Finite Group Databasebeta

← #350889

Subgroups — #350889

Order 512.

688 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33225C211yesmaximal, characteristic
2#33196C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic
4C4×C8×Q8C211yesmaximal, characteristic
5#9898C211yesmaximal, characteristic
6#9694C211yesmaximal, characteristic
7#9562C214yesmaximal
8#9220C214yesmaximal
9#8784C211yesmaximal, characteristic
10C2×#487—21no—
11C2×#487—21no—
12C2×#475—21no—
13C2×#475C2212yes—
14C2×C8×Q8C2212yes—
15#2868C2211yescharacteristic
16#2868C2211yescharacteristic
17C42×Q8C2211yescharacteristic
18#1988—21no—
19#1988C2212yes—
20#1956—22no—
21#1956C2214yes—
22#1948—21no—
23#1948—21no—
24#1948—21no—
25#1948C2212yes—
26#1842C2211yescharacteristic
27#1842C2211yescharacteristic
28#1803—21no—
29#1803—21no—
30#1803—21no—
31#1803C2212yes—
32C8×C4⋊C4—22no—
33C8×C4⋊C4C2212yes—
34C4×C4⋊C8—21no—
35C4×C4⋊C8C2211yescharacteristic
36C42×C8—21no—
37C42×C8C2211yescharacteristic
38C2×#332C2214yes—
39#1296C2214yes—
40#1277C2214yes—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8C2312yes—
47C2×C4×Q8D411yescharacteristic
48C2×C4×Q8D411yescharacteristic
49C2×C4×Q8C2311yescharacteristic
50#487—44no—
51#487—44no—
52#475—28no—
53#475—44no—
54C8×Q8—24no—
55C8×Q8—24no—
56C2×#106—22no—
57C2×#106—22no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×#103—22no—
67C2×#103—22no—
68C2×#103—24no—
69C2×#103—24no—
70C2×C4×C8—22no—
71C2×C4×C8—22no—
72C2×C4×C8—21no—
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D412yes—
75C2×C4×C8D411yescharacteristic
76C2×C4×C8C2312yes—
77#384—21no—
78#384—21no—
79#384—21no—
80#384—21no—
81#384—21no—
82#384—21no—
83#384—21no—
84#384C2312yes—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4D411yescharacteristic
89C4×C4⋊C4D411yescharacteristic
90C4×C4⋊C4C2311yescharacteristic
91C43—21no—
92C43C2311yescharacteristic
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#336—22no—
98#336—21no—
99#332—28no—
100C22×Q8—22no—
101C22×Q8—41no—
102C22×Q8C2×D411yescharacteristic
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8C2×D412yes—
107C22×C8C2×D412yes—
108C4⋊Q8—24no—
109C4⋊Q8—24no—
110C4⋊Q8—44no—
111C4⋊Q8—42no—
112C4⋊Q8—42no—
113C4⋊Q8—42no—
114C4⋊Q8—42no—
115C4⋊Q8—42no—
116C4⋊Q8—42no—
117C4⋊Q8—82no—
118C4×Q8—22no—
119C4×Q8—22no—
120C4×Q8—22no—
121C4×Q8SD1614yes—
122C4×Q8Q1614yes—
123C4×Q8#4114yes—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4C2×D411yescharacteristic
140C2×C4⋊C4C2×D411yescharacteristic
141C2×C4⋊C4C2×D411yescharacteristic
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42#4111yescharacteristic
153C2×C42#4111yescharacteristic
154C2×C42C2411yescharacteristic
155#106—44no—
156#106—44no—
157C4⋊C8—28no—
158C4⋊C8—22no—
159C4⋊C8—24no—
160C4⋊C8—24no—
161C4⋊C8—42no—
162C4⋊C8—42no—
163C4⋊C8—42no—
164C4⋊C8—42no—
165C4⋊C8#4114yes—
166#103—48no—
167#103—44no—
168#103—44no—
169#103—48no—
170C4×C8—28no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—42no—
175C4×C8—44no—
176C4×C8—42no—
177C4×C8SD1614yes—
178#95—21no—
179#95—21no—
180#95—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—44no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—44no—
188C2×Q8—82no—
189C2×Q8—82no—
190C2×Q8—82no—
191C2×Q8#12412yes—
192C2×Q8C2×SD1612yes—
193C2×Q8C2×Q1612yes—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4#12111yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4C22×D411yescharacteristic
205C22×C4C22×D411yescharacteristic
206C22×C4C2×#4111yescharacteristic
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—24no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8C2×SD1614yes—
222C4⋊C4—24no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—24no—
227C4⋊C4—24no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—82no—
236C4⋊C4—82no—
237C4⋊C4—82no—
238C4⋊C4—82no—
239C4⋊C4—82no—
240C4⋊C4—82no—
241C4⋊C4—82no—
242C4⋊C4—82no—
243C4⋊C4#12412yes—
244C4⋊C4C2×SD1612yes—
245C4⋊C4C2×Q1612yes—
246C42—24no—
247C42—22no—
248C42—24no—
249C42—24no—
250C42—22no—
251C42—22no—
252C42—42no—
253C42—42no—
254C42—42no—
255C42C2×SD1612yes—
256C42C2×SD1612yes—
257C42C2×Q1612yes—
258C42#13512yes—
259C42#13512yes—
260C42AGL(1,ℤ/8)12yes—
261C42C2×#4112yes—
262C42C2×#4112yes—
263C23C2×#12111yescharacteristic
264Q8—22no—
265Q8—22no—
266Q8—22no—
267Q8—84no—
268Q8—84no—
269Q8—164no—
270Q8#48714yescomplemented
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4#46012yes—
296C2×C4#46012yes—
297C2×C4#46512yes—
298C2×C4#46512yes—
299C2×C4#46712yes—
300C2×C4#46712yes—
301C2×C4#47112yes—
302C2×C4#47312yes—
303C2×C4#52512yes—
304C2×C4C2×#12411yescharacteristic
305C2×C4#52912yes—
306C2×C4#53211yescharacteristic
307C2×C4C22×SD1611yescharacteristic
308C2×C4C22×SD1611yescharacteristic
309C2×C4C22×Q1611yescharacteristic
310C2×C4C2×#13511yescharacteristic
311C2×C4C2×#13511yescharacteristic
312C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
313C2×C4#57512yes—
314C8—24no—
315C8—42no—
316C8—44no—
317C8—82no—
318C8—82no—
319C22#285011yescharacteristic
320C22C2×#46011yescharacteristic
321C22#304011yescharacteristic
322C22C2×#46511yescharacteristic
323C22C2×#46711yescharacteristic
324C22C2×#47111yescharacteristic
325C22C2×#47311yescharacteristic
326C4—24no—
327C4—22no—
328C4—22no—
329C4—22no—
330C4—42no—
331C4—42no—
332C4—42no—
333C4—162no—
334C4—162no—
335C4#166614yes—
336C4#167014yes—
337C4#310312yes—
338C4#311512yes—
339C4C2×#48711yescharacteristic
340C4#313111yescharacteristic
341C4#316711yescharacteristic
342C4#317011yescharacteristic
343C2C2×#166611yescharacteristic
344C2C2×#167011yescharacteristic
345C2#1927511yescharacteristic
346C2#1927711yescharacteristic
347C2#3302511yescharacteristic
348C2#3304311yescharacteristic
349C2#3306111yescharacteristic
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
270. Q8#487semidirect

Maximal subgroups

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

subgroupindex
#332252
#331962
C4×C8×Q82
#98982
#96942
#95622
#92202
#87842