Finite Group Databasebeta

← #194750

Subgroups — #194750

Order 512.

1262 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2581C211yesmaximal, characteristic
2#34059C211yesmaximal, characteristic
3#20741C212yesmaximal
4C2×#1467C211yesmaximal, characteristic
5#9023C212yesmaximal
6#9022C212yesmaximal
7#9015C212yesmaximal
8#9014C212yesmaximal
9#8017C212yesmaximal
10C22×C4×Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#432C2212yes—
13#2581—24no—
14#2581C418yescomplemented
15C2×#400C2211yescharacteristic
16C2×#400C2211yescharacteristic
17C2×#395C2211yescharacteristic
18C2×#395C2211yescharacteristic
19#2320—24no—
20#2316C2212yes—
21#2313—21no—
22#2313C2212yes—
23#2310—21no—
24#2295—24no—
25C2×#391—21no—
26C2×#389—21no—
27C2×#387—21no—
28C2×#387—21no—
29#2283—21no—
30#2283C2212yes—
31C2×#382C2211yescharacteristic
32C4×#117—21no—
33#1824C2214yes—
34#1805—24no—
35#1787C2214yes—
36#1763—24no—
37#1467—24no—
38C2×#324C2212yes—
39C2×#324C2212yes—
40#1294C2214yes—
41#1294C2214yes—
42C23×Q8D411yescharacteristic
43C23×Q8C2311yescharacteristic
44C2×C4×Q8—22no—
45C2×C4×Q8—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8—24no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—24no—
51C2×C4×Q8C2×C414yes—
52C2×C4×Q8C2×C414yes—
53C2×C4×Q8D414yes—
54C2×#117—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C42—21no—
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42C2311yescharacteristic
67#432—24no—
68#432—24no—
69C2×C4⋊C8—22no—
70C2×#103—28no—
71C2×C22⋊C8C2314yes—
72C2×#97—22no—
73#400—22no—
74#400—22no—
75#400C2×C414yes—
76#400C2×C414yes—
77#395—22no—
78#395—22no—
79#395C2×C414yes—
80#395C2×C414yes—
81#391—28no—
82#391—28no—
83#389—28no—
84#389—28no—
85#387—28no—
86#387—28no—
87#387—28no—
88#387—28no—
89#383—22no—
90#383—22no—
91#382—22no—
92#382—22no—
93#382C2×C414yes—
94#382D414yes—
95#381—22no—
96#381—22no—
97#380—22no—
98#380C2314yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4D414yes—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105#376—22no—
106#376—22no—
107#376—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95C2311yescharacteristic
116C2×#95C2311yescharacteristic
117C43—22no—
118#340—28no—
119#324—24no—
120#324—24no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—24no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—42no—
128C22×Q8—44no—
129C22×Q8C22⋊C414yes—
130C22×Q8C22×C412yes—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2411yescharacteristic
142C22×C8—24no—
143C4×Q8—28no—
144C4×Q8—28no—
145C4×Q8—28no—
146C4×Q8—28no—
147C4×Q8—28no—
148C4×Q8—28no—
149C4×Q8—44no—
150C4×Q8—44no—
151#117—44no—
152#117—44no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4C22⋊C414yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D418yes—
181C2×C22⋊C4—24no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—42no—
193C2×C42—42no—
194C2×C42C22⋊C414yes—
195C2×C42C22⋊C414yes—
196C2×C42C22×C412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42#4114yes—
201C4⋊C8—44no—
202#103—416no—
203C22⋊C8—24no—
204C22⋊C8—24no—
205#97—44no—
206#95—24no—
207#95—24no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—21no—
221#95—21no—
222#95—24no—
223#95C22×C412yes—
224#95C22×C412yes—
225C24C22×D411yescharacteristic
226C2×Q8—28no—
227C2×Q8—28no—
228C2×Q8—28no—
229C2×Q8—24no—
230C2×Q8—44no—
231C2×Q8—48no—
232C2×Q8—48no—
233C2×Q8—44no—
234C2×Q8—44no—
235C2×Q8—44no—
236C2×Q8—48no—
237C2×Q8—42no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—28no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4#12011yescharacteristic
280C22×C4#12012yes—
281C22×C4#12011yescharacteristic
282C22×C4#12312yes—
283C22×C4#12312yes—
284C22×C4#13714yes—
285C22×C4C23×C411yescharacteristic
286C22×C4C22×D412yes—
287C22×C4C22×D411yescharacteristic
288C22×C4C2×#4112yes—
289C2×C8—48no—
290C2×C8—44no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—28no—
296C4⋊C4—24no—
297C4⋊C4—48no—
298C4⋊C4—48no—
299C4⋊C4—48no—
300C4⋊C4—48no—
301C4⋊C4—48no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—84no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C42—24no—
309C42—24no—
310C42—24no—
311C42—24no—
312C42—48no—
313C42—48no—
314C42—48no—
315C42—84no—
316C42C4×D414yes—
317C42C4×D414yes—
318C42#12114yes—
319C42#12114yes—
320C23—22no—
321C23—22no—
322C23C22×C22⋊C411yescharacteristic
323C23C2×#12011yescharacteristic
324C23C2×#12311yescharacteristic
325C23C2×#13712yes—
326C23#57512yes—
327Q8—28no—
328Q8—416no—
329Q8—48no—
330Q8—44no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—48no—
371C2×C4—48no—
372C2×C4—48no—
373C2×C4—42no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4#38612yes—
381C2×C4#38612yes—
382C2×C4#38614yes—
383C2×C4#41914yes—
384C2×C4C2×C4×D412yes—
385C2×C4C2×#12112yes—
386C8—84no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—24no—
392C22—24no—
393C22C2×#38611yescharacteristic
394C22C2×#41911yescharacteristic
395C22#289611yescharacteristic
396C22#300712yes—
397C22#302012yes—
398C22#302512yes—
399C22#309512yes—
400C4—24no—
401C4—24no—
402C4—22no—
403C4—22no—
404C4—22no—
405C4—22no—
406C4—42no—
407C4—44no—
408C4—42no—
409C4—42no—
410C4—82no—
411C4—82no—
412C4—82no—
413C4—88no—
414C2—21no—
415C2—22no—
416C2—21no—
417C2#1816114yes—
418C2#2091111yescharacteristic
419C2#2095411yescharacteristic
420C2#2095911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2581C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25812
#340592
#207412
C2×#14672
#90232
#90222
#90152
#90142
#80172