Finite Group Databasebeta

← #194861

Subgroups — #194861

Order 512.

1692 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2591C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic, complemented
3#21764C212yesmaximal
4C2×#1463C211yesmaximal, characteristic, complemented
5#9023C214yesmaximal, complemented
6#9019C214yesmaximal, complemented
7#8019C212yesmaximal, complemented
8C22×#121C2211yescharacteristic
9C22×#117—21no—
10C23×C4⋊C4C2211yescharacteristic
11C2×#407C2212yes—
12#2591—24no—
13#2591C418yescomplemented
14C2×#400C2212yes—
15C2×#394C2212yes—
16#2316—24no—
17#2316—22no—
18#2316C2214yes—
19C2×#389—24no—
20C2×#388—22no—
21C2×#386—21no—
22C2×#386—21no—
23C2×#380—21no—
24C2×#380C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26#1895—24no—
27#1874—24no—
28#1824C2214yes—
29#1786C2214yes—
30#1463—24no—
31C2×#324C2214yes—
32#1294C2218yes—
33C23×D4—21no—
34C23×D4C2311yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4D411yescharacteristic
37C24×C4C2311yescharacteristic
38C22×M16—22no—
39C2×#121—22no—
40C2×#121—22no—
41C2×#121—22no—
42C2×#121—24no—
43C2×#121C2×C414yes—
44C2×#117—24no—
45C2×#117—24no—
46C2×#117—48no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2×C414yes—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4D414yes—
55C22×C4⋊C4D414yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C42—22no—
64C22×C42—21no—
65C2×#102—28no—
66#407—24no—
67#407—24no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8C2314yes—
70#400—24no—
71#400C2×C418yescomplemented
72#394—24no—
73#394C2×C418yescomplemented
74#389—216no—
75#389—216no—
76#388—216no—
77#386—28no—
78#386—28no—
79#380—24no—
80#380—22no—
81#380—22no—
82#380—24no—
83#380D414yes—
84#380C2314yes—
85C4×C22⋊C4—22no—
86C4×C22⋊C4C2×C414yes—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95C2312yes—
92#340—28no—
93#324—28no—
94C25—21no—
95C25C2×D411yescharacteristic
96C22×D4—24no—
97C22×D4—22no—
98C22×D4—21no—
99C22×D4—22no—
100C22×D4—48no—
101C22×D4—41no—
102C22×D4—41no—
103C22×D4—42no—
104C22×D4—82no—
105C22×D4C22×C412yes—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—24no—
110C23×C4—24no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—24no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—82no—
118C23×C4—82no—
119C23×C4C22⋊C414yes—
120C23×C4C22⋊C414yes—
121C23×C4C22×C412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4#4114yes—
129C23×C4C2411yescharacteristic
130C2×M16—44no—
131C2×M16—84no—
132C22×C8—24no—
133#121—28no—
134#121—28no—
135#121—44no—
136#121—44no—
137#117—416no—
138#117—48no—
139#117—48no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—48no—
155C2×C4⋊C4—84no—
156C2×C4⋊C4—84no—
157C2×C4⋊C4C22⋊C418yes—
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4C22×C414yes—
160C2×C4⋊C4C2×D414yes—
161C2×C4⋊C4C2×D418yes—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—48no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—84no—
175C2×C42—28no—
176C2×C42—24no—
177C2×C42—48no—
178C2×C42—82no—
179#102—88no—
180C22⋊C8—24no—
181C22⋊C8—24no—
182C22⋊C8—44no—
183#95—28no—
184#95—28no—
185#95—28no—
186#95—24no—
187#95—24no—
188#95—22no—
189#95C22×C414yes—
190C24—22no—
191C24—22no—
192C24—24no—
193C24—22no—
194C24—42no—
195C24—44no—
196C24—44no—
197C24—41no—
198C24—41no—
199C24—42no—
200C24—42no—
201C24C2×C22⋊C412yes—
202C24#12011yescharacteristic
203C24#12011yescharacteristic
204C24#12312yes—
205C24C22×D411yescharacteristic
206C2×D4—24no—
207C2×D4—48no—
208C2×D4—416no—
209C2×D4—44no—
210C2×D4—44no—
211C2×D4—42no—
212C2×D4—82no—
213C2×D4—84no—
214C2×D4—82no—
215C2×D4—88no—
216C2×D4—84no—
217C2×D4—84no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—216no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—216no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—28no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—48no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—48no—
247C22×C4—44no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—48no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—84no—
256C22×C4—84no—
257C22×C4—82no—
258C22×C4—84no—
259C22×C4C2×C22⋊C414yes—
260C22×C4C2×C22⋊C414yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C4×D414yes—
263C22×C4C4×D414yes—
264C22×C4#12012yes—
265C22×C4#12114yes—
266C22×C4#12114yes—
267C22×C4#12312yes—
268C22×C4AGL(1,ℤ/8)14yes—
269C22×C4#13714yes—
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D412yes—
272C22×C4C22×D411yescharacteristic
273C22×C4C2×#4112yes—
274M16—88no—
275C2×C8—48no—
276C2×C8—44no—
277C4⋊C4—28no—
278C4⋊C4—28no—
279C4⋊C4—216no—
280C4⋊C4—24no—
281C4⋊C4—48no—
282C4⋊C4—416no—
283C4⋊C4—42no—
284C4⋊C4—84no—
285C4⋊C4—88no—
286C22⋊C4—48no—
287C22⋊C4—48no—
288C22⋊C4—48no—
289C22⋊C4—48no—
290C22⋊C4—416no—
291C22⋊C4—88no—
292C42—416no—
293C42—84no—
294C23—21no—
295C23—21no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—24no—
300C23—24no—
301C23—24no—
302C23—24no—
303C23—21no—
304C23—21no—
305C23—22no—
306C23—22no—
307C23—42no—
308C23—42no—
309C23—48no—
310C23—48no—
311C23—44no—
312C23—44no—
313C23—44no—
314C23—42no—
315C23—82no—
316C23—82no—
317C23—88no—
318C23—88no—
319C23—82no—
320C23—81no—
321C23—81no—
322C23—82no—
323C23—82no—
324C23#38612yes—
325C23#38612yes—
326C23#45414yes—
327C23C22×C22⋊C411yescharacteristic
328C23C2×#12011yescharacteristic
329C23C2×#12311yescharacteristic
330C23C2×AGL(1,ℤ/8)12yes—
331C23C2×#13712yes—
332D4—48no—
333D4—84no—
334D4—88no—
335D4—88no—
336C2×C4—216no—
337C2×C4—216no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—48no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—48no—
364C2×C4—48no—
365C2×C4—48no—
366C2×C4—42no—
367C2×C4—82no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—816no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4#38614yes—
375C2×C4#41718yes—
376C2×C4C2×C4×D412yes—
377C2×C4C2×#12112yes—
378C8—84no—
379C22—24no—
380C22—24no—
381C22—24no—
382C22—24no—
383C22—22no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—21no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—48no—
393C22—48no—
394C22—42no—
395C22—42no—
396C22—88no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—164no—
401C22—162no—
402C22—162no—
403C22#188814yes—
404C22C2×#38611yescharacteristic
405C22C2×#41712yes—
406C22#300311yescharacteristic
407C22#300411yescharacteristic
408C22C2×#45412yes—
409C22#302612yes—
410C22#309212yes—
411C4—28no—
412C4—22no—
413C4—22no—
414C4—24no—
415C4—44no—
416C4—84no—
417C4—88no—
418C4—82no—
419C2—22no—
420C2—22no—
421C2—24no—
422C2—42no—
423C2—162no—
424C2#1832412yes—
425C2#1832612yes—
426C2#2090911yescharacteristic
427C2C2×#188811yescharacteristic
428C2#2096011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2316C2semidirect
4. C2×#1463C2semidirect
5. #9023C2semidirect
6. #9019C2semidirect
7. #8019C2semidirect
13. #2591C4semidirect
71. #400C2×C4semidirect
73. #394C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25912
C2×#23162
#217642
C2×#14632
#90232
#90192
#80192