Finite Group Databasebeta

← #288399

Subgroups — #288399

Order 512.

622 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28683C211yesmaximal, characteristic
2#25278C212yesmaximal
3#23162C211yesmaximal, characteristic
4C2×#1710C211yesmaximal, characteristic
5C2×#1696C211yesmaximal, characteristic
6#10541C212yesmaximal
7#10059C212yesmaximal
8#9740C211yesmaximal, characteristic
9#9593C211yesmaximal, characteristic
10#9060C212yesmaximal
11#8207C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#500C2211yescharacteristic
14C2×#487C2212yes—
15C2×#487C2212yes—
16C2×#477C2212yes—
17C2×#446C2211yescharacteristic
18#2546C2212yes—
19#2369C2212yes—
20#2087—22no—
21#2068—22no—
22#2043—22no—
23#1988C2212yes—
24#1987C2212yes—
25#1981—22no—
26#1981C2212yes—
27#1944—21no—
28#1944—21no—
29#1943—21no—
30#1939C2212yes—
31#1928—22no—
32#1918—22no—
33#1776—21no—
34C4×#106C2211yescharacteristic
35#1770C2211yescharacteristic
36C4×#103C2211yescharacteristic
37#1710—24no—
38#1696—24no—
39C2×#334C2211yescharacteristic
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—
61#446—22no—
62#446—22no—
63C2×#107—21no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107—21no—
67C2×#107—22no—
68C2×#106D411yescharacteristic
69C2×#106D411yescharacteristic
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×#103D412yes—
82C2×#97—21no—
83C2×#97—21no—
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#398—42no—
88#391—22no—
89#391—42no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4C2311yescharacteristic
97C4×C4⋊C4C2312yes—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340D412yes—
103#340D412yes—
104#336—21no—
105#336—22no—
106#334—22no—
107#333—24no—
108#332—42no—
109Q8⋊C8—22no—
110C82—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8C2×D411yescharacteristic
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—22no—
117C22×C8—21no—
118C22×C8C2×D412yes—
119C22×C8C2×D411yescharacteristic
120C22×C8C2×D411yescharacteristic
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—44no—
129C4⋊Q8—82no—
130C4⋊Q8C2×D414yes—
131C4×Q8—21no—
132C4×Q8—21no—
133C4×Q8—21no—
134C4×Q8—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4C2×D411yescharacteristic
144C2×C4⋊C4C2×D412yes—
145C2×C4⋊C4C2×D412yes—
146C2×C4⋊C4C2×D412yes—
147C2×C42—21no—
148C2×C42—22no—
149C2×C42—21no—
150C2×C42—42no—
151C2×C42C2411yescharacteristic
152#107—24no—
153#107—24no—
154#107—44no—
155#107—42no—
156#107—42no—
157#106—22no—
158#106Q1614yes—
159C4⋊C8—21no—
160C4⋊C8—22no—
161C4⋊C8—21no—
162C4⋊C8—21no—
163C4⋊C8—21no—
164C4⋊C8—22no—
165C4⋊C8—42no—
166C4⋊C8—82no—
167#103—42no—
168#103—44no—
169#103—44no—
170#103—44no—
171#103—44no—
172#103—48no—
173#103—82no—
174#97—42no—
175#97—42no—
176C4×C8—21no—
177C4×C8—21no—
178C4×C8—21no—
179C4×C8C2×D412yes—
180C4×C8#4112yes—
181C4×C8#4112yes—
182#95—21no—
183#95—22no—
184#95—42no—
185#95—42no—
186#95—42no—
187#95—42no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—24no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—82no—
201C2×Q8—82no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4C22×D411yescharacteristic
210C22×C4C22×D411yescharacteristic
211C22×C4H(5,𝔽2)11yescharacteristic
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—42no—
219C2×C8—41no—
220C2×C8—42no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8#12112yes—
229C2×C8#12112yes—
230C2×C8C2×Q1612yes—
231C2×C8#13712yes—
232C4⋊C4—22no—
233C4⋊C4—24no—
234C4⋊C4—21no—
235C4⋊C4—24no—
236C4⋊C4—21no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—84no—
250C4⋊C4—84no—
251C4⋊C4—162no—
252C4⋊C4C2×Q1614yes—
253C42—41no—
254C42—41no—
255C42—41no—
256C42—42no—
257C42—42no—
258C42—41no—
259C42—162no—
260C42C22×D411yescharacteristic
261C42C2×#4111yescharacteristic
262C42C2×#4111yescharacteristic
263C42H(5,𝔽2)11yescharacteristic
264C23D4211yescharacteristic
265Q8—41no—
266Q8—41no—
267Q8—42no—
268Q8—84no—
269Q8—82no—
270Q8—82no—
271Q8—82no—
272Q8—82no—
273Q8—162no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4C2×#12111yescharacteristic
295C2×C4#53211yescharacteristic
296C2×C4#54011yescharacteristic
297C2×C4D4211yescharacteristic
298C2×C4#54611yescharacteristic
299C2×C4#55011yescharacteristic
300C2×C4C22×Q1611yescharacteristic
301C2×C4C2×#13711yescharacteristic
302C2×C4#57512yes—
303C2×C4#57712yes—
304C2×C4#57811yescharacteristic
305C2×C4#57811yescharacteristic
306C8—22no—
307C8—42no—
308C8—42no—
309C8—81no—
310C8—82no—
311C8—81no—
312C8#46214yes—
313C22#242811yescharacteristic
314C22#327512yes—
315C22#327611yescharacteristic
316C22D4×Q1611yescharacteristic
317C22#329012yes—
318C4—21no—
319C4—21no—
320C4—44no—
321C4—82no—
322C4—82no—
323C4—81no—
324C4—81no—
325C4—84no—
326C4—162no—
327C4C2×#46211yescharacteristic
328C4#304411yescharacteristic
329C4#317311yescharacteristic
330C4#318311yescharacteristic
331C4#327112yes—
332C4#339511yescharacteristic
333C4#339811yescharacteristic
334C2#1992912yes—
335C2#2029512yes—
336C2#2537011yescharacteristic
337C2#2542411yescharacteristic
338C2#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
#286832
#252782
#231622
C2×#17102
C2×#16962
#105412
#100592
#97402
#95932
#90602
#82072