Finite Group Databasebeta

← #318213

Subgroups — #318213

Order 512.

778 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28252C212yesmaximal
2C4×#500C211yesmaximal, characteristic, complemented
3#21681C211yesmaximal, characteristic
4C2×#1693C212yesmaximal, complemented
5#10477C214yesmaximal, complemented
6#8805C211yesmaximal, characteristic, complemented
7#8652C212yesmaximal, complemented
8#8586C212yesmaximal, complemented
9C2×#500C2211yescharacteristic
10C2×#460—22no—
11C2×#461C2214yes—
12C2×#436—21no—
13C2×#436—21no—
14C2×#436C2212yes—
15C4×C4⋊Q8C2212yes—
16C4×#127C2211yescharacteristic
17#2055—24no—
18#1990C2212yes—
19#1954—21no—
20#1954—21no—
21#1954C2212yes—
22#1942—21no—
23#1942—21no—
24#1845C2211yescharacteristic
25#1842C2212yes—
26#1842C2211yescharacteristic
27#1803—22no—
28#1802C2214yes—
29C4×#107—21no—
30C4×#107—21no—
31C42×C8C2211yescharacteristic
32#1693—28no—
33C2×#334C2212yes—
34C2×#334C2212yes—
35C2×#332C2214yes—
36#1296C2214yes—
37C22×SD16—24no—
38C22×M16—22no—
39C2×C4⋊Q8C2312yes—
40C2×#127D411yescharacteristic
41C2×C4×Q8—22no—
42C2×C4×D4—21no—
43C2×C4×D4—21no—
44C2×C4×D4C2312yes—
45#500—22no—
46#500D414yescomplemented
47#461—28no—
48#461—44no—
49#460—48no—
50#436—24no—
51#436—24no—
52#436—44no—
53#436—44no—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107—21no—
57C2×#107—21no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2314yes—
64C2×#103—24no—
65C2×#102—24no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C4×C8D411yescharacteristic
70C2×C4×C8D411yescharacteristic
71C2×C4×C8D411yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#391—22no—
76#390—21no—
77#390—21no—
78#390D412yes—
79#389—22no—
80#389—22no—
81#384D411yescharacteristic
82#384C2312yes—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C43C2311yescharacteristic
86#340—22no—
87#340—22no—
88#340—24no—
89#334—24no—
90#334—42no—
91#332—28no—
92C22×Q8—22no—
93C22×D4—21no—
94C22×D4—21no—
95C22×D4C2×D412yes—
96C23×C4—22no—
97C2×SD16—48no—
98C2×SD16—48no—
99C2×SD16—84no—
100C2×M16—44no—
101C2×M16—44no—
102C2×M16—44no—
103C22×C8—22no—
104C22×C8—24no—
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8C2×D414yes—
113#127—24no—
114C4×Q8—44no—
115C4×Q8—44no—
116C4×D4—22no—
117C4×D4—22no—
118C4×D4—22no—
119C4×D4—22no—
120C4×D4—42no—
121C4×D4—42no—
122C4×D4—42no—
123C4×D4—42no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4C2×D412yes—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C42—42no—
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42#4111yescharacteristic
142C2×C42#4111yescharacteristic
143C2×C42C2411yescharacteristic
144#107—24no—
145#107—24no—
146#107—42no—
147#107—42no—
148C4⋊C8—22no—
149C4⋊C8—24no—
150C4⋊C8—22no—
151C4⋊C8—24no—
152C4⋊C8—42no—
153C4⋊C8—42no—
154C4⋊C8—44no—
155C4⋊C8—82no—
156#103—48no—
157#102—48no—
158C22⋊C8—44no—
159C22⋊C8—44no—
160C22⋊C8—44no—
161C4×C8—22no—
162C4×C8—22no—
163C4×C8—21no—
164C4×C8—21no—
165C4×C8—41no—
166C4×C8SD1614yescomplemented
167C4×C8C2×D412yes—
168C4×C8#4112yes—
169#95—42no—
170#95—42no—
171C24—22no—
172C2×Q8—24no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×D4—22no—
179C2×D4—22no—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C22×C4—22no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4#12111yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4C22×D411yescharacteristic
202C22×C4C22×D411yescharacteristic
203C22×C4C2×#4111yescharacteristic
204SD16—816no—
205M16—88no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—42no—
214C2×C8—44no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—44no—
218C2×C8—44no—
219C2×C8#12112yes—
220C2×C8#12112yes—
221C2×C8C2×SD1612yes—
222C2×C8C2×SD1612yes—
223C2×C8AGL(1,ℤ/8)12yes—
224C4⋊C4—24no—
225C4⋊C4—22no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—82no—
237C4⋊C4—84no—
238C22⋊C4—44no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—44no—
242C42—21no—
243C42—21no—
244C42—22no—
245C42—22no—
246C42—41no—
247C42—84no—
248C42C2×SD1612yes—
249C42AGL(1,ℤ/8)14yes—
250C42#13712yes—
251C42C22×D411yescharacteristic
252C42C2×#4111yescharacteristic
253C42H(5,𝔽2)11yescharacteristic
254C422−1+411yescharacteristic
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23C2×#12111yescharacteristic
263Q8—44no—
264Q8—44no—
265Q8—82no—
266Q8—82no—
267D4—44no—
268D4—44no—
269D4—84no—
270D4—84no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4#46112yes—
294C2×C4#46112yes—
295C2×C4#46912yes—
296C2×C4#46912yes—
297C2×C4#47312yes—
298C2×C4C2×#12111yescharacteristic
299C2×C4#53711yescharacteristic
300C2×C4D4211yescharacteristic
301C2×C4#54711yescharacteristic
302C2×C4#54711yescharacteristic
303C2×C4#55511yescharacteristic
304C2×C4C22×SD1611yescharacteristic
305C2×C4C2×AGL(1,ℤ/8)12yes—
306C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
307C2×C4C2×#13711yescharacteristic
308C2×C4#57511yescharacteristic
309C2×C4#57712yes—
310C8—22no—
311C8—22no—
312C8—41no—
313C8—84no—
314C8—82no—
315C8#46014yescomplemented
316C22—82no—
317C22—82no—
318C22—82no—
319C22—82no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22#253711yescharacteristic
324C22C2×#46111yescharacteristic
325C22#303611yescharacteristic
326C22C2×#46911yescharacteristic
327C22#306011yescharacteristic
328C22C2×#47311yescharacteristic
329C22#306711yescharacteristic
330C4—22no—
331C4—21no—
332C4—21no—
333C4—22no—
334C4—41no—
335C4—84no—
336C4—82no—
337C4—162no—
338C4#169314yescomplemented
339C4C2×#46011yescharacteristic
340C4#303711yescharacteristic
341C4#324611yescharacteristic
342C4#325411yescharacteristic
343C4#327212yes—
344C4#335412yes—
345C2—162no—
346C2C2×#169311yescharacteristic
347C2#1919411yescharacteristic
348C2#1928011yescharacteristic
349C2#1933811yescharacteristic
350C2#2823111yescharacteristic
351C2#2824111yescharacteristic
352C2#2825211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#500C2semidirect
4. C2×#1693C2semidirect
5. #10477C2semidirect
6. #8805C2semidirect
7. #8652C2semidirect
8. #8586C2semidirect
46. #500D4semidirect
166. C4×C8SD16semidirect
315. C8#460semidirect
338. C4#1693semidirect

Maximal subgroups

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

subgroupindex
#282522
C4×#5002
#216812
C2×#16932
#104772
#88052
#86522
#85862