Finite Group Databasebeta

← #242499

Subgroups — #242499

Order 512.

756 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21974C211yesmaximal, characteristic
2#21833C211yesmaximal, characteristic
3#21362C211yesmaximal, characteristic
4#21313C211yesmaximal, characteristic
5C2×#1574C211yesmaximal, characteristic
6C2×#1548C211yesmaximal, characteristic
7#10601C214yesmaximal, complemented
8#10589C214yesmaximal, complemented
9#8041C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×#423C2211yescharacteristic
15#2285C2211yescharacteristic
16C4×#117C2211yescharacteristic
17#1990C2214yes—
18#1988—21no—
19#1988—21no—
20#1988—21no—
21#1988—21no—
22#1987C2214yes—
23#1948C2212yes—
24#1947C2212yes—
25#1939C2214yes—
26#1854—22no—
27#1851—22no—
28#1842—21no—
29#1831—21no—
30#1823—22no—
31C4×#107—21no—
32C4×#106—21no—
33#1736—22no—
34#1574—24no—
35C2×#335C2212yes—
36C2×#334C2212yes—
37#1548—24no—
38C2×#333C2212yes—
39C2×#333C2212yes—
40#1296C2214yes—
41C23×Q8—21no—
42C22×M16—21no—
43C22×M16—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8Q814yes—
51C2×C4⋊Q8C2312yes—
52C2×#117—21no—
53C2×#117D412yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C42C2311yescharacteristic
57#431—22no—
58#431—22no—
59#431—22no—
60#431D414yes—
61C2×#107—22no—
62C2×#106—22no—
63#423—22no—
64#423—22no—
65#423—22no—
66#423—22no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8D411yescharacteristic
71C2×C4⋊C8D411yescharacteristic
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×C22⋊C8—21no—
77C2×C22⋊C8D412yes—
78C2×C4×C8—21no—
79C2×C4×C8C2312yes—
80#384C2312yes—
81#382—21no—
82#380—21no—
83#380—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4D411yescharacteristic
87C4×C4⋊C4D411yescharacteristic
88C4×C4⋊C4C2312yes—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—21no—
91#376—21no—
92C43—21no—
93#340—24no—
94#340—24no—
95#340—22no—
96#340—22no—
97#340—24no—
98#340—24no—
99#336—24no—
100#335—24no—
101#334—24no—
102#333—24no—
103#333—24no—
104C22×Q8—24no—
105C22×Q8—22no—
106C22×Q8—41no—
107C22×Q8—42no—
108C22×Q8—82no—
109C22×Q8—82no—
110C23×C4—41no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4#4111yescharacteristic
114C2×M16—44no—
115C2×M16—42no—
116C2×M16—42no—
117C2×M16—82no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8C2×D412yes—
123C22×C8C2×D412yes—
124C4⋊Q8—24no—
125C4⋊Q8—24no—
126C4⋊Q8—24no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—42no—
131C4⋊Q8—42no—
132C4⋊Q8—42no—
133C4⋊Q8C2×Q814yes—
134#117—24no—
135#117—42no—
136#117—82no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4C2×D412yes—
149C2×C4⋊C4C2×D412yes—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—41no—
152C2×C42—22no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—22no—
156C2×C42—41no—
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D412yes—
160C2×C42C2×Q812yes—
161C2×C42#4112yes—
162C2×C42#4112yes—
163C2×C42#4111yescharacteristic
164C2×C42C2411yescharacteristic
165#107—44no—
166#106—44no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—42no—
176C4⋊C8—42no—
177C4⋊C8—84no—
178C4⋊C8C2×D412yes—
179C4⋊C8C2×D412yes—
180C22⋊C8—24no—
181C22⋊C8—24no—
182C4×C8—24no—
183C4×C8—22no—
184C4×C8—22no—
185#95—21no—
186#95—21no—
187#95—22no—
188#95—41no—
189#95—41no—
190C24#12111yescharacteristic
191C2×Q8—24no—
192C2×Q8—42no—
193C2×Q8—44no—
194C2×Q8—88no—
195C2×Q8—88no—
196C2×Q8—82no—
197C2×Q8—82no—
198C2×Q8—82no—
199C2×Q8—82no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—42no—
209C22×C4—44no—
210C22×C4—42no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4#12112yes—
214C22×C4#12112yes—
215C22×C4#12111yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4C22⋊Q812yes—
219C22×C4C22⋊Q812yes—
220C22×C4C22⋊Q812yes—
221C22×C4#12612yes—
222C22×C4C22×D411yescharacteristic
223C22×C4C22×D411yescharacteristic
224C22×C4C2×#4111yescharacteristic
225M16—84no—
226M16—84no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—24no—
232C2×C8—24no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—44no—
238C2×C8—44no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—82no—
253C4⋊C4#13714yes—
254C22⋊C4—44no—
255C22⋊C4—82no—
256C42—44no—
257C42—42no—
258C42—42no—
259C42—42no—
260C42—42no—
261C42—82no—
262C42#12112yes—
263C42#12112yes—
264C42C22⋊Q812yes—
265C42C22⋊Q812yes—
266C42C22⋊Q812yes—
267C42#12612yes—
268C42C22×D411yescharacteristic
269C42C22×Q811yescharacteristic
270C42C2×#4111yescharacteristic
271C42C2×#4111yescharacteristic
272C23—22no—
273C23—22no—
274C23—21no—
275C23—21no—
276C23#39612yes—
277C23C2×#12111yescharacteristic
278Q8—84no—
279Q8—84no—
280Q8—88no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—88no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4#39612yes—
307C2×C4#39612yes—
308C2×C4#39612yes—
309C2×C4#47112yes—
310C2×C4#47212yes—
311C2×C4C2×#12111yescharacteristic
312C2×C4C2×#12111yescharacteristic
313C2×C4C2×C22⋊Q811yescharacteristic
314C2×C4C2×C22⋊Q811yescharacteristic
315C2×C4C2×C22⋊Q811yescharacteristic
316C2×C4C2×#12611yescharacteristic
317C2×C4C2×#13712yes—
318C2×C4#57512yes—
319C2×C4#57712yes—
320C2×C4#57812yes—
321C8—42no—
322C8—42no—
323C8—42no—
324C8—42no—
325C8—82no—
326C8—82no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22C2×#39611yescharacteristic
333C22#304612yes—
334C22#305612yes—
335C22C2×#47111yescharacteristic
336C22C2×#47211yescharacteristic
337C4—21no—
338C4—21no—
339C4—22no—
340C4—22no—
341C4—44no—
342C4—82no—
343C4—84no—
344C4—162no—
345C4#206314yes—
346C4#303911yescharacteristic
347C4#304011yescharacteristic
348C4#305711yescharacteristic
349C4#305911yescharacteristic
350C4#307812yes—
351C4#308212yes—
352C2—42no—
353C2#1949112yes—
354C2#1950012yes—
355C2C2×#206311yescharacteristic
356C2#2361311yescharacteristic
357C2#2361911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×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
7. #10601C2semidirect
8. #10589C2semidirect

Maximal subgroups

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

subgroupindex
#219742
#218332
#213622
#213132
C2×#15742
C2×#15482
#106012
#105892
#80412