Finite Group Databasebeta

← #194832

Subgroups — #194832

Order 512.

1426 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic
3#21759C212yesmaximal
4C22×#324C211yesmaximal, characteristic
5#9022C214yesmaximal, complemented
6#9014C214yesmaximal, complemented
7#8023C212yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11#2627—24no—
12#2627C418yescomplemented
13C2×#400C2212yes—
14C2×#398—22no—
15C2×#395C2212yes—
16#2313—24no—
17#2313—22no—
18#2313C2214yes—
19C2×#387—24no—
20C2×#385—21no—
21C2×#381—21no—
22C2×#381—21no—
23C2×#380C2211yescharacteristic
24C2×#380C2212yes—
25C2×C4×C22⋊C4—21no—
26#1892—24no—
27#1872—24no—
28#1824C2214yes—
29#1787C2214yes—
30C2×#324—24no—
31C2×#324C2214yes—
32#1294C2218yes—
33C23×Q8C2311yescharacteristic
34C24×C4D411yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4C2311yescharacteristic
37C23×C8—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8C2×C414yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—24no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2×C414yes—
50C22×C4⋊C4D412yes—
51C22×C4⋊C4D414yes—
52C22×C4⋊C4D414yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C42—22no—
59C2×#103—28no—
60C2×C22⋊C8—24no—
61C2×C22⋊C8—24no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8C2314yes—
64#400—24no—
65#400C2×C418yescomplemented
66#398—216no—
67#395—24no—
68#395C2×C418yescomplemented
69#387—216no—
70#387—216no—
71#385—28no—
72#381—24no—
73#381—28no—
74#381—24no—
75#380—22no—
76#380—22no—
77#380—22no—
78#380C2×C414yes—
79#380D414yes—
80#380C2314yes—
81C4×C22⋊C4—24no—
82C4×C22⋊C4—24no—
83C2×#95—22no—
84C2×#95—24no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95C2312yes—
90#340—28no—
91#324—28no—
92C25C2×D411yescharacteristic
93C22×Q8—21no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8—82no—
97C22×Q8C22×C412yes—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4—24no—
103C23×C4—24no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—24no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—82no—
111C23×C4—82no—
112C23×C4C22⋊C414yes—
113C23×C4C22⋊C414yes—
114C23×C4C22×C412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D412yes—
121C23×C4#4114yes—
122C23×C4C2411yescharacteristic
123C22×C8—24no—
124C22×C8—44no—
125C22×C8—84no—
126C22⋊Q8—28no—
127C22⋊Q8—28no—
128C22⋊Q8—44no—
129C22⋊Q8—44no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—84no—
148C2×C4⋊C4—82no—
149C2×C4⋊C4—84no—
150C2×C4⋊C4C22⋊C418yes—
151C2×C4⋊C4C22×C412yes—
152C2×C4⋊C4C22×C414yes—
153C2×C4⋊C4C2×D414yes—
154C2×C4⋊C4C2×D418yes—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—44no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4—44no—
165C2×C42—28no—
166C2×C42—84no—
167#103—88no—
168C22⋊C8—24no—
169C22⋊C8—24no—
170C22⋊C8—44no—
171#95—28no—
172#95—216no—
173#95—28no—
174#95—28no—
175#95—28no—
176#95—24no—
177#95—24no—
178#95—22no—
179#95C22×C414yes—
180C24—42no—
181C24—42no—
182C24C2×C22⋊C412yes—
183C24#12011yescharacteristic
184C24#12011yescharacteristic
185C24#12312yes—
186C24#13514yes—
187C24#13714yes—
188C24C22×D411yescharacteristic
189C2×Q8—24no—
190C2×Q8—48no—
191C2×Q8—42no—
192C2×Q8—82no—
193C2×Q8—82no—
194C2×Q8—84no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—28no—
204C22×C4—28no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—28no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—48no—
219C22×C4—44no—
220C22×C4—48no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—48no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—44no—
231C22×C4—48no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—84no—
235C22×C4—84no—
236C22×C4—82no—
237C22×C4—84no—
238C22×C4C2×C22⋊C414yes—
239C22×C4C2×C22⋊C414yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C4×D414yes—
242C22×C4C4×D414yes—
243C22×C4#12012yes—
244C22×C4#12114yes—
245C22×C4#12114yes—
246C22×C4#12312yes—
247C22×C4C23×C411yescharacteristic
248C22×C4C22×D412yes—
249C22×C4C22×D411yescharacteristic
250C22×C4C2×#4112yes—
251C2×C8—48no—
252C2×C8—44no—
253C2×C8—88no—
254C4⋊C4—28no—
255C4⋊C4—28no—
256C4⋊C4—216no—
257C4⋊C4—24no—
258C4⋊C4—48no—
259C4⋊C4—42no—
260C4⋊C4—48no—
261C4⋊C4—84no—
262C4⋊C4—84no—
263C4⋊C4—88no—
264C22⋊C4—44no—
265C22⋊C4—44no—
266C22⋊C4—44no—
267C22⋊C4—44no—
268C22⋊C4—48no—
269C22⋊C4—48no—
270C42—88no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—48no—
282C23—48no—
283C23#38612yes—
284C23#38612yes—
285C23#41614yes—
286C23#41914yes—
287C23#45214yes—
288C23#45214yes—
289C23#48518yes—
290C23C22×C22⋊C411yescharacteristic
291C23C2×#12011yescharacteristic
292C23C2×#12311yescharacteristic
293C23C2×#13512yes—
294C23C2×#13712yes—
295Q8—48no—
296Q8—84no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—48no—
313C2×C4—44no—
314C2×C4—48no—
315C2×C4—48no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—48no—
325C2×C4—48no—
326C2×C4—48no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—82no—
330C2×C4—84no—
331C2×C4—88no—
332C2×C4—816no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4#38614yes—
338C2×C4C2×C4×D412yes—
339C2×C4C2×#12112yes—
340C8—84no—
341C22—24no—
342C22—28no—
343C22—24no—
344C22—28no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—22no—
353C22—44no—
354C22—44no—
355C22—48no—
356C22#187314yes—
357C22#187314yes—
358C22C2×#38611yescharacteristic
359C22C2×#41611yescharacteristic
360C22C2×#41911yescharacteristic
361C22C2×#45212yes—
362C22#302012yes—
363C22#302612yes—
364C22C2×#48512yes—
365C4—22no—
366C4—22no—
367C4—24no—
368C4—44no—
369C4—44no—
370C4—84no—
371C4—88no—
372C4—82no—
373C4—82no—
374C2—21no—
375C2—22no—
376C2—21no—
377C2—24no—
378C2—24no—
379C2#1828914yes—
380C2C2×#187311yescharacteristic
381C2#2095411yescharacteristic
382C2#2096011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
5. #9022C2semidirect
6. #9014C2semidirect
12. #2627C4semidirect
65. #400C2×C4semidirect
68. #395C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
C2×#23132
#217592
C22×#3242
#90222
#90142
#80232