Finite Group Databasebeta

← #215921

Subgroups — #215921

Order 512.

930 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2#21912C211yesmaximal, characteristic
3#13949C211yesmaximal, characteristic
4#9911C212yesmaximal
5#9908C212yesmaximal
6#9885C212yesmaximal
7#9884C212yesmaximal
8#8717C212yesmaximal
9#8717C212yesmaximal
10C2×#544—21no—
11#2857C2212yes—
12#2857C2212yes—
13#2659—21no—
14#2659—21no—
15#2552C414yescomplemented
16#2551C414yescomplemented
17#2550—24no—
18#2316—21no—
19C2×#387—21no—
20#2285C2212yes—
21C2×C4×C4⋊C4C2211yescharacteristic
22#1916—24no—
23#1817C2212yes—
24#1797C2212yes—
25C4×C22⋊C8C2212yes—
26#1442—21no—
27#1441—21no—
28#1436—21no—
29C4⋊#95C2211yescharacteristic
30C4⋊#95C2211yescharacteristic
31#1434C2211yescharacteristic
32#1434C2211yescharacteristic
33#1434C2211yescharacteristic
34C4×#95C2211yescharacteristic
35#1294C2214yes—
36#1294C2214yes—
37#1294C2214yes—
38#1294C2214yes—
39#544—44no—
40#544—44no—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8C2×C412yes—
44C2×#125—24no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47C2×#117—22no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C42—21no—
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C2×C4⋊C8—22no—
57C2×C22⋊C8C2314yes—
58C2×C4×C8—22no—
59#398—24no—
60#398—24no—
61#395—24no—
62#395—24no—
63#391—22no—
64#391—22no—
65#389—24no—
66#389—24no—
67#389—22no—
68#389—22no—
69#389C2×C414yes—
70#389C2×C414yes—
71#387—44no—
72#384C2×C412yes—
73#384C2×C412yes—
74#384C2×C412yes—
75#384C2×C412yes—
76#382—22no—
77#380—22no—
78#380—22no—
79#380—22no—
80#380C2314yes—
81C4×C4⋊C4—22no—
82C4×C4⋊C4C2×C414yes—
83C4×C4⋊C4C2×C414yes—
84C4×C4⋊C4D414yes—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95D411yescharacteristic
95C2×#95D411yescharacteristic
96C2×#95C2311yescharacteristic
97C2×#95C2311yescharacteristic
98C2×#95C2311yescharacteristic
99C2×#95C2311yescharacteristic
100#336—22no—
101#336—24no—
102#336—22no—
103C22×Q8—22no—
104C22×Q8—22no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2411yescharacteristic
116C22×C8—24no—
117C4⋊Q8—416no—
118C4⋊Q8—82no—
119C4⋊Q8—82no—
120#125—416no—
121C22⋊Q8—44no—
122C22⋊Q8—44no—
123C22⋊Q8—44no—
124C22⋊Q8—44no—
125#117—44no—
126#117—44no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—81no—
146C2×C4⋊C4—81no—
147C2×C4⋊C4C22⋊C414yes—
148C2×C4⋊C4C22⋊C414yes—
149C2×C4⋊C4SD1614yes—
150C2×C4⋊C4Q1614yes—
151C2×C4⋊C4C22×C414yes—
152C2×C4⋊C4C22×C414yes—
153C2×C22⋊C4—24no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—24no—
157C2×C42—42no—
158C2×C42—42no—
159C2×C42—81no—
160C2×C42—81no—
161C2×C42—81no—
162C2×C42C22×C412yes—
163C2×C42C22×C412yes—
164C2×C42C22×C412yes—
165C2×C42C2×D412yes—
166C2×C42C2×D412yes—
167C2×C42C2×D412yes—
168C4⋊C8—82no—
169C22⋊C8—24no—
170C22⋊C8—24no—
171C4×C8—82no—
172#95—24no—
173#95—24no—
174#95—24no—
175#95—24no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—42no—
181#95—42no—
182#95—42no—
183#95—42no—
184#95—42no—
185#95—42no—
186#95—42no—
187#95—42no—
188#95—81no—
189#95C22⋊C414yes—
190#95C22⋊C414yes—
191C24C22×D411yescharacteristic
192C2×Q8—44no—
193C2×Q8—44no—
194C2×Q8—82no—
195C2×Q8—82no—
196C2×Q8—82no—
197C2×Q8—82no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
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—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
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×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×SD1612yes—
255C22×C4C2×SD1612yes—
256C22×C4C2×SD1612yes—
257C22×C4C2×Q1612yes—
258C22×C4C2×Q1612yes—
259C22×C4C2×Q1612yes—
260C22×C4AGL(1,ℤ/8)12yes—
261C22×C4AGL(1,ℤ/8)12yes—
262C22×C4C23×C411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C42−1+412yes—
265C22×C42−1+412yes—
266C2×C8—44no—
267C2×C8—44no—
268C2×C8—44no—
269C4⋊C4—28no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—48no—
276C4⋊C4—44no—
277C4⋊C4—48no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—84no—
285C4⋊C4—84no—
286C4⋊C4#103116yes—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C42—24no—
290C42—24no—
291C42—42no—
292C42—42no—
293C42—42no—
294C42—42no—
295C42—48no—
296C42—84no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23C22×C22⋊C411yescharacteristic
302C23#53611yescharacteristic
303C23#53611yescharacteristic
304C23C22×SD1611yescharacteristic
305C23C22×Q1611yescharacteristic
306C23C2×AGL(1,ℤ/8)11yescharacteristic
307C23C2×AGL(1,ℤ/8)11yescharacteristic
308Q8—164no—
309Q8—164no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4C2×#10314yes—
379C2×C4C2×#10314yes—
380C2×C4C2×#10314yes—
381C2×C4#41814yes—
382C2×C4#51912yes—
383C2×C4#53612yes—
384C8—84no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22#231011yescharacteristic
398C22#288311yescharacteristic
399C22#288411yescharacteristic
400C22C22×#10311yescharacteristic
401C22C2×#41811yescharacteristic
402C22#290111yescharacteristic
403C22#290111yescharacteristic
404C22#322711yescharacteristic
405C22#322711yescharacteristic
406C22#323011yescharacteristic
407C22#323011yescharacteristic
408C4—22no—
409C4—22no—
410C4—24no—
411C4—24no—
412C4—24no—
413C4—24no—
414C4—42no—
415C4—42no—
416C4—82no—
417C4—82no—
418C4—84no—
419C4—84no—
420C4—161no—
421C4—161no—
422C2—21no—
423C2—21no—
424C2—21no—
425C2—21no—
426C2#1761411yescharacteristic
427C2#1761911yescharacteristic
428C2#1762011yescharacteristic
429C2#1763111yescharacteristic
430C2#2182811yescharacteristic
431C2#2183411yescharacteristic
432C2#2184611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ 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
15. #2552C4semidirect
16. #2551C4semidirect

Maximal subgroups

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

subgroupindex
#351672
#219122
#139492
#99112
#99082
#98852
#98842
#87172