Finite Group Databasebeta

← #192040

Subgroups — #192040

Order 512.

1121 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2973C211yesmaximal, characteristic
2#21931C212yesmaximal, complemented
3#21740C211yesmaximal, characteristic
4#20722C211yesmaximal, characteristic
5#8852C218yesmaximal, complemented
6#7927C212yesmaximal, complemented
7C22×C4×D4C2211yescharacteristic
8#2973—24no—
9#2973—22no—
10#2973C2214yes—
11C2×#435—24no—
12C2×#432—21no—
13C2×#407—21no—
14C2×#407C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17#2280C2212yes—
18#1870—22no—
19#1869C2214yes—
20#1850—24no—
21#1850C2218yescomplemented
22#1819C412yes—
23#1817—21no—
24#1799C414yes—
25#1798—24no—
26#1781C412yes—
27#1779—21no—
28C4×#99C2214yes—
29#1282C2218yescomplemented
30C23×D4C2×C411yescharacteristic
31C24×C4D411yescharacteristic
32C24×C4C2311yescharacteristic
33C22×M16—22no—
34C22×M16—21no—
35C23×C8D414yes—
36C23×C8C2312yes—
37C2×C4×D4—22no—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4—22no—
41C2×C4×D4C2×C412yes—
42C2×C4×D4D412yes—
43C2×C4×D4Q812yes—
44C2×C4×D4C2312yes—
45C2×#117—22no—
46C22×C4⋊C4Q811yescharacteristic
47C22×C22⋊C4C2×C411yescharacteristic
48C22×C22⋊C4Q811yescharacteristic
49C22×C42D411yescharacteristic
50#435—216no—
51#435—216no—
52#432—24no—
53#432—24no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8—28no—
56C2×#99C2×C418yes—
57#407—24no—
58#407—24no—
59#407—22no—
60#407—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8D414yes—
73C2×C22⋊C8C2312yes—
74C2×C22⋊C8C2314yes—
75C2×#97—22no—
76#381—22no—
77#380—22no—
78#380C2×C414yes—
79#379C2×C414yes—
80C4×C22⋊C4—22no—
81C4×C22⋊C4C2314yes—
82#342C2×C418yes—
83#336—28no—
84#336—24no—
85C25C4⋊C411yescharacteristic
86C25C22×C411yescharacteristic
87C22×D4—22no—
88C22×D4—22no—
89C22×D4—44no—
90C22×D4C4⋊C412yes—
91C22×D4C22×C412yes—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—82no—
97C23×C4—82no—
98C23×C4C4⋊C411yescharacteristic
99C23×C4C4⋊C411yescharacteristic
100C23×C4C4⋊C411yescharacteristic
101C23×C4C4⋊C412yes—
102C23×C4C22×C411yescharacteristic
103C23×C4C2×D412yes—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4C2×Q811yescharacteristic
109C23×C4C2×Q811yescharacteristic
110C23×C4#4112yes—
111C23×C4#4112yes—
112C23×C4C2411yescharacteristic
113C2×M16—28no—
114C2×M16—42no—
115C2×M16—84no—
116C2×M16—82no—
117C22×C8—24no—
118C22×C8—24no—
119C22×C8—22no—
120C22×C8—24no—
121C22×C8—22no—
122C22×C8—82no—
123C22×C8—84no—
124C22×C8C2×D418yes—
125C4×D4—24no—
126C4×D4—24no—
127C4×D4—24no—
128C4×D4—24no—
129C4×D4—24no—
130C4×D4—24no—
131C4×D4—24no—
132C4×D4—24no—
133#117—84no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4C4⋊C412yes—
138C2×C4⋊C4C2×Q812yes—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4C4⋊C412yes—
146C2×C22⋊C4C4⋊C412yes—
147C2×C22⋊C4C22×C418yes—
148C2×C22⋊C4C22×C412yes—
149C2×C22⋊C4C2×Q812yes—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—24no—
155C2×C42C4⋊C412yes—
156C2×C42C2×D412yes—
157C4⋊C8—28no—
158C4⋊C8—88no—
159#99—48no—
160C22⋊C8—28no—
161C22⋊C8—24no—
162C22⋊C8—24no—
163C22⋊C8—24no—
164C22⋊C8—24no—
165C22⋊C8—24no—
166C22⋊C8—41no—
167C22⋊C8—42no—
168C22⋊C8—41no—
169C22⋊C8—41no—
170C22⋊C8—41no—
171C22⋊C8—84no—
172#97—28no—
173#95—24no—
174#95—24no—
175C24—24no—
176C24—21no—
177C24—21no—
178C24—42no—
179C24—42no—
180C24—42no—
181C24—81no—
182C24—81no—
183C24C2×C4⋊C411yescharacteristic
184C24C2×C4⋊C412yes—
185C24C2×C4⋊C411yescharacteristic
186C24C4×D412yes—
187C24C4×D412yes—
188C24C23×C411yescharacteristic
189C2×D4—24no—
190C2×D4—24no—
191C2×D4—24no—
192C2×D4—24no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—48no—
196C2×D4—48no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—28no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—42no—
217C22×C4—82no—
218C22×C4—81no—
219C22×C4—81no—
220C22×C4—81no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—81no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4C2×C4⋊C412yes—
228C22×C4C2×C4⋊C412yes—
229C22×C4C2×C4⋊C412yes—
230C22×C4C2×C4⋊C411yescharacteristic
231C22×C4C2×C4⋊C411yescharacteristic
232C22×C4C4×D412yes—
233C22×C4C4×D412yes—
234C22×C4#12011yescharacteristic
235C22×C4#12012yes—
236C22×C4#12011yescharacteristic
237C22×C4C22⋊Q812yes—
238C22×C4C22⋊Q812yes—
239C22×C4C22⋊Q812yes—
240C22×C4C22⋊Q812yes—
241C22×C4#12311yescharacteristic
242C22×C4#12311yescharacteristic
243C22×C4#12312yes—
244C22×C4C22×D412yes—
245C22×C4C22×D411yescharacteristic
246C22×C4C22×Q811yescharacteristic
247C22×C4C2×#4112yes—
248M16—88no—
249M16—84no—
250C2×C8—28no—
251C2×C8—48no—
252C2×C8—48no—
253C2×C8—44no—
254C2×C8—44no—
255C2×C8—48no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—84no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—84no—
262C22⋊C4—24no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—24no—
266C22⋊C4—216no—
267C22⋊C4—48no—
268C22⋊C4—84no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—48no—
274C23—24no—
275C23—22no—
276C23—22no—
277C23—24no—
278C23—22no—
279C23—24no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—41no—
284C23—42no—
285C23—41no—
286C23—41no—
287C23—42no—
288C23—41no—
289C23—41no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—81no—
298C23—81no—
299C23—88no—
300C23—88no—
301C23#38512yes—
302C23#38511yescharacteristic
303C23#38511yescharacteristic
304C23C22×C4⋊C411yescharacteristic
305C23C2×C4×D412yes—
306C23#56814yes—
307D4—48no—
308D4—48no—
309C2×C4—24no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—48no—
341C2×C4—48no—
342C2×C4—48no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4#38511yescharacteristic
348C2×C4#38511yescharacteristic
349C2×C4#38512yes—
350C2×C4C2×#12011yescharacteristic
351C2×C4C2×C22⋊Q812yes—
352C2×C4C2×#12311yescharacteristic
353C8—48no—
354C8—82no—
355C8—82no—
356C22—21no—
357C22—22no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—21no—
362C22—21no—
363C22—22no—
364C22—21no—
365C22—22no—
366C22—21no—
367C22—22no—
368C22—44no—
369C22—48no—
370C22—44no—
371C22—44no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—84no—
376C22—82no—
377C22—82no—
378C22—84no—
379C22—84no—
380C22#186812yes—
381C22C2×#38511yescharacteristic
382C22#290512yes—
383C22#293014yes—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—22no—
388C4—22no—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—88no—
393C4#302612yes—
394C2—24no—
395C2—22no—
396C2—22no—
397C2—42no—
398C2—41no—
399C2—82no—
400C2#2083511yescharacteristic
401C2#2088911yescharacteristic
402C2C2×#186811yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21931C2semidirect
5. #8852C2semidirect
6. #7927C2semidirect
21. #1850C22semidirect
29. #1282C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29732
#219312
#217402
#207222
#88522
#79272