Finite Group Databasebeta

← #206895

Subgroups — #206895

Order 512.

854 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2666C211yesmaximal, characteristic
2#21955C211yesmaximal, characteristic
3#20533C211yesmaximal, characteristic
4#9880C212yesmaximal
5#9880C212yesmaximal
6#9879C212yesmaximal
7#9878C212yesmaximal
8#7916C212yesmaximal
9#7914C212yesmaximal
10C2×#432C2212yes—
11C2×#431C2212yes—
12#2666—24no—
13#2666—24no—
14C2×#400C411yescharacteristic
15C2×#400C411yescharacteristic
16C2×#398C411yescharacteristic
17C2×#398C411yescharacteristic
18C2×#398C411yescharacteristic
19C2×#398C411yescharacteristic
20C2×#395C411yescharacteristic
21C2×#395C411yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#387C2211yescharacteristic
25C2×#387C2211yescharacteristic
26C2×#383C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1940—24no—
30#1797C2212yes—
31#1797C2212yes—
32C8×C4⋊C4—24no—
33C4×C22⋊C8C2212yes—
34C4×C22⋊C8C2212yes—
35#1280C2214yes—
36#1280C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×Q8C2×C411yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56#432—28no—
57#431—24no—
58#431Q818yescomplemented
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C22⋊C8C2314yes—
64C2×C22⋊C8C2314yes—
65C2×#97—22no—
66C2×C4×C8—22no—
67C2×C4×C8—22no—
68C2×C4×C8—22no—
69#400—24no—
70#400—24no—
71#398—24no—
72#398—24no—
73#398—24no—
74#398—24no—
75#395—24no—
76#395—24no—
77#391—22no—
78#391—22no—
79#389—22no—
80#389—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#383—22no—
86#383—22no—
87#382—22no—
88#382—22no—
89C4×C4⋊C4D414yes—
90C4×C4⋊C4D414yes—
91C2×#95C2×C411yescharacteristic
92C2×#95C2×C411yescharacteristic
93C2×#95C2×C411yescharacteristic
94C2×#95C2×C411yescharacteristic
95C2×#95C2×C411yescharacteristic
96C2×#95C2×C411yescharacteristic
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2311yescharacteristic
106C2×#95C2311yescharacteristic
107C2×#95C2311yescharacteristic
108C2×#95C2311yescharacteristic
109#336—22no—
110#336—22no—
111#336—22no—
112#336—22no—
113C22×Q8—22no—
114C22×Q8—24no—
115C22×Q8—81no—
116C22×Q8—81no—
117C23×C4C22×C411yescharacteristic
118C23×C4C22×C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C2411yescharacteristic
132C22×C8—24no—
133C22×C8—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
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—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—81no—
151C2×C4⋊C4—81no—
152C2×C4⋊C4—81no—
153C2×C4⋊C4—81no—
154C2×C4⋊C4—81no—
155C2×C4⋊C4—81no—
156C2×C4⋊C4—81no—
157C2×C4⋊C4—81no—
158C2×C4⋊C4C22⋊C414yes—
159C2×C4⋊C4C22⋊C414yes—
160C2×C42—81no—
161C2×C42—81no—
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C2×C42C2×D412yes—
167C2×C42C2×D412yes—
168C2×C42C2×Q814yes—
169C2×C42#4114yes—
170C4⋊C8—28no—
171C4⋊C8—44no—
172C4⋊C8—82no—
173C4⋊C8—82no—
174C22⋊C8—24no—
175C22⋊C8—28no—
176C22⋊C8C2×Q818yes—
177#97—44no—
178C4×C8—28no—
179C4×C8—82no—
180C4×C8—82no—
181#95—22no—
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—21no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95C22⋊C414yes—
202#95C22⋊C414yes—
203C24C23×C411yescharacteristic
204C2×Q8—44no—
205C2×Q8—82no—
206C2×Q8—82no—
207C2×Q8—88no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C4×Q814yes—
263C22×C4C4×Q814yes—
264C22×C4#13114yes—
265C22×C4#13114yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×Q812yes—
269C22×C4C2×#4112yes—
270C2×C8—28no—
271C2×C8—44no—
272C2×C8—44no—
273C2×C8—44no—
274C2×C8—44no—
275C2×C8—44no—
276C4⋊C4—28no—
277C4⋊C4—28no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C42—42no—
290C42—42no—
291C42—42no—
292C42—42no—
293C42C22⋊Q814yes—
294C42C22⋊Q814yes—
295C42#12414yes—
296C42C4⋊Q814yes—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23C22×C22⋊C411yescharacteristic
302C23C2×C4×Q812yes—
303C23C2×#13112yes—
304C23C2×#13112yes—
305Q8—88no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—28no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4#39114yes—
365C2×C4#39114yes—
366C2×C4#40814yes—
367C2×C4#40814yes—
368C2×C4C2×C22⋊Q812yes—
369C2×C4C2×#12411yescharacteristic
370C2×C4C2×C4⋊Q811yescharacteristic
371C8—48no—
372C8—84no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22C2×#39111yescharacteristic
380C22C2×#40811yescharacteristic
381C22C2×#40811yescharacteristic
382C22#288311yescharacteristic
383C22#288411yescharacteristic
384C22#288811yescharacteristic
385C22#288911yescharacteristic
386C22#296014yes—
387C4—24no—
388C4—22no—
389C4—21no—
390C4—21no—
391C4—24no—
392C4—24no—
393C4—24no—
394C4—42no—
395C4—42no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4—81no—
404C2—24no—
405C2#1738611yescharacteristic
406C2#1738711yescharacteristic
407C2#1739011yescharacteristic
408C2#1739111yescharacteristic
409C2#2130011yescharacteristic
410C2#2133411yescharacteristic
411C2#2133811yescharacteristic
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

Complements

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

NH ≅ G/Ntype
58. #431Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26662
#219552
#205332
#98802
#98792
#98782
#79162
#79142