Finite Group Databasebeta

← #480270

Subgroups — #480270

Order 512.

1532 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34173C211yesmaximal, characteristic
2#17371C212yesmaximal, complemented
3#11504C214yesmaximal, complemented
4#11501C214yesmaximal
5#11428C212yesmaximal, complemented
6#11425C212yesmaximal
7C24×Q8C411yescharacteristic
8C2×#519C411yescharacteristic
9C2×#519C2211yescharacteristic
10#2983—22no—
11Q8×C22⋊C4—22no—
12Q8×C22⋊C4—24no—
13Q8×C22⋊C4—22no—
14#2310C412yes—
15#2310C411yescharacteristic
16#2310C2211yescharacteristic
17#2310C2212yes—
18#2280C412yes—
19#2280C411yescharacteristic
20#2280C2211yescharacteristic
21#2280C2212yes—
22#1467—24no—
23#1464C2218yes—
24#1461—22no—
25#1458C2214yes—
26#1317C2214yes—
27#1317C2218yes—
28#1316C2214yes—
29C23×Q8—22no—
30C23×Q8—21no—
31C23×Q8—21no—
32C23×Q8—24no—
33C23×Q8—24no—
34C23×Q8—44no—
35C23×Q8—44no—
36C23×Q8C2×C411yescharacteristic
37C23×Q8C2×C412yes—
38C23×Q8C2×C411yescharacteristic
39C24×C4C2×C412yes—
40C24×C4C2×C411yescharacteristic
41#519—44no—
42#519—44no—
43C2×C4×Q8—22no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8D412yes—
47C2×C4×Q8D414yes—
48C2×C4×Q8D412yes—
49C2×#117C2×C412yes—
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117C2×C412yes—
53C2×#117C2×C412yes—
54C2×#117C2×C411yescharacteristic
55C2×#117C2311yescharacteristic
56C2×#117C2312yes—
57#446—48no—
58#432—28no—
59#431—24no—
60#391—24no—
61#391—22no—
62#391—22no—
63#391—24no—
64#391—22no—
65#391—24no—
66#391—22no—
67#391—24no—
68#381C2×C411yescharacteristic
69#381C2×C412yes—
70#381C2×C412yes—
71#381C2×C412yes—
72#381C2×C412yes—
73#381C2312yes—
74#381C2311yescharacteristic
75#380—24no—
76#380—22no—
77#380—22no—
78#380—24no—
79#379C2×C411yescharacteristic
80#379C2×C412yes—
81#379C2311yescharacteristic
82C4×C22⋊C4—22no—
83C4×C22⋊C4—24no—
84C4×C22⋊C4—22no—
85C4×C22⋊C4—24no—
86#324—24no—
87#324—28no—
88#323C2318yes—
89C25C22×C411yescharacteristic
90C22×Q8—24no—
91C22×Q8—24no—
92C22×Q8—216no—
93C22×Q8—28no—
94C22×Q8—28no—
95C22×Q8—21no—
96C22×Q8—22no—
97C22×Q8—22no—
98C22×Q8—21no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—24no—
102C22×Q8—24no—
103C22×Q8—22no—
104C22×Q8—42no—
105C22×Q8—44no—
106C22×Q8—42no—
107C22×Q8—44no—
108C22×Q8—432no—
109C22×Q8—432no—
110C22×Q8—48no—
111C22×Q8—44no—
112C22×Q8—48no—
113C22×Q8—44no—
114C22×Q8—44no—
115C22×Q8—44no—
116C22×Q8C22⋊C412yes—
117C22×Q8C22⋊C411yescharacteristic
118C22×Q8C22⋊C411yescharacteristic
119C22×Q8M1614yes—
120C23×C4—24no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—24no—
126C23×C4—22no—
127C23×C4—44no—
128C23×C4—42no—
129C23×C4—44no—
130C23×C4—42no—
131C23×C4M1614yes—
132C23×C4C22×C412yes—
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C412yes—
135C23×C4C22×C411yescharacteristic
136C4×Q8—28no—
137C4×Q8—44no—
138C4×Q8—48no—
139C4×Q8—48no—
140C4×Q8—44no—
141C4×Q8—44no—
142#117—42no—
143#117—44no—
144#117—44no—
145#117—42no—
146#117—44no—
147#117—42no—
148#117—42no—
149#117—44no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4C22⋊C414yes—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D414yes—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4C22×C412yes—
162C2×C22⋊C4C22×C411yescharacteristic
163C2×C22⋊C4C22×C411yescharacteristic
164C2×C22⋊C4C22×C411yescharacteristic
165C2×C22⋊C4C22×C412yes—
166C2×C22⋊C4C2411yescharacteristic
167C2×C42—21no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—21no—
171C2×C42C22⋊C412yes—
172C2×C42C22⋊C414yes—
173C2×C42C2×D412yes—
174C2×C42C2×D414yes—
175C4⋊C8—44no—
176C4⋊C8—48no—
177C22⋊C8—28no—
178#97—48no—
179C4×C8—44no—
180#95—24no—
181#95—24no—
182#95—24no—
183#95—22no—
184#95—24no—
185#95—22no—
186#95—24no—
187C24—21no—
188C24—21no—
189C24—41no—
190C24—41no—
191C24—44no—
192C24C2×M1612yes—
193C24C23×C411yescharacteristic
194C2×Q8—216no—
195C2×Q8—216no—
196C2×Q8—28no—
197C2×Q8—28no—
198C2×Q8—24no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—24no—
202C2×Q8—24no—
203C2×Q8—24no—
204C2×Q8—22no—
205C2×Q8—24no—
206C2×Q8—22no—
207C2×Q8—48no—
208C2×Q8—416no—
209C2×Q8—416no—
210C2×Q8—42no—
211C2×Q8—44no—
212C2×Q8—42no—
213C2×Q8—44no—
214C2×Q8—48no—
215C2×Q8—464no—
216C2×Q8—432no—
217C2×Q8—432no—
218C2×Q8—44no—
219C2×Q8—42no—
220C2×Q8—44no—
221C2×Q8—42no—
222C2×Q8#9914yes—
223C2×Q8#10014yes—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—28no—
231C22×C4—28no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—416no—
242C22×C4—416no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—48no—
248C22×C4—48no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C414yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×M1612yes—
264C22×C4C2×M1612yes—
265C22×C4C2×M1614yes—
266C22×C4C2×M1612yes—
267C22×C4C22×D412yes—
268C2×C8—48no—
269C4⋊C4—28no—
270C4⋊C4—24no—
271C4⋊C4—42no—
272C4⋊C4—44no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C22⋊C4—41no—
278C22⋊C4—42no—
279C22⋊C4—41no—
280C22⋊C4—44no—
281C22⋊C4—48no—
282C22⋊C4—44no—
283C22⋊C4—42no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C42—24no—
287C42—28no—
288C42—42no—
289C42—44no—
290C42—42no—
291C42—44no—
292C42—44no—
293C42—42no—
294C23—24no—
295C23—22no—
296C23—21no—
297C23—21no—
298C23—42no—
299C23—44no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—44no—
304C23—44no—
305C23—48no—
306C23—44no—
307C23—41no—
308C23—41no—
309C23C22×C22⋊C411yescharacteristic
310C23C22×M1611yescharacteristic
311C23C22×M1611yescharacteristic
312Q8—216no—
313Q8—28no—
314Q8—28no—
315Q8—48no—
316Q8—416no—
317Q8—416no—
318Q8—48no—
319C2×C4—28no—
320C2×C4—28no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—48no—
335C2×C4—416no—
336C2×C4—416no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—48no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4#40712yes—
356C2×C4#40712yes—
357C2×C4C2×#9914yes—
358C2×C4C2×#9912yes—
359C2×C4C2×#10014yes—
360C2×C4C2×#10012yes—
361C8—88no—
362C22—22no—
363C22—24no—
364C22—21no—
365C22—21no—
366C22—41no—
367C22—41no—
368C22—44no—
369C22—42no—
370C22—42no—
371C22—44no—
372C22—44no—
373C22—48no—
374C22—44no—
375C22—42no—
376C22—42no—
377C22C2×#40711yescharacteristic
378C22C22×#9911yescharacteristic
379C22C22×#10011yescharacteristic
380C4—22no—
381C4—24no—
382C4—24no—
383C4—42no—
384C4—42no—
385C4—44no—
386C4—44no—
387C4—42no—
388C4—82no—
389C4—81no—
390C4—81no—
391C4#146214yes—
392C2—21no—
393C2—21no—
394C2—44no—
395C2—41no—
396C2—41no—
397C2C2×#146211yescharacteristic
398C2#3334912yes—
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#519 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17371C2semidirect
3. #11504C2semidirect
5. #11428C2semidirect

Maximal subgroups

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

subgroupindex
#341732
#173712
#115042
#115012
#114282
#114252