Finite Group Databasebeta

← #215393

Subgroups — #215393

Order 512.

1066 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic
3#13757C211yesmaximal, characteristic
4#9711C214yesmaximal
5#9683C214yesmaximal
6#8333C214yesmaximal
7C2×#533C2212yes—
8#2868C411yescharacteristic, complemented
9#2868C411yescharacteristic, complemented
10#2862C412yescomplemented
11#2859—21no—
12#2859—21no—
13#2857—21no—
14#2857—21no—
15#2857C2212yes—
16#2857C2212yes—
17C4×C4⋊Q8—21no—
18C4×C4⋊Q8—21no—
19C4×C4⋊Q8C412yescomplemented
20C4×C4⋊Q8C412yescomplemented
21#2285—21no—
22#2285—21no—
23C2×#376—21no—
24C2×C43C2211yescharacteristic
25#1839—21no—
26#1831—21no—
27#1817C2212yes—
28#1797C2212yes—
29#1767—21no—
30#1749C2212yes—
31#1745—21no—
32C2×#328C22116yes—
33#1434—21no—
34#1434—21no—
35#1434C2212yes—
36#1434C2212yes—
37C4×#95—21no—
38C4×#95C2212yes—
39#533—48no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2×C412yes—
46C2×#125—22no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8C2314yes—
49C2×C4×Q8—22no—
50C2×C4×Q8C2×C414yes—
51C2×#117—22no—
52C22×C42—22no—
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×C4⋊C8—22no—
61C2×C22⋊C8C2314yes—
62C2×#97—22no—
63#391—22no—
64#391—22no—
65#391C2×C414yes—
66#391C2×C414yes—
67#389—22no—
68#389—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#384—21no—
76#384—21no—
77#384—21no—
78#384—21no—
79#384—21no—
80#384—21no—
81#384C2×C412yes—
82#384C2×C412yes—
83#382—22no—
84#382—22no—
85#380—22no—
86#380—22no—
87#380—22no—
88#380—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93#376—28no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95C2314yes—
100C43C2×C411yescharacteristic
101C43C2×C411yescharacteristic
102C43C2×C411yescharacteristic
103C43C2×C411yescharacteristic
104C43D411yescharacteristic
105C43D411yescharacteristic
106C43D411yescharacteristic
107C43D411yescharacteristic
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112#328—416no—
113C22×Q8—22no—
114C22×Q8C22×C414yes—
115C23×C4—24no—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2411yescharacteristic
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—216no—
132C22⋊Q8—48no—
133C4×Q8—48no—
134C4×Q8—48no—
135#117—48no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
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⋊C4—24no—
149C2×C4⋊C4C22×C414yes—
150C2×C22⋊C4—24no—
151C2×C42—24no—
152C2×C42—24no—
153C2×C42—24no—
154C2×C42—24no—
155C2×C42—28no—
156C2×C42—24no—
157C2×C42—42no—
158C2×C42—42no—
159C2×C42—42no—
160C2×C42—44no—
161C2×C42—44no—
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×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C411yescharacteristic
178C2×C42C22×C411yescharacteristic
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C4⋊C8—44no—
191C22⋊C8—28no—
192#97—44no—
193#95—28no—
194#95—28no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—28no—
200#95—44no—
201#95—44no—
202#95—44no—
203C24C22×D411yescharacteristic
204C2×Q8—28no—
205C2×Q8—44no—
206C2×Q8—44no—
207C2×Q8—44no—
208C2×Q8—44no—
209C2×Q8—44no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
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—28no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4#13714yes—
250C22×C4#13714yes—
251C22×C4C23×C411yescharacteristic
252C22×C4C22×D411yescharacteristic
253C22×C4H(5,𝔽2)14yes—
254C2×C8—44no—
255C2×C8—44no—
256C2×C8—44no—
257C4⋊C4—28no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C4⋊C4—48no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—48no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C22⋊C4—48no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—41no—
294C42—41no—
295C42—41no—
296C42—41no—
297C42—41no—
298C42—41no—
299C42—416no—
300C42—48no—
301C42#10414yes—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23C22×C22⋊C411yescharacteristic
307C23#53512yes—
308C23C2×#13712yes—
309C23C2×#13712yes—
310Q8—88no—
311Q8—88no—
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—28no—
331C2×C4—28no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—48no—
350C2×C4—48no—
351C2×C4—48no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4#41012yes—
363C2×C4#41012yes—
364C2×C4#41912yes—
365C2×C4#41912yes—
366C2×C4#41912yes—
367C2×C4#41912yes—
368C2×C4C2×#10412yes—
369C2×C4C2×#10412yes—
370C2×C4C2×#10412yes—
371C2×C4#42112yes—
372C2×C4#51812yes—
373C2×C4#53512yes—
374C8—84no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22#230911yescharacteristic
382C22C2×#41011yescharacteristic
383C22C2×#41011yescharacteristic
384C22C2×#41911yescharacteristic
385C22C2×#41911yescharacteristic
386C22C22×#10411yescharacteristic
387C22C2×#42111yescharacteristic
388C22#321814yes—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—22no—
397C4—22no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—88no—
403C4—84no—
404C4—84no—
405C4#150614yes—
406C2—24no—
407C2C2×#150611yescharacteristic
408C2#1754511yescharacteristic
409C2#1756211yescharacteristic
410C2#1758411yescharacteristic
411C2#2179411yescharacteristic
412C2#2181111yescharacteristic
413C2#2181511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #2868C4semidirect
9. #2868C4semidirect
10. #2862C4semidirect
19. C4×C4⋊Q8C4semidirect
20. C4×C4⋊Q8C4semidirect

Maximal subgroups

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

subgroupindex
#366452
#207782
#137572
#97112
#96832
#83332