Finite Group Databasebeta

← #215643

Subgroups — #215643

Order 512.

766 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2606C211yesmaximal, characteristic
2#21914C211yesmaximal, characteristic
3#21914C211yesmaximal, characteristic
4#9883C212yesmaximal
5#9881C212yesmaximal
6#9880C212yesmaximal
7#9879C212yesmaximal
8#8705C212yesmaximal
9#8699C212yesmaximal
10#2606—22no—
11#2606—22no—
12#2606—22no—
13#2606—22no—
14C2×#398C412yes—
15C2×#398C412yes—
16C2×#389C412yes—
17C2×#389C2212yes—
18C2×#387C412yes—
19C2×#387C2212yes—
20C2×#383C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×#376C2211yescharacteristic
23#1917—22no—
24#1917—22no—
25#1916—22no—
26#1916—22no—
27#1817C2212yes—
28#1817C2212yes—
29#1797C2212yes—
30#1797C2212yes—
31#1749C2211yescharacteristic
32#1749C2211yescharacteristic
33C4×C22⋊C8C2211yescharacteristic
34C4×C22⋊C8C2211yescharacteristic
35#1280C2212yes—
36#1280C2212yes—
37#1280C2212yes—
38#1280C2212yes—
39#1280C2212yes—
40#1280C2212yes—
41#1280C2212yes—
42#1280C2212yes—
43C22×C4⋊C4C2×C412yes—
44C22×C4⋊C4C2×C412yes—
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C412yes—
50C22×C42C2×C412yes—
51C22×C42C2311yescharacteristic
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C2×C4⋊C8—22no—
55C2×C4⋊C8—22no—
56C2×C22⋊C8C2312yes—
57C2×C22⋊C8C2312yes—
58C2×C22⋊C8C2312yes—
59C2×C22⋊C8C2312yes—
60C2×#97—21no—
61C2×#97—21no—
62C2×C4×C8—21no—
63C2×C4×C8—21no—
64#398—22no—
65#398—22no—
66#398—22no—
67#398—22no—
68#398—22no—
69#398—22no—
70#398—22no—
71#398—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#383—22no—
89#383—22no—
90C4×C4⋊C4D412yes—
91C4×C4⋊C4D412yes—
92C4×C4⋊C4D412yes—
93C4×C4⋊C4D412yes—
94#376—22no—
95#376—22no—
96C2×#95C2×C411yescharacteristic
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C412yes—
99C2×#95C2×C412yes—
100C2×#95C2×C412yes—
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C412yes—
104C2×#95C2312yes—
105C2×#95C2312yes—
106#336—22no—
107#336—22no—
108#336—24no—
109#336—24no—
110#336—21no—
111#336—21no—
112#336—21no—
113#336—21no—
114C23×C4C22×C411yescharacteristic
115C23×C4C22×C412yes—
116C23×C4C22×C412yes—
117C23×C4C22×C412yes—
118C23×C4C22×C412yes—
119C23×C4C22×C412yes—
120C23×C4C22×C411yescharacteristic
121C23×C4C22×C412yes—
122C23×C4C2411yescharacteristic
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—82no—
147C2×C4⋊C4—82no—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C22⋊C412yes—
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22⋊C412yes—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—24no—
157C2×C42—42no—
158C2×C42—81no—
159C2×C42—81no—
160C2×C42C2×D412yes—
161C2×C42C2×D412yes—
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D412yes—
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D412yes—
168C4⋊C8—82no—
169C4⋊C8—82no—
170C22⋊C8—22no—
171C22⋊C8—22no—
172C22⋊C8—22no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178#97—81no—
179#97—81no—
180C4×C8—81no—
181C4×C8—81no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95C22⋊C412yes—
197#95C22⋊C412yes—
198#95C22⋊C412yes—
199#95C22⋊C412yes—
200C24C23×C411yescharacteristic
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4#13114yes—
244C22×C4C22×D411yescharacteristic
245C22×C4C22×D411yescharacteristic
246C22×C42−1+412yes—
247C22×C42−1+412yes—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—82no—
276C42—24no—
277C42—24no—
278C42—41no—
279C42—41no—
280C42—41no—
281C42—41no—
282C42—42no—
283C42—42no—
284C42—44no—
285C42—44no—
286C23—21no—
287C23—21no—
288C23—22no—
289C23C22×C22⋊C411yescharacteristic
290C23#51911yescharacteristic
291C23#51911yescharacteristic
292C23C2×#13112yes—
293C23#56812yes—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4#41314yes—
347C2×C4#53612yes—
348C2×C4#53612yes—
349C8—82no—
350C8—82no—
351C8—82no—
352C8—82no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22#231011yescharacteristic
362C22#288012yes—
363C22#288311yescharacteristic
364C22#288411yescharacteristic
365C22C2×#41312yes—
366C22#298512yes—
367C22#298611yescharacteristic
368C22#298711yescharacteristic
369C4—24no—
370C4—22no—
371C4—22no—
372C4—44no—
373C4—42no—
374C4—42no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—82no—
379C4—82no—
380C4—82no—
381C2—22no—
382C2—21no—
383C2—21no—
384C2#1742312yes—
385C2#1742712yes—
386C2#2182411yescharacteristic
387C2#2182811yescharacteristic
388C2#2183111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#26062
#219142
#98832
#98812
#98802
#98792
#87052
#86992