Finite Group Databasebeta

← #218877

Subgroups — #218877

Order 512.

575 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#23698C211yesmaximal, characteristic
2#22104C212yesmaximal
3#9548C212yesmaximal, complemented
4#9537C212yesmaximal, complemented
5#9077C212yesmaximal
6#8950C212yesmaximal, complemented
7#8848C212yesmaximal, complemented
8#8599C211yesmaximal, characteristic
9#7850C211yesmaximal, characteristic
10C2×#446C2212yes—
11C2×C4×Q16C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15C4×C22⋊Q8C2212yes—
16#2087C411yescharacteristic
17#2076C411yescharacteristic
18#2068C412yes—
19#2056C411yescharacteristic
20#2036C412yes—
21#2001C411yescharacteristic
22#1984C2212yes—
23#1940—22no—
24#1939C2212yes—
25#1924—22no—
26#1924—22no—
27#1917—22no—
28#1797C2212yes—
29C4×#107C2211yescharacteristic
30#1770C2212yes—
31C4×#103C2212yes—
32C2×#335C2211yescharacteristic
33C2×#335C2211yescharacteristic
34C2×Q8⋊C8C2212yes—
35C2×Q8⋊C8C2212yes—
36#1279C2212yes—
37#1279C2212yes—
38#1277C2212yes—
39#1273C2212yes—
40#1273C2212yes—
41C22×Q16C2×C411yescharacteristic
42C22×M16—21no—
43C2×C22⋊Q8C2×C412yes—
44C2×C4×Q8C2312yes—
45C22×C42C2311yescharacteristic
46#446—42no—
47#446—42no—
48C4×Q16—41no—
49C4×Q16—41no—
50#431—41no—
51#431—41no—
52#431—41no—
53#431—41no—
54C2×#107C2×C412yes—
55C2×#107C2×C411yescharacteristic
56C2×C4⋊C8—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8C2×C411yescharacteristic
59C2×C4⋊C8C2×C411yescharacteristic
60C2×C4⋊C8D411yescharacteristic
61C2×C4⋊C8D411yescharacteristic
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×#103C2×C412yes—
65C2×#103C2×C412yes—
66C2×#103C2×C412yes—
67C2×C22⋊C8—22no—
68C2×C22⋊C8C2×C411yescharacteristic
69C2×C22⋊C8C2×C411yescharacteristic
70C2×C22⋊C8D412yes—
71C4×M16—41no—
72C4×M16—41no—
73C2×#97—22no—
74C2×#97—21no—
75C2×#97—22no—
76C2×C4×C8C2312yes—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2312yes—
80#391C2×C412yes—
81#389C2×C412yes—
82#387—22no—
83#385—22no—
84#380C2×C412yes—
85C4×C4⋊C4—22no—
86C4×C4⋊C4C2312yes—
87C4×C22⋊C4—22no—
88#340C2×C411yescharacteristic
89#340C2×C412yes—
90#340C2×C411yescharacteristic
91#340C2×C412yes—
92#336—22no—
93#336—22no—
94#336—22no—
95#336—22no—
96#336—22no—
97#335—41no—
98#335—41no—
99Q8⋊C8—42no—
100Q8⋊C8—42no—
101C22×Q8C22×C412yes—
102C23×C4C22×C411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4#4111yescharacteristic
105C2×Q16—42no—
106C2×Q16—41no—
107C2×M16—44no—
108C2×M16—42no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8C22×C411yescharacteristic
113C22×C8C22×C411yescharacteristic
114C22×C8C22×C411yescharacteristic
115C22×C8C22×C411yescharacteristic
116C22×C8C2×D412yes—
117C22×C8C2×D412yes—
118C22⋊Q8—42no—
119C22⋊Q8—42no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×Q8—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4C22×C412yes—
128C2×C4⋊C4C22×C412yes—
129C2×C4⋊C4C22×C412yes—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—22no—
137C2×C42—22no—
138C2×C42C22×C411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42#4111yescharacteristic
141C2×C42C2411yescharacteristic
142#107—42no—
143#107—41no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8—41no—
152C4⋊C8—44no—
153C4⋊C8—82no—
154#103—42no—
155#103—42no—
156#103—42no—
157C22⋊C8—41no—
158C22⋊C8—41no—
159C22⋊C8—42no—
160C22⋊C8—82no—
161#97—42no—
162#97—44no—
163#97—82no—
164C4×C8—21no—
165C4×C8—21no—
166C4×C8—21no—
167C4×C8—21no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—22no—
171C4×C8—22no—
172#95—22no—
173#95—22no—
174#95—22no—
175#95—22no—
176#95—22no—
177C24C4×D411yescharacteristic
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—42no—
201C22×C4C4×D411yescharacteristic
202C22×C4C4×D411yescharacteristic
203C22×C4C4×D411yescharacteristic
204C22×C4C23×C411yescharacteristic
205C22×C4C22×D411yescharacteristic
206C22×C4C2×#4111yescharacteristic
207Q16—44no—
208M16—48no—
209C2×C8—24no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—44no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—82no—
242C22⋊C4—44no—
243C22⋊C4—82no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—42no—
251C42—42no—
252C42—42no—
253C42—42no—
254C42H(5,𝔽2)11yescharacteristic
255C42H(5,𝔽2)11yescharacteristic
256C42H(5,𝔽2)11yescharacteristic
257C422−1+411yescharacteristic
258C23—21no—
259C23—22no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23C2×C4×D411yescharacteristic
264Q8—42no—
265Q8—42no—
266Q8—42no—
267Q8—42no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4#51811yescharacteristic
298C2×C4#52011yescharacteristic
299C2×C4#53411yescharacteristic
300C2×C4#53711yescharacteristic
301C2×C4#53811yescharacteristic
302C2×C4#53911yescharacteristic
303C2×C4#56812yes—
304C2×C4#56812yes—
305C2×C4#57611yescharacteristic
306C2×C4#57811yescharacteristic
307C2×C4#57811yescharacteristic
308C2×C4#57811yescharacteristic
309C8—44no—
310C8—42no—
311C8—42no—
312C8—41no—
313C8—41no—
314C8—41no—
315C8—41no—
316C8—82no—
317C8—82no—
318C22—42no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22#232511yescharacteristic
324C22#293012yes—
325C22#295011yescharacteristic
326C22#295111yescharacteristic
327C22#295612yes—
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—82no—
334C4—82no—
335C4—82no—
336C4—82no—
337C4#297811yescharacteristic
338C4#298011yescharacteristic
339C4#299512yes—
340C4#320211yescharacteristic
341C4#320311yescharacteristic
342C4#326111yescharacteristic
343C4#326211yescharacteristic
344C2—81no—
345C2#1795312yes—
346C2#1795412yes—
347C2#2217711yescharacteristic
348C2#2219711yescharacteristic
349C2#2220711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9548C2semidirect
4. #9537C2semidirect
6. #8950C2semidirect
7. #8848C2semidirect

Maximal subgroups

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

subgroupindex
#236982
#221042
#95482
#95372
#90772
#89502
#88482
#85992
#78502