Finite Group Databasebeta

← #203768

Subgroups — #203768

Order 512.

768 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21981C211yesmaximal, characteristic
2#21170C211yesmaximal, characteristic
3C2×#1948C211yesmaximal, characteristic
4#20533C211yesmaximal, characteristic
5C2×#1570C211yesmaximal, characteristic
6#9613C214yesmaximal, complemented
7#9612C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic
9#8039C211yesmaximal, characteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#428—21no—
13C22×#106—21no—
14C2×#423C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16#2285C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1949—22no—
20#1948—22no—
21#1948—22no—
22#1948C2212yes—
23#1948C2214yes—
24#1947C2212yes—
25#1853—22no—
26#1844C2212yes—
27#1834C2212yes—
28C8×C4⋊C4—22no—
29C8×C4⋊C4C2214yes—
30C4×C22⋊C8—22no—
31#1736—21no—
32C2×#340—21no—
33#1570C418yescomplemented
34C2×#334C2214yes—
35#1296C2212yes—
36#1296C2212yes—
37#1296C2214yes—
38C22×M16—21no—
39C23×C8—21no—
40C2×#117—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4D411yescharacteristic
46C22×C42—21no—
47C22×C42C2311yescharacteristic
48#431—22no—
49#431C2×C414yes—
50#431Q814yes—
51#431Q814yes—
52#428—44no—
53C2×#107—22no—
54C2×#106—24no—
55C2×#106—22no—
56C2×#106—24no—
57C2×#106—44no—
58#423—22no—
59#423C2×C414yes—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8C2×C414yes—
64C2×C4⋊C8D412yes—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C22⋊C8—21no—
69C2×C22⋊C8D412yes—
70C2×C4×C8—21no—
71C2×C4×C8—24no—
72C2×C4×C8C2312yes—
73#384—21no—
74#384—21no—
75#384—21no—
76#384C2311yescharacteristic
77#384C2312yes—
78#384C2311yescharacteristic
79#382—21no—
80#380—21no—
81#380—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4D412yes—
87C4×C4⋊C4C2312yes—
88C2×#95—21no—
89#340—24no—
90#340—22no—
91#340—24no—
92#340—22no—
93#336—24no—
94#334C2×C4116yescomplemented
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—41no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4#4111yescharacteristic
101C2×M16—24no—
102C2×M16—42no—
103C2×M16—42no—
104C22×C8—24no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—42no—
109C22×C8—42no—
110C22×C8C2×D414yes—
111#117—42no—
112#117—42no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—82no—
132C2×C4⋊C4—82no—
133C2×C4⋊C4SD1614yes—
134C2×C4⋊C4SD1614yes—
135C2×C4⋊C4C2×D412yes—
136C2×C4⋊C4C2×D412yes—
137C2×C22⋊C4—41no—
138C2×C42—22no—
139C2×C42—22no—
140C2×C42—41no—
141C2×C42—82no—
142C2×C42C22×C412yes—
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×Q812yes—
146C2×C42C2×Q812yes—
147C2×C42#4112yes—
148C2×C42#4111yescharacteristic
149C2×C42C2411yescharacteristic
150#107—44no—
151#106—48no—
152#106—44no—
153#106—84no—
154C4⋊C8—24no—
155C4⋊C8—24no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8C4⋊C418yes—
160C4⋊C8C22×C414yes—
161C4⋊C8C22×C414yes—
162C4⋊C8C2×Q814yes—
163C22⋊C8—24no—
164C22⋊C8C4⋊C418yes—
165C4×C8—24no—
166C4×C8—84no—
167C4×C8C22×C414yes—
168C4×C8C2×Q814yes—
169#95—22no—
170#95—22no—
171#95—41no—
172#95—41no—
173C24#12111yescharacteristic
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4—82no—
190C22×C4—82no—
191C22×C4—82no—
192C22×C4C2×C4⋊C412yes—
193C22×C4C4×D412yes—
194C22×C4C4×Q812yes—
195C22×C4#12111yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4C22⋊Q812yes—
199C22×C4#12512yes—
200C22×C4C4⋊Q812yes—
201C22×C4C2×SD1612yes—
202C22×C4C2×SD1612yes—
203C22×C4#13512yes—
204C22×C4#13512yes—
205C22×C4C22×D411yescharacteristic
206C22×C4C22×D411yescharacteristic
207C22×C4C2×#4111yescharacteristic
208M16—48no—
209C2×C8—24no—
210C2×C8—24no—
211C2×C8—24no—
212C2×C8—24no—
213C2×C8—24no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—48no—
221C2×C8C2×C4⋊C418yes—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—24no—
225C4⋊C4—24no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—42no—
235C4⋊C4—81no—
236C4⋊C4—81no—
237C4⋊C4—82no—
238C4⋊C4—82no—
239C4⋊C4—82no—
240C4⋊C4—82no—
241C4⋊C4C2×SD1614yes—
242C4⋊C4C2×SD1614yes—
243C22⋊C4—82no—
244C42—44no—
245C42—82no—
246C42—82no—
247C42C2×C4⋊C412yes—
248C42C4×D412yes—
249C42C4×Q812yes—
250C42C22⋊Q812yes—
251C42#12512yes—
252C42C4⋊Q812yes—
253C42C23×C411yescharacteristic
254C42C22×Q811yescharacteristic
255C42C22×Q811yescharacteristic
256C42C2×#4111yescharacteristic
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23#38912yes—
262C23#46512yes—
263C23#46512yes—
264C23C2×#12111yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—84no—
282C2×C4—84no—
283C2×C4—81no—
284C2×C4—82no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—164no—
293C2×C4#38912yes—
294C2×C4#38912yes—
295C2×C4#38912yes—
296C2×C4C4×SD1614yes—
297C2×C4C4×SD1614yes—
298C2×C4#46512yes—
299C2×C4#46512yes—
300C2×C4#49814yes—
301C2×C4#49914yes—
302C2×C4C22×C4⋊C411yescharacteristic
303C2×C4C2×C4×D411yescharacteristic
304C2×C4C2×C4×Q811yescharacteristic
305C2×C4C2×C22⋊Q811yescharacteristic
306C2×C4C2×#12511yescharacteristic
307C2×C4C2×C4⋊Q811yescharacteristic
308C2×C4C22×SD1611yescharacteristic
309C2×C4C22×SD1611yescharacteristic
310C2×C4C2×#13511yescharacteristic
311C2×C4C2×#13511yescharacteristic
312C2×C4#57712yes—
313C2×C4#57712yes—
314C8—28no—
315C8—44no—
316C8—44no—
317C8—44no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22#194114yes—
325C22C2×#38911yescharacteristic
326C22#304211yescharacteristic
327C22#304311yescharacteristic
328C22C2×#46511yescharacteristic
329C22C2×#46511yescharacteristic
330C22#306811yescharacteristic
331C22#306911yescharacteristic
332C4—22no—
333C4—21no—
334C4—21no—
335C4—22no—
336C4—44no—
337C4—44no—
338C4—161no—
339C4—162no—
340C4—161no—
341C4#194114yes—
342C4#291212yes—
343C4C2×C4×SD1611yescharacteristic
344C4C2×C4×SD1611yescharacteristic
345C4#308412yes—
346C4C2×#49811yescharacteristic
347C4C2×#49911yescharacteristic
348C2—24no—
349C2#1939011yescharacteristic
350C2#1939111yescharacteristic
351C2#1939611yescharacteristic
352C2#1939711yescharacteristic
353C2C2×#194111yescharacteristic
354C2#2117411yescharacteristic
355C2#2118211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9613C2semidirect
7. #9612C2semidirect
33. #1570C4semidirect
94. #334C2×C4semidirect

Maximal subgroups

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

subgroupindex
#219812
#211702
C2×#19482
#205332
C2×#15702
#96132
#96122
#80432
#80392