Finite Group Databasebeta

← #238529

Subgroups — #238529

Order 512.

1078 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35173C211yesmaximal, characteristic
2#21385C212yesmaximal
3C2×#1518C212yesmaximal
4#14150C211yesmaximal, characteristic
5#10407C214yesmaximal
6#10406C214yesmaximal
7#7908C211yesmaximal, characteristic
8C2×#533C2212yes—
9C2×#431C2212yes—
10#2857—22no—
11#2659C2214yes—
12#2584—24no—
13#2567—22no—
14#2546—22no—
15#2316—21no—
16#2316—21no—
17C2×#389—21no—
18C2×#382C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#1987—22no—
22#1981—22no—
23#1872C2214yes—
24#1872C2214yes—
25C4×#103—24no—
26#1518—28no—
27C2×#328C2218yes—
28#1444C2212yes—
29#1442C2212yes—
30#1436—22no—
31C4⋊#95—21no—
32#1280C2214yes—
33#533—24no—
34#533—24no—
35#533—24no—
36#533—24no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8D412yes—
39C2×C4⋊Q8D412yes—
40C2×#125—22no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8C2314yes—
43C2×C4×Q8—22no—
44C2×C4×Q8D414yes—
45C2×#117—22no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C42—21no—
53C22×C42D412yes—
54C22×C42C2311yescharacteristic
55#431—24no—
56#431—24no—
57C2×C4⋊C8—22no—
58C2×#103—24no—
59C2×#103—24no—
60C2×C22⋊C8C2314yes—
61C2×C4×C8—22no—
62#398—28no—
63#395—28no—
64#391—24no—
65#389—24no—
66#389—24no—
67#389—24no—
68#389—24no—
69#389—24no—
70#389—24no—
71#389—44no—
72#384—22no—
73#382—22no—
74#382D414yes—
75#380—24no—
76#380—24no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95C2312yes—
87C2×#95C2312yes—
88#340—24no—
89#340—24no—
90#328—216no—
91C22×Q8—22no—
92C22×Q8C2×D414yes—
93C23×C4—21no—
94C23×C4—22no—
95C23×C4—21no—
96C23×C4C2×D412yes—
97C23×C4C2×D412yes—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4C2411yescharacteristic
101C22×C8—24no—
102C4⋊Q8—24no—
103C4⋊Q8—24no—
104C4⋊Q8—24no—
105C4⋊Q8—24no—
106C4⋊Q8—84no—
107#125—44no—
108#125—44no—
109C22⋊Q8—24no—
110C22⋊Q8—24no—
111C22⋊Q8—24no—
112C22⋊Q8—24no—
113C22⋊Q8—44no—
114C22⋊Q8—44no—
115C4×Q8—28no—
116C4×Q8—28no—
117C4×Q8—44no—
118C4×Q8—44no—
119#117—44no—
120#117—44no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—82no—
144C2×C4⋊C4SD1614yes—
145C2×C4⋊C4Q1614yes—
146C2×C4⋊C4C2×D414yes—
147C2×C22⋊C4—24no—
148C2×C42—24no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—42no—
152C2×C42—42no—
153C2×C42—42no—
154C2×C42—82no—
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C2×C42#4114yes—
159C2×C42#4112yes—
160C4⋊C8—44no—
161#103—48no—
162#103—48no—
163C22⋊C8—24no—
164C22⋊C8—24no—
165C4×C8—44no—
166#95—28no—
167#95—28no—
168#95—24no—
169#95—24no—
170#95—22no—
171#95—44no—
172#95—41no—
173#95—41no—
174#95—44no—
175#95—82no—
176#95C2×D414yes—
177C24C22×D411yescharacteristic
178C2×Q8—24no—
179C2×Q8—24no—
180C2×Q8—44no—
181C2×Q8—44no—
182C2×Q8—44no—
183C2×Q8—44no—
184C2×Q8—44no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4#12011yescharacteristic
221C22×C4#12011yescharacteristic
222C22×C4#12012yes—
223C22×C4#12112yes—
224C22×C4#12112yes—
225C22×C4#12412yes—
226C22×C4#12412yes—
227C22×C4C2×SD1614yes—
228C22×C4C2×SD1612yes—
229C22×C4C2×Q1614yes—
230C22×C4C2×Q1612yes—
231C22×C4#13712yes—
232C22×C4#13712yes—
233C22×C4C22×D412yes—
234C22×C4C22×D411yescharacteristic
235C22×C4C2×#4112yes—
236C22×C4C2×#4111yescharacteristic
237C2×C8—44no—
238C2×C8—44no—
239C2×C8—44no—
240C4⋊C4—28no—
241C4⋊C4—28no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—48no—
253C4⋊C4—44no—
254C4⋊C4—48no—
255C4⋊C4—84no—
256C4⋊C4—84no—
257C4⋊C4—84no—
258C4⋊C4—84no—
259C4⋊C4—88no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C42—44no—
263C42—48no—
264C42—82no—
265C42—84no—
266C42#12114yes—
267C42#12114yes—
268C42#12314yes—
269C42#12314yes—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23C2×#12011yescharacteristic
275C23C2×#12111yescharacteristic
276C23C2×#12411yescharacteristic
277C23C22×SD1611yescharacteristic
278C23C22×Q1611yescharacteristic
279C23C2×#13711yescharacteristic
280C23C2×#13711yescharacteristic
281Q8—48no—
282Q8—84no—
283Q8—88no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4#39412yes—
333C2×C4#39412yes—
334C2×C4#39414yes—
335C2×C4#44814yes—
336C2×C4#45114yes—
337C2×C4#46114yes—
338C2×C4#46214yes—
339C2×C4#48714yes—
340C2×C4#49114yes—
341C2×C4C2×#12112yes—
342C2×C4C2×#12312yes—
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×#39411yescharacteristic
357C22C2×#44811yescharacteristic
358C22C2×#45111yescharacteristic
359C22#301911yescharacteristic
360C22#302011yescharacteristic
361C22#302511yescharacteristic
362C22#302511yescharacteristic
363C22C2×#46111yescharacteristic
364C22C2×#46211yescharacteristic
365C22C2×#48711yescharacteristic
366C22C2×#49111yescharacteristic
367C4—22no—
368C4—22no—
369C4—22no—
370C4—22no—
371C4—44no—
372C4—44no—
373C4—84no—
374C4—82no—
375C4—82no—
376C4—84no—
377C4—84no—
378C4#204318yes—
379C2—21no—
380C2—21no—
381C2—21no—
382C2—21no—
383C2#1821211yescharacteristic
384C2#1821311yescharacteristic
385C2#1822011yescharacteristic
386C2#1822111yescharacteristic
387C2#2349911yescharacteristic
388C2C2×#204311yescharacteristic
389C2#2351411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ 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
#351732
#213852
C2×#15182
#141502
#104072
#104062
#79082