Finite Group Databasebeta

← #318873

Subgroups — #318873

Order 512.

520 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29290C211yesmaximal, characteristic
2#29287C211yesmaximal, characteristic
3#22311C211yesmaximal, characteristic
4#22248C211yesmaximal, characteristic
5#10597C212yesmaximal
6#10589C212yesmaximal
7#9739C212yesmaximal
8#9622C212yesmaximal
9#8809C211yesmaximal, characteristic
10#8590C211yesmaximal, characteristic
11#8590C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C4×C4⋊Q8C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16#2088—22no—
17#2087—22no—
18#1990C2211yescharacteristic
19#1990C2211yescharacteristic
20#1988—21no—
21#1987—21no—
22#1987C2212yes—
23#1987C2212yes—
24#1985—22no—
25#1948—21no—
26#1948C2212yes—
27#1947—21no—
28#1947C2212yes—
29#1939C2212yes—
30#1921—22no—
31#1921—22no—
32#1917—22no—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1834C2211yescharacteristic
36#1831C2211yescharacteristic
37#1770C2212yes—
38#1744C2211yescharacteristic
39C2×#334C2212yes—
40C2×#334C2212yes—
41C2×#333C2212yes—
42C2×#333C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C2×C4⋊Q8D411yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51#500—22no—
52#500Q814yescomplemented
53C2×#107—22no—
54C2×#107—22no—
55C2×#106—22no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×#97—22no—
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2311yescharacteristic
67#391—21no—
68#391—21no—
69#391—21no—
70#389—21no—
71#389—21no—
72#389—21no—
73#389—21no—
74#389—21no—
75#389D412yes—
76#384D411yescharacteristic
77#384C2311yescharacteristic
78#384C2311yescharacteristic
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4C2312yes—
85C43C2311yescharacteristic
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#336—22no—
96#336—22no—
97#336—21no—
98#336—21no—
99#336D412yes—
100#336D412yes—
101#334—24no—
102#334—42no—
103#333—42no—
104#333—42no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8C2×D412yes—
112C22×C8C2×D412yes—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8C2×Q812yes—
122C4⋊Q8C2×Q812yes—
123C4×Q8—82no—
124C4×Q8—82no—
125C4×Q8—82no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4C2×D412yes—
143C2×C4⋊C4C2×D412yes—
144C2×C42—22no—
145C2×C42—41no—
146C2×C42—41no—
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42#4111yescharacteristic
152C2×C42#4111yescharacteristic
153C2×C42C2411yescharacteristic
154#107—28no—
155#107—82no—
156#106—82no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—42no—
164C4⋊C8—42no—
165C4⋊C8—81no—
166C4⋊C8—81no—
167C4⋊C8—82no—
168#97—82no—
169C4×C8—21no—
170C4×C8—21no—
171C4×C8C2×Q812yes—
172C4×C8#4112yes—
173#95—22no—
174#95—22no—
175#95—41no—
176#95—41no—
177#95—41no—
178#95—41no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—42no—
184C2×Q8—81no—
185C2×Q8—81no—
186C2×Q8—81no—
187C2×Q8—81no—
188C2×Q8—81no—
189C2×Q8—81no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4C22×D411yescharacteristic
201C22×C4C22×D411yescharacteristic
202C22×C4C2×#4111yescharacteristic
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8C22⋊Q814yes—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—81no—
238C4⋊C4—81no—
239C4⋊C4—82no—
240C4⋊C4—81no—
241C4⋊C4—81no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C42—21no—
245C42—21no—
246C42—22no—
247C42—22no—
248C42—41no—
249C42—41no—
250C42—42no—
251C42—42no—
252C42—82no—
253C42—82no—
254C42#13712yes—
255C42#13712yes—
256C42C22×Q811yescharacteristic
257C42C2×#4111yescharacteristic
258C42H(5,𝔽2)11yescharacteristic
259C422−1+411yescharacteristic
260C23C2×#12111yescharacteristic
261Q8—82no—
262Q8—82no—
263Q8—82no—
264Q8—82no—
265Q8—84no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4#47012yes—
297C2×C4#47012yes—
298C2×C4C2×C22⋊Q811yescharacteristic
299C2×C4#53711yescharacteristic
300C2×C4D4×Q811yescharacteristic
301C2×C4#55011yescharacteristic
302C2×C4#55411yescharacteristic
303C2×C4#55611yescharacteristic
304C2×C4C2×#13711yescharacteristic
305C2×C4C2×#13711yescharacteristic
306C2×C4#57712yes—
307C2×C4#57711yescharacteristic
308C2×C4#57711yescharacteristic
309C2×C4#57812yes—
310C8—24no—
311C8—42no—
312C8—82no—
313C8—82no—
314C8—82no—
315C22#253811yescharacteristic
316C22#304211yescharacteristic
317C22#304411yescharacteristic
318C22C2×#47011yescharacteristic
319C22#306011yescharacteristic
320C22#306811yescharacteristic
321C22#306811yescharacteristic
322C4—21no—
323C4—21no—
324C4—22no—
325C4—22no—
326C4—41no—
327C4—41no—
328C4—82no—
329C4—82no—
330C4—82no—
331C4—82no—
332C4—161no—
333C4—161no—
334C4#308212yes—
335C4#325812yes—
336C4#338711yescharacteristic
337C4#338711yescharacteristic
338C4#340011yescharacteristic
339C4#340011yescharacteristic
340C2#1942911yescharacteristic
341C2#1944111yescharacteristic
342C2#1951611yescharacteristic
343C2#1953211yescharacteristic
344C2#2827411yescharacteristic
345C2#2829111yescharacteristic
346C2#2829911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
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
52. #500Q8semidirect

Maximal subgroups

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

subgroupindex
#292902
#292872
#223112
#222482
#105972
#105892
#97392
#96222
#88092
#85902