Finite Group Databasebeta

← #204987

Subgroups — #204987

Order 512.

1146 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35173C211yesmaximal, characteristic
2#21951C211yesmaximal, characteristic
3C2×#1518C212yesmaximal
4C2×#1503C212yesmaximal
5#13804C211yesmaximal, characteristic
6#9714C214yesmaximal
7#9711C214yesmaximal
8C2×#533C2212yes—
9C2×#431C2212yes—
10#2857C2214yes—
11#2659—22no—
12#2584—24no—
13#2567C414yescomplemented
14#2546C414yescomplemented
15#2316—21no—
16#2316—21no—
17C2×#382C2211yescharacteristic
18C2×C4×C4⋊C4—21no—
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#1939—24no—
22#1797C2214yes—
23#1518—28no—
24#1503—28no—
25C2×#328C2214yes—
26C2×#328C2214yes—
27C2×#328C2218yes—
28#1436—22no—
29C4⋊#95—21no—
30#1434C2212yes—
31C4×#95C2212yes—
32#533—24no—
33#533—24no—
34#533—24no—
35#533—24no—
36C2×C4⋊Q8C2×C412yes—
37C2×C4⋊Q8C2×C414yes—
38C2×C4⋊Q8C2×C412yes—
39C2×#125—22no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8C2314yes—
42C2×C4×Q8—22no—
43C2×C4×Q8C2×C414yes—
44C2×#117—22no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4D411yescharacteristic
49C22×C42—21no—
50C22×C42—22no—
51C22×C42C2312yes—
52C22×C42C2311yescharacteristic
53#431—24no—
54#431—24no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—24no—
57C2×C22⋊C8C2314yes—
58C2×C4×C8—22no—
59#398—28no—
60#395—28no—
61#391C2×C418yes—
62#389—24no—
63#389—24no—
64#389—24no—
65#389—24no—
66#389—24no—
67#389—24no—
68#384C2×C414yes—
69#382—22no—
70#382D414yes—
71#380—24no—
72#380—24no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—44no—
75C4×C4⋊C4C2×C414yes—
76C4×C4⋊C4D414yes—
77C4×C4⋊C4D414yes—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95D412yes—
83C2×#95C2311yescharacteristic
84C2×#95C2312yes—
85C2×#95C2311yescharacteristic
86#336—24no—
87#328—28no—
88#328—28no—
89#328—216no—
90C22×Q8—22no—
91C22×Q8C22×C414yes—
92C23×C4—21no—
93C23×C4—22no—
94C23×C4—21no—
95C23×C4C2×D412yes—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D412yes—
98C23×C4C2×D411yescharacteristic
99C23×C4C2411yescharacteristic
100C22×C8—24no—
101C4⋊Q8—24no—
102C4⋊Q8—24no—
103C4⋊Q8—24no—
104C4⋊Q8—24no—
105C4⋊Q8—84no—
106#125—44no—
107#125—44no—
108C22⋊Q8—24no—
109C22⋊Q8—24no—
110C22⋊Q8—24no—
111C22⋊Q8—24no—
112C22⋊Q8—44no—
113C22⋊Q8—44no—
114C4×Q8—28no—
115C4×Q8—28no—
116C4×Q8—44no—
117C4×Q8—44no—
118#117—44no—
119#117—44no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4C22⋊C414yes—
138C2×C4⋊C4C22⋊C414yes—
139C2×C4⋊C4SD1614yes—
140C2×C4⋊C4Q1614yes—
141C2×C4⋊C4C22×C414yes—
142C2×C22⋊C4—24no—
143C2×C42—24no—
144C2×C42—24no—
145C2×C42—42no—
146C2×C42—42no—
147C2×C42—44no—
148C2×C42—44no—
149C2×C42—82no—
150C2×C42—82no—
151C2×C42C22×C414yes—
152C2×C42C22×C412yes—
153C2×C42C2×D414yes—
154C2×C42C2×D412yes—
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C2×C42#4112yes—
158C2×C42#4112yes—
159C4⋊C8—44no—
160C4⋊C8—84no—
161C22⋊C8—24no—
162C22⋊C8—24no—
163C4×C8—44no—
164#95—28no—
165#95—28no—
166#95—24no—
167#95—21no—
168#95—21no—
169#95—22no—
170#95—44no—
171#95—44no—
172#95—82no—
173#95C22⋊C418yes—
174#95C2×D412yes—
175#95C2×D412yes—
176#95C2×D414yes—
177C24C22×D411yescharacteristic
178C2×Q8—24no—
179C2×Q8—24no—
180C2×Q8—44no—
181C2×Q8—44no—
182C2×Q8—44no—
183C2×Q8—82no—
184C2×Q8—82no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—22no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4C2×C22⋊C414yes—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C414yes—
221C22×C4C2×C22⋊C412yes—
222C22×C4#12112yes—
223C22×C4#12112yes—
224C22×C4#12112yes—
225C22×C4#12112yes—
226C22×C4C2×SD1614yes—
227C22×C4C2×SD1612yes—
228C22×C4C2×Q1614yes—
229C22×C4C2×Q1612yes—
230C22×C4#13712yes—
231C22×C4#13712yes—
232C22×C4C23×C411yescharacteristic
233C22×C4C22×D411yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4C22×D411yescharacteristic
236C22×C4C2×#4111yescharacteristic
237C22×C4C2×#4111yescharacteristic
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—44no—
241C4⋊C4—28no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—48no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—48no—
250C4⋊C4—44no—
251C4⋊C4—48no—
252C4⋊C4—84no—
253C4⋊C4—84no—
254C4⋊C4—84no—
255C4⋊C4—84no—
256C4⋊C4#103116yes—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C42—44no—
260C42—44no—
261C42—48no—
262C42—48no—
263C42—88no—
264C42C4×D414yes—
265C42C4×D414yes—
266C42#12414yes—
267C42#12714yes—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23C22×C22⋊C411yescharacteristic
273C23C2×#12111yescharacteristic
274C23C2×#12111yescharacteristic
275C23C22×SD1611yescharacteristic
276C23C22×Q1611yescharacteristic
277C23C2×#13711yescharacteristic
278C23C2×#13711yescharacteristic
279Q8—84no—
280Q8—84no—
281Q8—88no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—81no—
324C2×C4—82no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—82no—
328C2×C4—81no—
329C2×C4—82no—
330C2×C4#39014yes—
331C2×C4#39014yes—
332C2×C4C2×#10318yes—
333C2×C4C2×#10314yes—
334C2×C4#41914yes—
335C2×C4#46114yes—
336C2×C4#46114yes—
337C2×C4#46214yes—
338C2×C4#46214yes—
339C2×C4C2×C4×D411yescharacteristic
340C2×C4C2×C4×D411yescharacteristic
341C2×C4C2×#12411yescharacteristic
342C2×C4C2×#12711yescharacteristic
343C8—84no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22C2×#39011yescharacteristic
357C22#288311yescharacteristic
358C22#288411yescharacteristic
359C22C22×#10311yescharacteristic
360C22C2×#41911yescharacteristic
361C22#290111yescharacteristic
362C22#290111yescharacteristic
363C22C2×#46111yescharacteristic
364C22C2×#46111yescharacteristic
365C22C2×#46211yescharacteristic
366C22C2×#46211yescharacteristic
367C4—28no—
368C4—24no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—44no—
374C4—44no—
375C4—82no—
376C4—84no—
377C4—84no—
378C4—84no—
379C4#198118yes—
380C2—21no—
381C2—21no—
382C2—21no—
383C2—21no—
384C2#1761411yescharacteristic
385C2#1761911yescharacteristic
386C2#1762711yescharacteristic
387C2#1763211yescharacteristic
388C2#2121011yescharacteristic
389C2#2121411yescharacteristic
390C2C2×#198111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2567C4semidirect
14. #2546C4semidirect

Maximal subgroups

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

subgroupindex
#351732
#219512
C2×#15182
C2×#15032
#138042
#97142
#97112