Finite Group Databasebeta

← #215672

Subgroups — #215672

Order 512.

888 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#36645C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic
3#20778C211yesmaximal, characteristic
4#9895C214yesmaximal, complemented
5#9894C214yesmaximal, complemented
6#8722C214yesmaximal
7C2×#533C412yes—
8#2868C411yescharacteristic
9#2868C411yescharacteristic
10#2862C2212yes—
11#2859C412yes—
12#2859C2212yes—
13#2857—21no—
14#2857—21no—
15#2857—21no—
16#2857—21no—
17C4×C4⋊Q8—21no—
18C4×C4⋊Q8—21no—
19C4×C4⋊Q8—21no—
20C4×C4⋊Q8—21no—
21#2285C412yes—
22#2285C2212yes—
23C2×C43C2211yescharacteristic
24#1839—21no—
25#1839—21no—
26#1831—21no—
27#1831—21no—
28#1817—21no—
29#1817—21no—
30#1797—21no—
31#1797—21no—
32#1767C2212yes—
33#1767C2212yes—
34#1749C2212yes—
35#1749C2212yes—
36#1745C2212yes—
37#1745C2212yes—
38#1295C22116yes—
39#533—48no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2×C412yes—
46C2×#125C2×C414yes—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×#117C2×C414yes—
52C22×C42C2×C411yescharacteristic
53C22×C42C2×C411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42C2311yescharacteristic
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×#97C2314yes—
64C2×#97C2314yes—
65#391—22no—
66#391—22no—
67#391—22no—
68#391—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#384—21no—
78#384—21no—
79#384—21no—
80#384—21no—
81#384C2×C412yes—
82#384C2×C412yes—
83#384C2×C412yes—
84#384C2×C412yes—
85#382C2×C414yes—
86#382C2314yes—
87#380—22no—
88#380—22no—
89#380—22no—
90#380—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C43C2×C411yescharacteristic
96C43C2×C411yescharacteristic
97C43D411yescharacteristic
98C43D411yescharacteristic
99C43D411yescharacteristic
100C43D411yescharacteristic
101C43C2311yescharacteristic
102C43C2311yescharacteristic
103#336—22no—
104#336—22no—
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C22⋊C411yescharacteristic
116C23×C4C22⋊C411yescharacteristic
117C23×C4C22×C411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C22×C8—24no—
121C22×C8—24no—
122C4⋊Q8—44no—
123C4⋊Q8—44no—
124C4⋊Q8—44no—
125C4⋊Q8—44no—
126C4⋊Q8—44no—
127C4⋊Q8—44no—
128#125—48no—
129C22⋊Q8—48no—
130C22⋊Q8—48no—
131C4×Q8—48no—
132C4×Q8—48no—
133#117—48no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4C22×C414yes—
147C2×C4⋊C4C22×C414yes—
148C2×C22⋊C4—24no—
149C2×C42—24no—
150C2×C42—24no—
151C2×C42—24no—
152C2×C42—24no—
153C2×C42—42no—
154C2×C42—42no—
155C2×C42—42no—
156C2×C42—42no—
157C2×C42C22⋊C411yescharacteristic
158C2×C42C22⋊C411yescharacteristic
159C2×C42C22⋊C411yescharacteristic
160C2×C42C22⋊C411yescharacteristic
161C2×C42C22⋊C411yescharacteristic
162C2×C42C22⋊C411yescharacteristic
163C2×C42C22⋊C411yescharacteristic
164C2×C42C22⋊C411yescharacteristic
165C2×C42C22⋊C411yescharacteristic
166C2×C42C22⋊C411yescharacteristic
167C2×C42C22⋊C411yescharacteristic
168C2×C42C22⋊C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C22×C411yescharacteristic
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2411yescharacteristic
185C4⋊C8—44no—
186C4⋊C8—44no—
187C22⋊C8—44no—
188C22⋊C8—44no—
189#97—28no—
190#97—28no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—24no—
195C24C2×C22⋊C411yescharacteristic
196C2×Q8—44no—
197C2×Q8—44no—
198C2×Q8—44no—
199C2×Q8—44no—
200C2×Q8—44no—
201C2×Q8—44no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4#10014yes—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C23×C411yescharacteristic
237C22×C4C22×D411yescharacteristic
238C22×C4C22×D411yescharacteristic
239C2×C8—44no—
240C2×C8—44no—
241C2×C8—44no—
242C2×C8—44no—
243C2×C8—44no—
244C2×C8—44no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—48no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—48no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C22⋊C4—48no—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—41no—
279C42—41no—
280C42—41no—
281C42—41no—
282C42—41no—
283C42—41no—
284C42—41no—
285C42—41no—
286C42—48no—
287C42#10114yes—
288C422−1+414yes—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23C2×#10012yes—
296C23C22×C22⋊C411yescharacteristic
297Q8—88no—
298Q8—88no—
299C2×C4—28no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4C2×#10012yes—
343C2×C4C2×#10012yes—
344C2×C4C2×#10112yes—
345C2×C4C2×#10112yes—
346C2×C4C2×#10112yes—
347C2×C4#42112yes—
348C2×C4#42112yes—
349C2×C4#42112yes—
350C2×C4#42112yes—
351C2×C4#51912yes—
352C2×C4#53612yes—
353C2×C4#53612yes—
354C8—84no—
355C8—84no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22#231011yescharacteristic
364C22C22×#10011yescharacteristic
365C22C22×#10111yescharacteristic
366C22C2×#42111yescharacteristic
367C22C2×#42111yescharacteristic
368C22C2×#42111yescharacteristic
369C22C2×#42111yescharacteristic
370C4—22no—
371C4—22no—
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—22no—
381C4—22no—
382C4—41no—
383C4—41no—
384C4—84no—
385C4—84no—
386C2—42no—
387C2#1766411yescharacteristic
388C2#1767111yescharacteristic
389C2#1767211yescharacteristic
390C2#1767511yescharacteristic
391C2#2182911yescharacteristic
392C2#2184511yescharacteristic
393C2#2184511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9895C2semidirect
5. #9894C2semidirect

Maximal subgroups

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

subgroupindex
#366452
#207782
#98952
#98942
#87222