Finite Group Databasebeta

← #215709

Subgroups — #215709

Order 512.

888 conjugacy classes of subgroups.

The subgroup lattice has 397 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#9907C214yesmaximal, complemented
5#9903C212yesmaximal, complemented
6#9903C212yesmaximal, complemented
7#8726C214yesmaximal
8C2×#533—21no—
9#2868C2211yescharacteristic
10#2868C2211yescharacteristic
11#2862—21no—
12#2859—21no—
13#2859C2212yes—
14#2857—21no—
15#2857—21no—
16#2857C412yes—
17#2857C412yes—
18C4×C4⋊Q8—21no—
19C4×C4⋊Q8—21no—
20C4×C4⋊Q8C412yes—
21C4×C4⋊Q8C412yes—
22#2285—21no—
23#2285C2212yes—
24C2×C43C2211yescharacteristic
25#1839—21no—
26#1839—21no—
27#1831C2212yes—
28#1831C2212yes—
29#1817—21no—
30#1817—21no—
31#1797C2212yes—
32#1797C2212yes—
33#1767C2212yes—
34#1767C2212yes—
35#1749—21no—
36#1749—21no—
37#1745—21no—
38#1745—21no—
39#1296C2218yes—
40#1296C2218yes—
41#533—48no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8C2×C412yes—
48C2×#125—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×#117—22no—
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42C2311yescharacteristic
61C2×C4⋊C8C2314yes—
62C2×C4⋊C8C2314yes—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×#97—22no—
66C2×#97—22no—
67#391—22no—
68#391—22no—
69#391—22no—
70#391—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389C2×C414yes—
78#389C2×C414yes—
79#384—21no—
80#384—21no—
81#384—21no—
82#384—21no—
83#384C2×C412yes—
84#384C2×C412yes—
85#384C2312yes—
86#384C2312yes—
87#382—22no—
88#382—22no—
89#380—22no—
90#380—22no—
91#380—22no—
92#380C2×C414yes—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4C2×C414yes—
97C43C2×C411yescharacteristic
98C43C2×C411yescharacteristic
99C43D411yescharacteristic
100C43D411yescharacteristic
101C43D411yescharacteristic
102C43D411yescharacteristic
103C43C2311yescharacteristic
104C43C2311yescharacteristic
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112#336—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C23×C4C22⋊C411yescharacteristic
116C23×C4C22⋊C411yescharacteristic
117C23×C4C22⋊C411yescharacteristic
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C22×C8—24no—
123C22×C8—24no—
124C4⋊Q8—44no—
125C4⋊Q8—44no—
126C4⋊Q8—44no—
127C4⋊Q8—44no—
128C4⋊Q8—44no—
129C4⋊Q8—44no—
130#125—48no—
131C22⋊Q8—48no—
132C22⋊Q8—48no—
133C4×Q8—48no—
134C4×Q8—48no—
135#117—48no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
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⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4C22×C414yes—
149C2×C4⋊C4C22×C414yes—
150C2×C22⋊C4—24no—
151C2×C42—24no—
152C2×C42—24no—
153C2×C42—24no—
154C2×C42—24no—
155C2×C42—42no—
156C2×C42—42no—
157C2×C42—42no—
158C2×C42—42no—
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×C42C22×C411yescharacteristic
175C2×C42C22×C411yescharacteristic
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×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2411yescharacteristic
187C4⋊C8—28no—
188C4⋊C8—28no—
189C22⋊C8—44no—
190C22⋊C8—44no—
191#97—44no—
192#97—44no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95—24no—
197C24C2×C22⋊C411yescharacteristic
198C2×Q8—44no—
199C2×Q8—44no—
200C2×Q8—44no—
201C2×Q8—44no—
202C2×Q8—44no—
203C2×Q8—44no—
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—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
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×C4C2×C22⋊C411yescharacteristic
237C22×C4AGL(1,ℤ/8)14yes—
238C22×C4C23×C411yescharacteristic
239C22×C4C22×D411yescharacteristic
240C22×C4C22×D411yescharacteristic
241C2×C8—44no—
242C2×C8—44no—
243C2×C8—44no—
244C2×C8—44no—
245C2×C8—44no—
246C2×C8—44no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—48no—
250C4⋊C4—48no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C22⋊C4—48no—
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—22no—
279C42—22no—
280C42—41no—
281C42—41no—
282C42—41no—
283C42—41no—
284C42—41no—
285C42—41no—
286C42—41no—
287C42—41no—
288C42—48no—
289C42#13714yes—
290C422−1+414yes—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23#41812yes—
298C23C22×C22⋊C411yescharacteristic
299Q8—88no—
300Q8—88no—
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—28no—
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—21no—
331C2×C4—21no—
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×C4—44no—
343C2×C4—44no—
344C2×C4#41312yes—
345C2×C4#41312yes—
346C2×C4#41812yes—
347C2×C4#41912yes—
348C2×C4#41912yes—
349C2×C4#51912yes—
350C2×C4#53612yes—
351C2×C4#53612yes—
352C2×C4C2×AGL(1,ℤ/8)12yes—
353C2×C4C2×#13712yes—
354C2×C4#57512yes—
355C2×C4#57512yes—
356C8—84no—
357C8—84no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22#231011yescharacteristic
366C22C2×#41311yescharacteristic
367C22C2×#41311yescharacteristic
368C22C2×#41811yescharacteristic
369C22C2×#41911yescharacteristic
370C22#289611yescharacteristic
371C22#289611yescharacteristic
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—84no—
385C4—84no—
386C4#323814yes—
387C4#324014yes—
388C2—42no—
389C2#1773111yescharacteristic
390C2#1774411yescharacteristic
391C2#1774711yescharacteristic
392C2#1774711yescharacteristic
393C2#2183111yescharacteristic
394C2#2184011yescharacteristic
395C2#2184211yescharacteristic
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. #9907C2semidirect
5. #9903C2semidirect
6. #9903C2semidirect

Maximal subgroups

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

subgroupindex
#366452
#207782
#99072
#99032
#87262