Finite Group Databasebeta

← #281014

Subgroups — #281014

Order 512.

574 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28265C211yesmaximal, characteristic
2#24994C212yesmaximal
3C4⋊C4×Q16C211yesmaximal, characteristic
4#10626C212yesmaximal
5#10437C212yesmaximal
6#9656C211yesmaximal, characteristic
7#9615C211yesmaximal, characteristic
8#8888C212yesmaximal
9#8814C211yesmaximal, characteristic
10#8645C211yesmaximal, characteristic
11C8×#103C211yesmaximal, characteristic
12C2×C8×Q8C2211yescharacteristic
13C2×C4×Q16—21no—
14C2×C4×Q16C2211yescharacteristic
15#2582C2212yes—
16Q8×C4⋊C4C2212yes—
17#2099—22no—
18#2072—22no—
19#2068—22no—
20#1984C2212yes—
21#1969—21no—
22#1948C2212yes—
23#1942—21no—
24#1942—21no—
25#1927—22no—
26#1920—22no—
27#1863—22no—
28#1845C2211yescharacteristic
29#1845C2211yescharacteristic
30#1844C2212yes—
31#1834C2212yes—
32#1803C2212yes—
33#1803C2212yes—
34#1803C2212yes—
35C8×C4⋊C4C2211yescharacteristic
36C4×#103C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×Q8⋊C8C2211yescharacteristic
39C2×Q8⋊C8C2212yes—
40C2×C82C2211yescharacteristic
41#1296C2212yes—
42#1296C2212yes—
43#1279C2211yescharacteristic
44#1277C2211yescharacteristic
45C22×Q16D411yescharacteristic
46C2×C4×Q8—22no—
47C2×C4×Q8C2311yescharacteristic
48C2×C4×Q8C2312yes—
49C8×Q8—22no—
50C8×Q8—22no—
51C4×Q16—22no—
52C4×Q16—22no—
53C4×Q16—44no—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107—21no—
57C2×#107—21no—
58C2×#107D411yescharacteristic
59C2×#106—22no—
60C2×#106—42no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2311yescharacteristic
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2312yes—
65C2×#103—22no—
66C2×#103—22no—
67C2×#103—22no—
68C2×#103D412yes—
69C2×#103D411yescharacteristic
70C2×#103D411yescharacteristic
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#398—42no—
77#391—22no—
78#391—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—42no—
83#384—22no—
84#384C2312yes—
85#384C2312yes—
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2311yescharacteristic
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340D411yescharacteristic
95#340D411yescharacteristic
96#336—22no—
97#336—22no—
98#336—21no—
99#335—22no—
100Q8⋊C8—22no—
101Q8⋊C8—24no—
102C82—22no—
103C22×Q8C2×D412yes—
104C22×Q8C2×D411yescharacteristic
105C2×Q16—22no—
106C2×Q16—22no—
107C2×Q16SD1614yes—
108C22×C8—21no—
109C22×C8—22no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C4×Q8—22no—
117C4×Q8—21no—
118C4×Q8—22no—
119C4×Q8—22no—
120C4×Q8—21no—
121C4×Q8—21no—
122C4×Q8—21no—
123C4×Q8—22no—
124C4×Q8—44no—
125C4×Q8—82no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4C2×D411yescharacteristic
138C2×C4⋊C4C2×D412yes—
139C2×C4⋊C4C2×D411yescharacteristic
140C2×C4⋊C4C2×D411yescharacteristic
141C2×C42—21no—
142C2×C42—22no—
143C2×C42—21no—
144C2×C42—42no—
145C2×C42C2411yescharacteristic
146#107—22no—
147#107—42no—
148#107—42no—
149#107—81no—
150#107—81no—
151#106—44no—
152#106—84no—
153C4⋊C8—22no—
154C4⋊C8—21no—
155C4⋊C8—22no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—42no—
160#103—22no—
161#103—24no—
162#103—22no—
163#103—44no—
164#103—44no—
165#103—82no—
166C4×C8—21no—
167C4×C8—21no—
168C4×C8—21no—
169C4×C8—42no—
170C4×C8—42no—
171C4×C8#4112yes—
172C4×C8#4112yes—
173C4×C8#4112yes—
174#95—21no—
175#95—42no—
176#95—42no—
177#95—42no—
178#95—42no—
179C2×Q8—22no—
180C2×Q8—21no—
181C2×Q8—22no—
182C2×Q8—21no—
183C2×Q8—22no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—21no—
189C2×Q8C2×SD1614yes—
190C2×Q8#13512yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D411yescharacteristic
201C22×C4H(5,𝔽2)11yescharacteristic
202Q16—28no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—41no—
215C2×C8—42no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8#12312yes—
224C2×C8#12312yes—
225C2×C8C2×SD1612yes—
226C2×C8#13712yes—
227C4⋊C4—21no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—22no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—42no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—84no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—162no—
250C4⋊C4—162no—
251C4⋊C4#13512yes—
252C42—41no—
253C42—42no—
254C42—41no—
255C42—42no—
256C42—41no—
257C42—41no—
258C42—84no—
259C42C2×#4111yescharacteristic
260C42C2×#4111yescharacteristic
261C42C2×#4111yescharacteristic
262C422−1+411yescharacteristic
263C23D4211yescharacteristic
264Q8—24no—
265Q8—22no—
266Q8—24no—
267Q8—22no—
268Q8—41no—
269Q8—42no—
270Q8—42no—
271Q8—41no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4C2×#12311yescharacteristic
299C2×C4#53211yescharacteristic
300C2×C4#54211yescharacteristic
301C2×C4#54711yescharacteristic
302C2×C4#54711yescharacteristic
303C2×C4#55311yescharacteristic
304C2×C4C22×SD1611yescharacteristic
305C2×C4C2×#13511yescharacteristic
306C2×C4C2×#13511yescharacteristic
307C2×C4C2×#13711yescharacteristic
308C2×C4#57711yescharacteristic
309C2×C4#57711yescharacteristic
310C2×C4#57811yescharacteristic
311C2×C4#57811yescharacteristic
312C8—24no—
313C8—22no—
314C8—41no—
315C8—41no—
316C8—81no—
317C8—81no—
318C8—82no—
319C8#48314yes—
320C22#242211yescharacteristic
321C22#327811yescharacteristic
322C22D4×SD1611yescharacteristic
323C22#328511yescharacteristic
324C22#328511yescharacteristic
325C22#329211yescharacteristic
326C22#329311yescharacteristic
327C4—21no—
328C4—21no—
329C4—42no—
330C4—44no—
331C4—42no—
332C4—81no—
333C4—81no—
334C4—82no—
335C4—162no—
336C4—162no—
337C4C2×#48311yescharacteristic
338C4#309911yescharacteristic
339C4#316811yescharacteristic
340C4#317711yescharacteristic
341C4#333511yescharacteristic
342C4#333611yescharacteristic
343C4#334211yescharacteristic
344C4#335611yescharacteristic
345C2#1994211yescharacteristic
346C2#1994411yescharacteristic
347C2#1998311yescharacteristic
348C2#1998611yescharacteristic
349C2#2501411yescharacteristic
350C2#2503311yescharacteristic
351C2#2505311yescharacteristic
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
#282652
#249942
C4⋊C4×Q162
#106262
#104372
#96562
#96152
#88882
#88142
#86452
C8×#1032