Finite Group Databasebeta

← #329402

Subgroups — #329402

Order 512.

786 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33172C211yesmaximal, characteristic
2#32954C212yesmaximal
3C2×#1984C211yesmaximal, characteristic
4#10617C212yesmaximal, complemented
5#10486C214yesmaximal, complemented
6#10276C212yesmaximal, complemented
7#9131C212yesmaximal
8#8595C211yesmaximal, characteristic
9C22×C4×Q8C2211yescharacteristic
10C2×#494C2211yescharacteristic
11C2×#475C2214yes—
12C2×#467—21no—
13C2×#467—21no—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2857C2212yes—
17#1984—21no—
18#1984—21no—
19#1984C2212yes—
20#1984C2212yes—
21#1969—21no—
22#1969—21no—
23#1956C2212yes—
24#1865—24no—
25#1853—22no—
26#1845C2211yescharacteristic
27#1803C2214yes—
28#1797C2212yes—
29#1787—22no—
30C2×#336—21no—
31C2×#336—21no—
32C2×#334C2212yes—
33C2×#332C2214yes—
34#1296C2214yes—
35#1277C2212yes—
36C23×Q8D411yescharacteristic
37C22×Q16—21no—
38C22×Q16—21no—
39C23×C8—21no—
40C23×C8D411yescharacteristic
41C23×C8D411yescharacteristic
42C2×C4⋊Q8C2312yes—
43C2×C22⋊Q8—22no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—42no—
47C2×C4×Q8—42no—
48C2×C4×Q8D412yes—
49C2×C4×Q8C2312yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D411yescharacteristic
52C22×C42—21no—
53C22×C42C2311yescharacteristic
54#494—42no—
55#475—44no—
56#475—44no—
57#467—44no—
58#467—44no—
59C2×#107—21no—
60C2×#107—21no—
61C2×#106—24no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—42no—
64C2×C4⋊C8C2314yes—
65C2×C4⋊C8C2312yes—
66C2×#103—24no—
67C2×#103—24no—
68C2×#103—22no—
69C2×#103—22no—
70C2×C22⋊C8—22no—
71C2×C4×C8—42no—
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#391—22no—
77#389—22no—
78#389—22no—
79#384C2312yes—
80#380—22no—
81#380—22no—
82#340—24no—
83#336—22no—
84#336—22no—
85#336—22no—
86#336—22no—
87#336—22no—
88#334—42no—
89#332—44no—
90C22×Q8—24no—
91C22×Q8—21no—
92C22×Q8—24no—
93C22×Q8—22no—
94C22×Q8—42no—
95C22×Q8C2×D412yes—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4#4111yescharacteristic
101C2×Q16—42no—
102C2×Q16—42no—
103C2×Q16—42no—
104C2×Q16—42no—
105C2×Q16—42no—
106C2×Q16—42no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—24no—
111C22×C8—41no—
112C22×C8—41no—
113C22×C8—41no—
114C22×C8—42no—
115C22×C8—42no—
116C22×C8—42no—
117C22×C8SD1614yes—
118C22×C8C2×D411yescharacteristic
119C22×C8C2×D411yescharacteristic
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C4⋊Q8—24no—
123C4⋊Q8—24no—
124C22⋊Q8—44no—
125C22⋊Q8—44no—
126C4×Q8—24no—
127C4×Q8—24no—
128C4×Q8—44no—
129C4×Q8—42no—
130C4×Q8—42no—
131C4×Q8—44no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4C2×D412yes—
144C2×C22⋊C4—42no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—42no—
150C2×C42—42no—
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42#4111yescharacteristic
154C2×C42C2411yescharacteristic
155#107—42no—
156#107—42no—
157#106—48no—
158C4⋊C8—24no—
159C4⋊C8—24no—
160C4⋊C8—24no—
161C4⋊C8—42no—
162#103—44no—
163#103—48no—
164#103—48no—
165#103—44no—
166C22⋊C8—44no—
167C4×C8—22no—
168C4×C8—22no—
169C4×C8—41no—
170C4×C8—41no—
171#95—42no—
172#95—42no—
173C24#12111yescharacteristic
174C2×Q8—24no—
175C2×Q8—22no—
176C2×Q8—24no—
177C2×Q8—24no—
178C2×Q8—416no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8#13514yes—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—41no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4#12111yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4C2×SD1612yes—
211C22×C4AGL(1,ℤ/8)12yes—
212C22×C4#13714yes—
213C22×C4C22×D411yescharacteristic
214C22×C4C22×D411yescharacteristic
215C22×C4C2×#4111yescharacteristic
216Q16—84no—
217Q16—84no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—44no—
228C2×C8—44no—
229C2×C8—44no—
230C2×C8—42no—
231C2×C8—44no—
232C2×C8—42no—
233C2×C8—44no—
234C2×C8—42no—
235C2×C8—84no—
236C2×C8C2×SD1612yes—
237C2×C8C2×SD1612yes—
238C2×C8#13712yes—
239C4⋊C4—24no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—82no—
249C4⋊C4—84no—
250C22⋊C4—84no—
251C42—21no—
252C42—22no—
253C42—21no—
254C42—22no—
255C42—44no—
256C42—42no—
257C42—42no—
258C42H(5,𝔽2)12yes—
259C42H(5,𝔽2)12yes—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23#46012yes—
265C23#47012yes—
266C23#47312yes—
267C23C2×#12111yescharacteristic
268Q8—28no—
269Q8—48no—
270Q8—44no—
271Q8—82no—
272Q8—82no—
273Q8—82no—
274Q8—82no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—28no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—24no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—44no—
291C2×C4—48no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—84no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4#46012yes—
305C2×C4#47012yes—
306C2×C4#53411yescharacteristic
307C2×C4#53412yes—
308C2×C4#53511yescharacteristic
309C2×C4#53812yes—
310C2×C4C22×SD1611yescharacteristic
311C2×C4C2×#13512yes—
312C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
313C2×C4C2×#13712yes—
314C2×C4C2×#13711yescharacteristic
315C2×C4#57511yescharacteristic
316C8—22no—
317C8—22no—
318C8—22no—
319C8—41no—
320C8—84no—
321C8—82no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22#169014yes—
329C22#261711yescharacteristic
330C22C2×#46011yescharacteristic
331C22#303711yescharacteristic
332C22#305011yescharacteristic
333C22C2×#47011yescharacteristic
334C22C2×#47311yescharacteristic
335C22#306711yescharacteristic
336C4—22no—
337C4—22no—
338C4—22no—
339C4—44no—
340C4—41no—
341C4—82no—
342C4—162no—
343C4—162no—
344C4#318711yescharacteristic
345C4#319014yes—
346C4#319311yescharacteristic
347C4#320611yescharacteristic
348C4#321311yescharacteristic
349C2—24no—
350C2#1911411yescharacteristic
351C2C2×#169011yescharacteristic
352C2#1926511yescharacteristic
353C2#1928711yescharacteristic
354C2#2971311yescharacteristic
355C2#2973311yescharacteristic
356C2#2974911yescharacteristic
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
4. #10617C2semidirect
5. #10486C2semidirect
6. #10276C2semidirect

Maximal subgroups

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

subgroupindex
#331722
#329542
C2×#19842
#106172
#104862
#102762
#91312
#85952