Finite Group Databasebeta

← #205023

Subgroups — #205023

Order 512.

1244 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35011C211yesmaximal, characteristic
2#21934C211yesmaximal, characteristic
3C2×#1511C212yesmaximal
4C2×#1508C212yesmaximal
5#13807C211yesmaximal, characteristic
6#9714C214yesmaximal, complemented
7#9711C214yesmaximal, complemented
8C22×C22⋊Q8C2212yes—
9C2×#407C2212yes—
10#2857C2214yes—
11#2659—21no—
12#2659—21no—
13#2583—24no—
14#2502C414yescomplemented
15#2434C414yescomplemented
16#2315—22no—
17C2×#381C2211yescharacteristic
18C2×C4×C22⋊C4C2211yescharacteristic
19C2×C4×C22⋊C4C2211yescharacteristic
20#1851—24no—
21#1797C2214yes—
22#1511—28no—
23#1508—28no—
24C2×#328C2214yes—
25C2×#328C2214yes—
26C2×#328C2218yes—
27#1437—22no—
28#1434—21no—
29#1434C2212yes—
30C4×#95—21no—
31C4×#95C2212yes—
32C23×Q8C2×C412yes—
33C24×C4C2311yescharacteristic
34C22×M16—22no—
35C2×C4⋊Q8C2×C414yes—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—24no—
39C2×C22⋊Q8—24no—
40C2×C22⋊Q8—24no—
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8C2×C414yes—
43C2×C22⋊Q8C2314yes—
44C22×C4⋊C4—22no—
45C22×C4⋊C4C2×C412yes—
46C22×C22⋊C4—22no—
47C22×C22⋊C4—21no—
48C22×C22⋊C4—21no—
49C22×C22⋊C4D412yes—
50C22×C42—21no—
51C22×C42—22no—
52C22×C42C2312yes—
53C2×C4⋊C8—24no—
54#407—24no—
55#407—24no—
56C2×C22⋊C8—22no—
57C2×C22⋊C8C2314yes—
58#398—24no—
59#398—24no—
60#395—24no—
61#395—24no—
62#391C2×C418yes—
63#389—24no—
64#389—24no—
65#385—28no—
66#385—28no—
67#384C2×C414yes—
68#381—22no—
69#381D414yes—
70#380—24no—
71#380—24no—
72C4×C22⋊C4—22no—
73C4×C22⋊C4C2×C414yes—
74C4×C22⋊C4D414yes—
75C4×C22⋊C4D414yes—
76C2×#95—21no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95D411yescharacteristic
81C2×#95D411yescharacteristic
82C2×#95C2311yescharacteristic
83C2×#95C2312yes—
84C2×#95C2311yescharacteristic
85#336—24no—
86#328—28no—
87#328—28no—
88#328—216no—
89C25C2×D411yescharacteristic
90C22×Q8—22no—
91C22×Q8—24no—
92C22×Q8—24no—
93C22×Q8—84no—
94C22×Q8C22×C414yes—
95C23×C4—21no—
96C23×C4—22no—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—82no—
101C23×C4C22×C412yes—
102C23×C4C2×D412yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4#4112yes—
109C23×C4#4112yes—
110C23×C4C2411yescharacteristic
111C2×M16—44no—
112C2×M16—84no—
113C22×C8—24no—
114C4⋊Q8—84no—
115C22⋊Q8—28no—
116C22⋊Q8—28no—
117C22⋊Q8—24no—
118C22⋊Q8—24no—
119C22⋊Q8—24no—
120C22⋊Q8—24no—
121C22⋊Q8—44no—
122C22⋊Q8—44no—
123C22⋊Q8—44no—
124C22⋊Q8—44no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4C22×C414yes—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—44no—
146C2×C22⋊C4—44no—
147C2×C22⋊C4—82no—
148C2×C22⋊C4C22⋊C418yes—
149C2×C42—24no—
150C2×C42—44no—
151C2×C42—44no—
152C2×C42—82no—
153C2×C42—82no—
154C2×C42—82no—
155C2×C42C22×C414yes—
156C2×C42C2×D414yes—
157C4⋊C8—84no—
158C22⋊C8—24no—
159C22⋊C8—24no—
160C22⋊C8—44no—
161#95—28no—
162#95—28no—
163#95—24no—
164#95—21no—
165#95—21no—
166#95—22no—
167#95—44no—
168#95—44no—
169#95—82no—
170#95C22⋊C414yes—
171#95C22⋊C414yes—
172#95C2×D412yes—
173#95C2×D412yes—
174#95C2×D414yes—
175C24—22no—
176C24—22no—
177C24—41no—
178C24—41no—
179C24—42no—
180C24C2×C22⋊C412yes—
181C24#12112yes—
182C24#12112yes—
183C24C22×D411yescharacteristic
184C2×Q8—24no—
185C2×Q8—24no—
186C2×Q8—48no—
187C2×Q8—44no—
188C2×Q8—84no—
189C2×Q8—88no—
190C2×Q8—88no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4—84no—
229C22×C4C2×C22⋊C414yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C414yes—
232C22×C4C4×D414yes—
233C22×C4C4×D414yes—
234C22×C4#12112yes—
235C22×C4#12112yes—
236C22×C4#12414yes—
237C22×C4#12714yes—
238C22×C4#13714yes—
239C22×C4#13714yes—
240C22×C4C23×C411yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D411yescharacteristic
243C22×C4C22×D411yescharacteristic
244C22×C4C2×#4111yescharacteristic
245C22×C4C2×#4111yescharacteristic
246M16—88no—
247C2×C8—48no—
248C2×C8—44no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—48no—
255C4⋊C4—84no—
256C4⋊C4—84no—
257C4⋊C4—84no—
258C4⋊C4—84no—
259C22⋊C4—216no—
260C22⋊C4—48no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—48no—
264C22⋊C4—84no—
265C42—44no—
266C42—44no—
267C42—84no—
268C42—88no—
269C23—22no—
270C23—21no—
271C23—21no—
272C23—24no—
273C23—24no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—24no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—82no—
284C23—82no—
285C23—82no—
286C23—82no—
287C23#39014yes—
288C23C22×C22⋊C411yescharacteristic
289C23C2×#12111yescharacteristic
290C23C2×#12111yescharacteristic
291C23C2×#13712yes—
292C23C2×#13712yes—
293Q8—84no—
294Q8—84no—
295Q8—88no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—48no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—88no—
341C2×C4—88no—
342C2×C4#39014yes—
343C2×C4#41918yes—
344C2×C4C2×C4×D411yescharacteristic
345C2×C4C2×C4×D411yescharacteristic
346C2×C4C2×#12411yescharacteristic
347C2×C4C2×#12711yescharacteristic
348C8—84no—
349C22—22no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—22no—
354C22—22no—
355C22—21no—
356C22—21no—
357C22—24no—
358C22—24no—
359C22—24no—
360C22—44no—
361C22—44no—
362C22—82no—
363C22—82no—
364C22—84no—
365C22C2×#39011yescharacteristic
366C22#288312yes—
367C22C2×#41912yes—
368C22#290111yescharacteristic
369C22#290111yescharacteristic
370C22#303911yescharacteristic
371C22#303912yes—
372C22#303911yescharacteristic
373C4—28no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—24no—
379C4—44no—
380C4—44no—
381C4—82no—
382C4—84no—
383C4—84no—
384C4—84no—
385C2—21no—
386C2—22no—
387C2—21no—
388C2—44no—
389C2#1762112yes—
390C2#1762712yes—
391C2#2121011yescharacteristic
392C2#2121311yescharacteristic
393C2#2125911yescharacteristic
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
14. #2502C4semidirect
15. #2434C4semidirect

Maximal subgroups

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

subgroupindex
#350112
#219342
C2×#15112
C2×#15082
#138072
#97142
#97112