacteristic
317C2×C4#54011yescharacteristic
318C2×C4D4211yescharacteristic
319C2×C4#54611yescharacteristic
320C2×C4#55011yescharacteristic
321C2×C4C22×Q1611yescharacteristic
322C2×C4C2×#13511yescharacteristic
323C2×C4C2×#13511yescharacteristic
324C2×C4C2×#13711yescharacteristic
325C2×C4#57611yescharacteristic
326C2×C4#57811yescharacteristic
327C2×C4#57811yescharacteristic
328C2×C4#57811yescharacteristic
329C8—22no—
330C8—42no—
331C8—41no—
332C8—41no—
333C8—81no—
334C8—82no—
335C8—81no—
336C8#46214yes—
337C22#242811yescharacteristic
338C22#327611yescharacteristic
339C22#328011yescharacteristic
340C22#328511yescharacteristic
341C22D4×Q1611yescharacteristic
342C22#329011yescharacteristic
343C22#329011yescharacteristic
344C4—21no—
345C4—21no—
346C4—44no—
347C4—42no—
348C4—42no—
349C4—81no—
350C4—84no—
351C4—81no—
352C4—82no—
353C4—162no—
354C4C2×#46211yescharacteristic
355C4#304411yescharacteristic
356C4#316811yescharacteristic
357C4#317511yescharacteristic
358C4#328511yescharacteristic
359C4#328711yescharacteristic
360C4#339511yescharacteristic
361C4#339811yescharacteristic
362C2#1990511yescharacteristic
363C2#1993811yescharacteristic
364C2#1999311yescharacteristic
365C2#2000811yescharacteristic
366C2#2537011yescharacteristic
367C2#2541211yescharacteristic
368C2#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

Maximal subgroups

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

subgroupindex
#286752
#252782
#231582
C2×#17102
C2×#16762
#105412
#100592
#97312
#95942
#90602
C8×#1032