Finite Group Databasebeta

← #6380293

Subgroups — #6380293

Order 512.

1882 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
1C2×#2961C211yesmaximal, characteristic
2#34213C212yesmaximal
3#22730C218yesmaximal, complemented
4#22728C218yesmaximal, complemented
5#21175C214yesmaximal
6#21170C212yesmaximal
7#20803C212yesmaximal
8#20799C211yesmaximal, characteristic
9#20454C212yesmaximal
10C2×#1776C211yesmaximal, characteristic
11C2×#533C2211yescharacteristic
12#2961C4116yescomplemented
13C2×#446C2218yes—
14C2×#428C2214yes—
15C2×#427C2212yes—
16C2×#427C2212yes—
17C22×#107C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C2×#424C2212yes—
20C2×#423C2211yescharacteristic
21C2×#405C2212yes—
22C22×#97C2211yescharacteristic
23#2370—24no—
24#2369—22no—
25#2365—22no—
26#2355C2218yes—
27#2355C2214yes—
28Q8×C4⋊C4C2214yes—
29#2329—24no—
30#2317C2214yes—
31C4×C22⋊Q8—24no—
32#2285C2212yes—
33#2274C2214yes—
34C2×C4×C4⋊C4C2212yes—
35#1949C2218yes—
36#1949C2218yes—
37#1949C2218yes—
38#1948C2214yes—
39#1948C2218yes—
40#1947C2214yes—
41#1947C2218yes—
42#1847C2214yes—
43#1847C2214yes—
44#1847C2218yes—
45#1846C2214yes—
46#1845C2214yes—
47#1776C2214yes—
48#1776C2218yes—
49#1776C2214yes—
50C4×#106C2218yes—
51#1736C2214yes—
52C2×#340C2212yes—
53C22×M16C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56#533C2×C418yes—
57#533Q818yes—
58C2×C4⋊Q8D412yes—
59C2×#125D412yes—
60C2×C22⋊Q8D414yes—
61C2×C4×Q8—24no—
62C2×C4×Q8C2314yes—
63C2×#117C2314yes—
64C2×#117C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C42—22no—
68C22×C42C2311yescharacteristic
69#446C2×C4164yescomplemented
70#428—48no—
71#427—28no—
72#427—44no—
73C2×#107—24no—
74C2×#107C2312yes—
75C2×#107C2314yes—
76C2×#107C2312yes—
77C2×#107C2318yes—
78C2×#107C2312yes—
79C2×#107C2312yes—
80C2×#106Q818yescomplemented
81C2×#106C2314yes—
82C2×#106C2318yes—
83C2×#106C2314yes—
84C2×#106C2314yes—
85#424C2×C4116yes—
86#423C2×C418yes—
87C2×C4⋊C8C2314yes—
88C2×C4⋊C8C2318yes—
89#405C2×C4116yes—
90C2×#97C2×C418yes—
91C2×#97C2314yes—
92C2×#97C2314yes—
93C2×C4×C8C2314yes—
94#391—28no—
95#391—24no—
96#389—24no—
97#389—24no—
98#389—28no—
99#389—24no—
100#387—28no—
101#387—28no—
102#387—28no—
103#385—28no—
104#384C2318yes—
105#384C2312yes—
106#384C2312yes—
107#382C2318yes—
108#382C2314yes—
109#380—22no—
110#380—22no—
111C4×C4⋊C4—24no—
112C4×C4⋊C4—28no—
113C4×C4⋊C4—24no—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2318yes—
120C4×C4⋊C4C2314yes—
121C4×C4⋊C4C2314yes—
122C4×C4⋊C4C2318yes—
123C4×C22⋊C4—28no—
124C2×#95—22no—
125#340C2318yes—
126#340C2318yes—
127#340C2318yes—
128C22×Q8C2×D414yes—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4C2×D412yes—
132C23×C4C2411yescharacteristic
133C2×M16—24no—
134C2×M16—24no—
135C2×M16C22×C418yes—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8C22×C414yes—
139C22×C8C22×C414yes—
140C22×C8C2×Q814yes—
141C22×C8#4114yes—
142C22×C8C2412yes—
143C22×C8C2414yes—
144C22×C8C2412yes—
145C4⋊Q8C4⋊C4116yes—
146#125C4⋊C4116yes—
147C22⋊Q8C4⋊C4132yes—
148C4×Q8—88no—
149C4×Q8C22×C4116yes—
150C4×Q8C2×Q8116yes—
151#117—28no—
152#117—28no—
153#117C22×C418yes—
154#117C2×Q818yes—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4C2×D414yes—
167C2×C4⋊C4C2×D418yes—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4C2×Q818yes—
170C2×C4⋊C4C2414yes—
171C2×C4⋊C4C2414yes—
172C2×C4⋊C4C2414yes—
173C2×C4⋊C4C2414yes—
174C2×C22⋊C4—24no—
175C2×C22⋊C4C2×D414yes—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—28no—
179C2×C42—28no—
180C2×C42—24no—
181C2×C42—84no—
182C2×C42C22×C414yes—
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×Q814yes—
186C2×C42C2414yes—
187C2×C42C2412yes—
188#107—28no—
189#107—24no—
190#107—24no—
191#107—42no—
192#107—42no—
193#107—48no—
194#106—28no—
195#106—44no—
196#106—48no—
197#106C2×Q8116yes—
198C4⋊C8C22×C4116yes—
199C4⋊C8C22×C4132yes—
200#97C22×C4116yes—
201#97C22×C4116yes—
202C4×C8C22×C4116yes—
203#95—24no—
204#95—24no—
205#95—28no—
206#95—28no—
207#95—24no—
208#95—24no—
209#95—28no—
210#95—28no—
211C24C22×D411yescharacteristic
212C2×Q8—44no—
213C2×Q8—44no—
214C2×Q8C2×C4⋊C4116yes—
215C22×C4—24no—
216C22×C4—28no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4C2×C4⋊C418yes—
228C22×C4C23×C412yes—
229C22×C4C22×D412yes—
230C22×C4C22×D412yes—
231C22×C4C22×D418yes—
232C22×C4C22×D411yescharacteristic
233C22×C4C22×Q812yes—
234C22×C4C22×Q812yes—
235C22×C4C2×#4112yes—
236C22×C4H(5,𝔽2)12yes—
237C22×C42−1+412yes—
238C22×C4C2511yescharacteristic
239M16—216no—
240C2×C8—28no—
241C2×C8—22no—
242C2×C8—24no—
243C2×C8—22no—
244C2×C8—24no—
245C2×C8C4×Q8116yescomplemented
246C2×C8C23×C418yes—
247C2×C8C23×C414yes—
248C2×C8C23×C414yes—
249C2×C8C22×Q814yes—
250C2×C8C2×#4114yes—
251C4⋊C4—28no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—82no—
263C4⋊C4—88no—
264C4⋊C4—82no—
265C4⋊C4—88no—
266C4⋊C4C2×C4⋊C4116yes—
267C4⋊C4C2×C4⋊C4132yes—
268C4⋊C4C2×C4⋊C4116yes—
269C4⋊C4C22×Q818yes—
270C22⋊C4—48no—
271C22⋊C4C2×C4⋊C4116yes—
272C42—48no—
273C42—48no—
274C42—88no—
275C42—84no—
276C42—88no—
277C42C2×C4⋊C414yes—
278C42C2×C4⋊C414yes—
279C42C23×C418yes—
280C42C23×C414yes—
281C42C22×Q814yes—
282C42C22×Q818yes—
283C23—22no—
284C23—22no—
285C23C22×C4⋊C412yes—
286C23D4×Q812yes—
287C23#55012yes—
288C23C23×D411yescharacteristic
289Q8—416no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—28no—
293C2×C4—28no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—28no—
297C2×C4—48no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—88no—
310C2×C4—88no—
311C2×C4C22×C4⋊C414yes—
312C2×C4C22×C4⋊C414yes—
313C2×C4C22×C4⋊C4116yes—
314C2×C4C22×C4⋊C412yes—
315C2×C4C2×C4×Q814yes—
316C2×C4#52014yes—
317C2×C4D4×Q812yes—
318C2×C4#55012yes—
319C2×C4#55714yes—
320C2×C4Q8214yes—
321C2×C4C24×C411yescharacteristic
322C2×C4C23×Q811yescharacteristic
323C2×C4C23×Q811yescharacteristic
324C2×C4C22×#4111yescharacteristic
325C2×C4C2×H(5,𝔽2)11yescharacteristic
326C2×C4C2×2−1+411yescharacteristic
327C8—24no—
328C8—28no—
329C8C2×C4×Q818yes—
330C22—24no—
331C22—24no—
332C22—22no—
333C22—22no—
334C22Q8×C4⋊C414yes—
335C22C23×C4⋊C411yescharacteristic
336C22C2×D4×Q811yescharacteristic
337C22C2×#55011yescharacteristic
338C22#358512yes—
339C22#358612yes—
340C4—24no—
341C4—24no—
342C4—216no—
343C4—48no—
344C4—84no—
345C4—84no—
346C4—84no—
347C4Q8×C4⋊C414yes—
348C4C22×C4×Q811yescharacteristic
349C4C2×#52011yescharacteristic
350C4C2×#55711yescharacteristic
351C4C2×Q8211yescharacteristic
352C2—24no—
353C2C2×Q8×C4⋊C411yescharacteristic
354C2#6010912yes—
355C2#6137712yes—
356C2#6141411yescharacteristic
357C2#6141611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22730C2semidirect
4. #22728C2semidirect
12. #2961C4semidirect
69. #446C2×C4semidirect
80. C2×#106Q8semidirect
245. C2×C8C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#29612
#342132
#227302
#227282
#211752
#211702
#208032
#207992
#204542
C2×#17762