Finite Group Databasebeta

← #204986

Subgroups — #204986

Order 512.

1364 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35011C211yesmaximal, characteristic
2#21930C211yesmaximal, characteristic
3C2×#1520C212yesmaximal
4C22×#328C212yesmaximal
5#13802C211yesmaximal, characteristic
6#9714C214yesmaximal, complemented
7#9711C214yesmaximal, complemented
8C22×C22⋊Q8C2212yes—
9C22×C22⋊C8C2212yes—
10#2857C2214yes—
11#2659—22no—
12#2583—24no—
13#2502C414yescomplemented
14#2434C414yescomplemented
15#2315—21no—
16#2315—21no—
17C2×#380C2211yescharacteristic
18C2×C4×C22⋊C4—21no—
19C2×C4×C22⋊C4C2211yescharacteristic
20C22×#95C2211yescharacteristic
21#1850—24no—
22#1797C2214yes—
23#1520—28no—
24C2×#328—28no—
25C2×#328C2214yes—
26C2×#328C2214yes—
27C2×#328C2218yes—
28#1436—22no—
29#1434C2212yes—
30C22⋊#95—21no—
31C4×#95C2212yes—
32C23×Q8C2×C412yes—
33C24×C4—21no—
34C24×C4C2311yescharacteristic
35C23×C8—22no—
36C2×C4⋊Q8C2×C414yescomplemented
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—24no—
40C2×C22⋊Q8—24no—
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8—24no—
43C2×C22⋊Q8C2×C414yes—
44C2×C22⋊Q8C2314yes—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4C2×C412yes—
48C22×C22⋊C4—22no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4D411yescharacteristic
51C22×C22⋊C4D411yescharacteristic
52C22×C42—22no—
53C22×C42C2312yes—
54C2×C4⋊C8—24no—
55C2×C22⋊C8—24no—
56C2×C22⋊C8—24no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8C2314yes—
59#398—28no—
60#395—28no—
61#391C2×C418yescomplemented
62#389—24no—
63#389—24no—
64#385—24no—
65#385—24no—
66#385—24no—
67#385—24no—
68#384C2×C414yescomplemented
69#380—24no—
70#380—24no—
71#380—22no—
72#380D414yes—
73C4×C22⋊C4—44no—
74C4×C22⋊C4C2×C414yes—
75C4×C22⋊C4D414yes—
76C2×#95—21no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95D414yes—
82C2×#95D412yes—
83C2×#95C2311yescharacteristic
84C2×#95C2312yes—
85C2×#95C2311yescharacteristic
86#336—24no—
87#328—28no—
88#328—28no—
89#328—216no—
90C25C2×D411yescharacteristic
91C22×Q8—22no—
92C22×Q8—24no—
93C22×Q8—24no—
94C22×Q8—84no—
95C22×Q8C22×C414yes—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4—21no—
99C23×C4—24no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—44no—
103C23×C4—42no—
104C23×C4—42no—
105C23×C4—82no—
106C23×C4—82no—
107C23×C4C22×C412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4#4112yes—
115C23×C4#4112yes—
116C23×C4C2411yescharacteristic
117C22×C8—24no—
118C22×C8—44no—
119C22×C8—84no—
120C4⋊Q8—84no—
121C22⋊Q8—28no—
122C22⋊Q8—28no—
123C22⋊Q8—24no—
124C22⋊Q8—24no—
125C22⋊Q8—24no—
126C22⋊Q8—24no—
127C22⋊Q8—44no—
128C22⋊Q8—44no—
129C22⋊Q8—44no—
130C22⋊Q8—44no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—82no—
144C2×C4⋊C4C22×C414yes—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4C22⋊C414yes—
155C2×C22⋊C4C22⋊C414yes—
156C2×C22⋊C4SD1614yes—
157C2×C22⋊C4Q1614yes—
158C2×C42—44no—
159C2×C42—44no—
160C2×C42—82no—
161C2×C42—82no—
162C2×C42C22×C414yes—
163C2×C42C2×D414yes—
164C4⋊C8—84no—
165C22⋊C8—24no—
166C22⋊C8—24no—
167C22⋊C8—44no—
168#95—28no—
169#95—28no—
170#95—24no—
171#95—21no—
172#95—21no—
173#95—22no—
174#95—44no—
175#95—44no—
176#95—82no—
177#95C22⋊C418yes—
178#95C2×D412yes—
179#95C2×D412yes—
180#95C2×D414yes—
181C24—42no—
182C24—42no—
183C24C2×C22⋊C412yes—
184C24#12112yes—
185C24#12112yes—
186C24C2×SD1612yes—
187C24C2×Q1612yes—
188C24#13712yes—
189C24#13712yes—
190C24C22×D411yescharacteristic
191C2×Q8—24no—
192C2×Q8—24no—
193C2×Q8—48no—
194C2×Q8—44no—
195C2×Q8—84no—
196C2×Q8—88no—
197C2×Q8—88no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—48no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—48no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4—84no—
236C22×C4—84no—
237C22×C4—84no—
238C22×C4—88no—
239C22×C4—88no—
240C22×C4C2×C22⋊C414yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C414yes—
243C22×C4C4×D414yes—
244C22×C4C4×D414yes—
245C22×C4#12112yes—
246C22×C4#12112yes—
247C22×C4#12414yes—
248C22×C4#12714yes—
249C22×C4C2×SD1614yes—
250C22×C4C2×Q1614yes—
251C22×C4C23×C411yescharacteristic
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C22×C4C2×#4111yescharacteristic
256C22×C4C2×#4111yescharacteristic
257C2×C8—48no—
258C2×C8—44no—
259C2×C8—88no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—48no—
266C4⋊C4—84no—
267C4⋊C4—84no—
268C4⋊C4—84no—
269C4⋊C4—84no—
270C4⋊C4—84no—
271C22⋊C4—28no—
272C22⋊C4—44no—
273C22⋊C4—44no—
274C22⋊C4—48no—
275C22⋊C4—48no—
276C22⋊C4#103116yes—
277C42—44no—
278C42—44no—
279C42—88no—
280C23—22no—
281C23—24no—
282C23—22no—
283C23—24no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—44no—
288C23—44no—
289C23—42no—
290C23—44no—
291C23—44no—
292C23#39014yes—
293C23C2×#10314yes—
294C23#41914yes—
295C23#46114yes—
296C23#46114yes—
297C23#46214yes—
298C23#46214yes—
299C23C22×C22⋊C411yescharacteristic
300C23C2×#12111yescharacteristic
301C23C2×#12111yescharacteristic
302C23C22×SD1611yescharacteristic
303C23C22×Q1611yescharacteristic
304C23C2×#13711yescharacteristic
305C23C2×#13711yescharacteristic
306Q8—84no—
307Q8—84no—
308Q8—88no—
309C2×C4—24no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—48no—
339C2×C4—48no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—816no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—88no—
354C2×C4—88no—
355C2×C4#39014yes—
356C2×C4C2×#10318yes—
357C2×C4C2×C4×D411yescharacteristic
358C2×C4C2×C4×D411yescharacteristic
359C2×C4C2×#12411yescharacteristic
360C2×C4C2×#12711yescharacteristic
361C8—84no—
362C22—22no—
363C22—22no—
364C22—28no—
365C22—22no—
366C22—22no—
367C22—28no—
368C22—21no—
369C22—21no—
370C22—24no—
371C22—21no—
372C22—21no—
373C22—24no—
374C22—44no—
375C22—44no—
376C22—44no—
377C22—44no—
378C22#198118yes—
379C22C2×#39011yescharacteristic
380C22#288312yes—
381C22C22×#10311yescharacteristic
382C22C2×#41911yescharacteristic
383C22#290112yes—
384C22C2×#46111yescharacteristic
385C22C2×#46111yescharacteristic
386C22C2×#46211yescharacteristic
387C22C2×#46211yescharacteristic
388C4—28no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—24no—
394C4—44no—
395C4—44no—
396C4—84no—
397C4—82no—
398C4—84no—
399C4—84no—
400C2—21no—
401C2—21no—
402C2—28no—
403C2—21no—
404C2—21no—
405C2#1761412yes—
406C2#1762112yes—
407C2#2121011yescharacteristic
408C2#2121311yescharacteristic
409C2C2×#198111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9714C2semidirect
7. #9711C2semidirect
13. #2502C4semidirect
14. #2434C4semidirect
36. C2×C4⋊Q8C2×C4semidirect
61. #391C2×C4semidirect
68. #384C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350112
#219302
C2×#15202
C22×#3282
#138022
#97142
#97112