Finite Group Databasebeta

← #393500

Subgroups — #393500

Order 512.

2128 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2501C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic, complemented
3#20912C212yesmaximal
4C2×#1821C211yesmaximal, characteristic, complemented
5#10958C214yesmaximal, complemented
6#10891C214yesmaximal, complemented
7#10733C212yesmaximal, complemented
8C22×#123—22no—
9C22×#121C2211yescharacteristic
10C22×#117C2212yes—
11C23×C4⋊C4C2211yescharacteristic
12C2×#418C2214yes—
13C2×#407C2212yes—
14#2501—24no—
15#2501C418yescomplemented
16C2×#399—22no—
17C2×#390C2212yes—
18C2×#384C2212yes—
19#2274—24no—
20#2274—22no—
21#2274C2214yes—
22C2×C4×C4⋊C4—24no—
23C2×C4×C22⋊C4—22no—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1875—24no—
26#1874—24no—
27#1821—24no—
28C2×#343C2214yes—
29#1736C2214yes—
30C2×#339C2218yes—
31C23×D4—21no—
32C23×D4C2311yescharacteristic
33C24×C4D412yes—
34C24×C4C2311yescharacteristic
35C22×M16C2314yes—
36C2×#123—216no—
37C2×#123—416no—
38C2×#121—22no—
39C2×#121—22no—
40C2×#121—22no—
41C2×#121—24no—
42C2×#121C2×C414yes—
43C2×#117—22no—
44C2×#117—22no—
45C2×#117—216no—
46C2×#117D414yes—
47C2×#117C2314yes—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4C2×C414yes—
53C22×C4⋊C4D414yes—
54C22×C4⋊C4D412yes—
55C22×C4⋊C4D414yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—24no—
62C22×C42—22no—
63C22×C42—24no—
64C22×C42C2312yes—
65#418—216no—
66C2×#102—28no—
67#407—24no—
68#407—24no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71#399—216no—
72#390—24no—
73#390C2×C418yescomplemented
74#384—24no—
75#384C2×C418yescomplemented
76C4×C4⋊C4—216no—
77C4×C4⋊C4—216no—
78C4×C22⋊C4—28no—
79C4×C22⋊C4—22no—
80C4×C22⋊C4—28no—
81C4×C22⋊C4C2×C414yes—
82C2×#95—24no—
83C2×#95—21no—
84#343—28no—
85#340—28no—
86#339—216no—
87C25—21no—
88C25C2×D411yescharacteristic
89C22×D4—24no—
90C22×D4—22no—
91C22×D4—21no—
92C22×D4—22no—
93C22×D4—28no—
94C22×D4—48no—
95C22×D4—41no—
96C22×D4—41no—
97C22×D4—42no—
98C22×D4C22×C412yes—
99C23×C4—28no—
100C23×C4—24no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—24no—
105C23×C4—21no—
106C23×C4—48no—
107C23×C4C22⋊C418yes—
108C23×C4C22×C412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D414yes—
114C23×C4#4114yes—
115C23×C4C2411yescharacteristic
116C2×M16—24no—
117C2×M16—28no—
118C2×M16—24no—
119C2×M16—28no—
120C22×C8—24no—
121#123—464no—
122#121—28no—
123#121—28no—
124#121—44no—
125#121—44no—
126#117—216no—
127#117—216no—
128#117—28no—
129#117—216no—
130#117#41116yes—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—216no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—216no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—416no—
146C2×C4⋊C4C22⋊C418yes—
147C2×C4⋊C4C22×C412yes—
148C2×C4⋊C4C22×C414yes—
149C2×C4⋊C4C2×D414yes—
150C2×C4⋊C4C2×D418yes—
151C2×C22⋊C4—216no—
152C2×C22⋊C4—28no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—216no—
158C2×C22⋊C4—416no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C42—216no—
162C2×C42—24no—
163C2×C42—24no—
164C2×C42—28no—
165C2×C42—22no—
166C2×C42—216no—
167C2×C42—88no—
168C2×C42C22×C414yes—
169#102—416no—
170C22⋊C8—44no—
171C22⋊C8—44no—
172#95—216no—
173#95—24no—
174C24—24no—
175C24—24no—
176C24—22no—
177C24—42no—
178C24—44no—
179C24—44no—
180C24—41no—
181C24—41no—
182C24—44no—
183C24C2×C22⋊C412yes—
184C24#12011yescharacteristic
185C24#12011yescharacteristic
186C24#12312yes—
187C24C22×D411yescharacteristic
188C2×D4—24no—
189C2×D4—28no—
190C2×D4—416no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—42no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—48no—
197C2×D4—44no—
198C2×D4—88no—
199C2×D4—84no—
200C2×D4—84no—
201C22×C4—24no—
202C22×C4—28no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—216no—
208C22×C4—28no—
209C22×C4—216no—
210C22×C4—28no—
211C22×C4—24no—
212C22×C4—28no—
213C22×C4—24no—
214C22×C4—28no—
215C22×C4—24no—
216C22×C4—28no—
217C22×C4—22no—
218C22×C4—44no—
219C22×C4—48no—
220C22×C4—44no—
221C22×C4—48no—
222C22×C4—48no—
223C22×C4—48no—
224C22×C4—416no—
225C22×C4—44no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—88no—
229C22×C4—84no—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C418yes—
232C22×C4C4×D414yes—
233C22×C4C4×D414yes—
234C22×C4#12012yes—
235C22×C4#12114yes—
236C22×C4#12114yes—
237C22×C4#12312yes—
238C22×C4C23×C411yescharacteristic
239C22×C4C22×D412yes—
240C22×C4C22×D411yescharacteristic
241C22×C4C2×#4112yes—
242C22×C4C2×#4118yes—
243M16—48no—
244M16—48no—
245C2×C8—44no—
246C2×C8—44no—
247C2×C8—44no—
248C4⋊C4—216no—
249C4⋊C4—28no—
250C4⋊C4—28no—
251C4⋊C4—28no—
252C4⋊C4—216no—
253C4⋊C4—24no—
254C4⋊C4—432no—
255C4⋊C4—42no—
256C4⋊C4#123116yescomplemented
257C22⋊C4—216no—
258C22⋊C4—216no—
259C22⋊C4—216no—
260C22⋊C4—432no—
261C22⋊C4—48no—
262C42—216no—
263C42—216no—
264C42—28no—
265C42—44no—
266C42—816no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—24no—
273C23—24no—
274C23—24no—
275C23—22no—
276C23—22no—
277C23—24no—
278C23—42no—
279C23—42no—
280C23—416no—
281C23—44no—
282C23—44no—
283C23—42no—
284C23—44no—
285C23—82no—
286C23—82no—
287C23—816no—
288C23—82no—
289C23—82no—
290C23—82no—
291C23—82no—
292C23#38612yes—
293C23#38612yes—
294C23#45314yes—
295C23C22×C22⋊C411yescharacteristic
296C23C2×#12011yescharacteristic
297C23C2×#12311yescharacteristic
298C23#52514yes—
299D4—48no—
300D4—88no—
301D4—88no—
302D4—84no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—28no—
308C2×C4—28no—
309C2×C4—28no—
310C2×C4—28no—
311C2×C4—28no—
312C2×C4—216no—
313C2×C4—28no—
314C2×C4—44no—
315C2×C4—416no—
316C2×C4—432no—
317C2×C4—416no—
318C2×C4—48no—
319C2×C4—48no—
320C2×C4—48no—
321C2×C4—44no—
322C2×C4—48no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—84no—
326C2×C4—88no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4#38614yes—
330C2×C4C2×C4×D412yes—
331C2×C4C4×#4114yes—
332C2×C4C2×#12112yes—
333C2×C4C2×#12314yes—
334C2×C4#52514yes—
335C2×C4#52918yes—
336C2×C4#53214yes—
337C8—84no—
338C22—22no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—28no—
345C22—28no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—48no—
350C22—48no—
351C22—44no—
352C22—88no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22—164no—
357C22—164no—
358C22#188714yes—
359C22C2×#38611yescharacteristic
360C22#229712yes—
361C22#284912yes—
362C22#285312yes—
363C22C2×#45312yes—
364C22#302012yes—
365C4—24no—
366C4—48no—
367C4—48no—
368C4—416no—
369C4—82no—
370C4C4×#12314yes—
371C4#285414yes—
372C2—22no—
373C2—22no—
374C2—24no—
375C2—42no—
376C2—162no—
377C2#1470611yescharacteristic
378C2C2×#188711yescharacteristic
379C2#2095411yescharacteristic
380C2#3299614yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2274C2semidirect
4. C2×#1821C2semidirect
5. #10958C2semidirect
6. #10891C2semidirect
7. #10733C2semidirect
15. #2501C4semidirect
73. #390C2×C4semidirect
75. #384C2×C4semidirect
256. C4⋊C4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#25012
C2×#22742
#209122
C2×#18212
#109582
#108912
#107332