Finite Group Databasebeta

← #189877

Subgroups — #189877

Order 512.

1352 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34181C211yesmaximal, characteristic
2#21742C211yesmaximal, characteristic
3#13940C211yesmaximal, characteristic
4#8716C214yesmaximal, complemented
5#8713C214yesmaximal, complemented
6#8019C212yesmaximal, complemented
7#8015C212yesmaximal, complemented
8C2×#518—21no—
9C22×C4×D4C2211yescharacteristic
10C2×#407C2212yes—
11#2349—24no—
12D4×C4⋊C4C418yescomplemented
13#2316—21no—
14#2315—21no—
15#2315—21no—
16#2285C2212yes—
17C2×#381C2211yescharacteristic
18C2×#380C2212yes—
19C2×#380C2212yes—
20C2×C4×C22⋊C4C2211yescharacteristic
21#1870—24no—
22#1817C2212yes—
23#1779C2212yes—
24#1440—21no—
25#1438—21no—
26#1437—21no—
27#1436—21no—
28#1434C2212yes—
29C22⋊#95C2212yes—
30#1294C2218yes—
31#1294C2218yes—
32C23×D4D411yescharacteristic
33C24×C4C2311yescharacteristic
34C24×C4C2311yescharacteristic
35C22×M16—22no—
36#518—48no—
37C2×C4×D4—24no—
38C2×C4×D4—24no—
39C2×C4×D4—28no—
40C2×C4×D4C2×C414yes—
41C2×C4×D4D414yes—
42C2×#117—22no—
43C22×C4⋊C4C2×C414yes—
44C22×C4⋊C4C2×C414yes—
45C22×C4⋊C4D411yescharacteristic
46C22×C22⋊C4—22no—
47C22×C22⋊C4—22no—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4D411yescharacteristic
52C22×C22⋊C4D411yescharacteristic
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55#407—24no—
56#407—24no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—24no—
59C2×C22⋊C8C2314yes—
60#389—28no—
61#385—28no—
62#385—28no—
63#384C2×C414yes—
64#382—22no—
65#381—24no—
66#380—22no—
67#380—22no—
68#380—22no—
69#380—22no—
70#380—22no—
71#380C2×C414yes—
72#380C2×C414yes—
73#380C2314yes—
74C4×C4⋊C4—22no—
75C4×C4⋊C4C2×C414yes—
76C4×C22⋊C4—24no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95C2312yes—
84C2×#95C2312yes—
85#336—24no—
86C25C2×D411yescharacteristic
87C25C2×D411yescharacteristic
88C22×D4—22no—
89C22×D4—22no—
90C22×D4—24no—
91C22×D4—24no—
92C22×D4C22⋊C414yes—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—24no—
97C23×C4—24no—
98C23×C4—22no—
99C23×C4—82no—
100C23×C4—82no—
101C23×C4C22×C412yes—
102C23×C4C22×C412yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D412yes—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D412yes—
109C23×C4#4114yes—
110C23×C4#4114yes—
111C23×C4C2411yescharacteristic
112C2×M16—44no—
113C2×M16—84no—
114C22×C8—24no—
115C4×D4—216no—
116C4×D4—28no—
117C4×D4—28no—
118C4×D4—432no—
119#117—48no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—84no—
127C2×C4⋊C4—84no—
128C2×C4⋊C4C22⋊C414yes—
129C2×C4⋊C4C22×C418yes—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—44no—
138C2×C22⋊C4—48no—
139C2×C22⋊C4—48no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—44no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—44no—
144C2×C22⋊C4—82no—
145C2×C22⋊C4—82no—
146C2×C22⋊C4C22⋊C414yes—
147C2×C22⋊C4C22⋊C414yes—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—24no—
151C2×C42—42no—
152C2×C42—82no—
153C2×C42C22×C412yes—
154C2×C42C2×D412yes—
155C22⋊C8—24no—
156C22⋊C8—24no—
157C22⋊C8—44no—
158C22⋊C8—84no—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—24no—
163#95—24no—
164#95—24no—
165#95—44no—
166#95—42no—
167#95—44no—
168#95—82no—
169C24—21no—
170C24—21no—
171C24—22no—
172C24—22no—
173C24—22no—
174C24—42no—
175C24—42no—
176C24—42no—
177C24—42no—
178C24C2×C22⋊C412yes—
179C24C2×C22⋊C412yes—
180C24#12312yes—
181C24#12312yes—
182C24#13714yes—
183C24C22×D411yescharacteristic
184C2×D4—216no—
185C2×D4—216no—
186C2×D4—28no—
187C2×D4—28no—
188C2×D4—28no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—28no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—48no—
215C22×C4—48no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—84no—
224C22×C4—84no—
225C22×C4—84no—
226C22×C4—84no—
227C22×C4—84no—
228C22×C4—84no—
229C22×C4—84no—
230C22×C4—84no—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4#11718yes—
236C22×C4#12314yes—
237C22×C4#12314yes—
238C22×C4#12312yes—
239C22×C4#12312yes—
240C22×C4AGL(1,ℤ/8)14yes—
241C22×C4C23×C411yescharacteristic
242C22×C4C22×D411yescharacteristic
243C22×C4C2×#4112yes—
244C22×C4C2×#4112yes—
245M16—88no—
246C2×C8—48no—
247C2×C8—44no—
248C4⋊C4—28no—
249C4⋊C4—48no—
250C4⋊C4—48no—
251C4⋊C4—84no—
252C4⋊C4—84no—
253C4⋊C4—84no—
254C22⋊C4—28no—
255C22⋊C4—28no—
256C22⋊C4—48no—
257C22⋊C4—48no—
258C22⋊C4—48no—
259C22⋊C4—48no—
260C22⋊C4—88no—
261C22⋊C4—84no—
262C42—24no—
263C42—24no—
264C42—48no—
265C42—84no—
266C23—22no—
267C23—22no—
268C23—24no—
269C23—24no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—24no—
274C23—24no—
275C23—22no—
276C23—24no—
277C23—24no—
278C23—24no—
279C23—24no—
280C23—42no—
281C23—42no—
282C23—44no—
283C23—42no—
284C23—42no—
285C23—44no—
286C23—81no—
287C23—81no—
288C23—82no—
289C23—82no—
290C23—82no—
291C23—82no—
292C23—82no—
293C23—82no—
294C23—81no—
295C23—81no—
296C23#38114yes—
297C23#41914yes—
298C23C22×C22⋊C411yescharacteristic
299C23C2×#12311yescharacteristic
300C23C2×#12311yescharacteristic
301C23C2×AGL(1,ℤ/8)12yes—
302C23C2×#13712yes—
303D4—232no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—48no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—48no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—48no—
327C2×C4—48no—
328C2×C4—48no—
329C2×C4—84no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—168no—
346C2×C4—168no—
347C2×C4#38114yes—
348C2×C4#41814yes—
349C2×C4C2×#11712yes—
350C2×C4C2×#12312yes—
351C8—84no—
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—21no—
359C22—21no—
360C22—24no—
361C22—28no—
362C22—24no—
363C22—24no—
364C22—24no—
365C22—42no—
366C22—44no—
367C22—44no—
368C22—44no—
369C22—42no—
370C22—84no—
371C22—84no—
372C22—84no—
373C22—84no—
374C22C2×#38111yescharacteristic
375C22#288312yes—
376C22C2×#41811yescharacteristic
377C22C2×#41911yescharacteristic
378C22#290112yes—
379C22#309212yes—
380C22#309212yes—
381C4—22no—
382C4—22no—
383C4—24no—
384C4—24no—
385C4—42no—
386C4—42no—
387C4—84no—
388C4—88no—
389C4—162no—
390C2—22no—
391C2—22no—
392C2—44no—
393C2—44no—
394C2#1762112yes—
395C2#1762612yes—
396C2#2068911yescharacteristic
397C2#2070011yescharacteristic
398C2#2070611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8716C2semidirect
5. #8713C2semidirect
6. #8019C2semidirect
7. #8015C2semidirect
12. D4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#341812
#217422
#139402
#87162
#87132
#80192
#80152