Finite Group Databasebeta

← #480276

Subgroups — #480276

Order 512.

1527 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
1#34259C211yesmaximal, characteristic
2#17372C212yesmaximal, complemented
3#11510C214yesmaximal, complemented
4#11507C214yesmaximal, complemented
5#11434C212yesmaximal, complemented
6#11431C212yesmaximal, complemented
7C22×2−1+4C411yescharacteristic
8C2×#520C2211yescharacteristic
9C2×C4×#41C411yescharacteristic
10#2983—22no—
11Q8×C22⋊C4—22no—
12#2339—22no—
13#2339—24no—
14#2311C412yes—
15#2311C411yescharacteristic
16#2310C2211yescharacteristic
17#2310C2212yes—
18#2297C2211yescharacteristic
19#2297C2212yes—
20C2×#391C412yes—
21C2×#391C411yescharacteristic
22#1466—24no—
23#1464C2218yes—
24#1460—22no—
25#1458C2214yes—
26#1319C2214yes—
27#1319C2218yes—
28#1318C2214yes—
29C2×2−1+4—24no—
30C2×2−1+4—24no—
31C2×2−1+4—44no—
32C2×2−1+4—44no—
33C22×#41—22no—
34C22×#41—21no—
35C22×#41C2×C412yes—
36C22×#41C2×C411yescharacteristic
37C22×#41C2×C411yescharacteristic
38C23×Q8—21no—
39C23×Q8C2×C412yes—
40C23×Q8C2×C411yescharacteristic
41#520—44no—
42C4×#41—44no—
43C2×C4×Q8—21no—
44C2×C4×Q8D412yes—
45C2×C4×D4—22no—
46C2×C4×D4—21no—
47C2×C4×D4D412yes—
48C2×C4×D4D414yes—
49C2×#117C2×C411yescharacteristic
50C2×#117C2×C412yes—
51C2×#117C2311yescharacteristic
52C2×#117C2312yes—
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C42C2×C412yes—
56C22×C42C2×C411yescharacteristic
57#446—48no—
58#432—28no—
59#431—24no—
60#391—24no—
61#391—22no—
62#391—22no—
63#391—24no—
64#391C2×C412yes—
65#391C2×C412yes—
66#391C2×C412yes—
67#391C2×C411yescharacteristic
68#391C2×C412yes—
69#391C2312yes—
70#391C2311yescharacteristic
71#390C2×C411yescharacteristic
72#390C2×C412yes—
73#390C2311yescharacteristic
74#386—22no—
75#386—24no—
76#386—22no—
77#386—24no—
78#381—22no—
79#381—24no—
80#380—24no—
81#380—22no—
82C4×C22⋊C4—22no—
83C4×C22⋊C4—24no—
84C2×#95—24no—
85C2×#95—22no—
86#324C2318yes—
87#323—24no—
88#323—28no—
892−1+4—432no—
902−1+4—432no—
91C2×#41—216no—
92C2×#41—28no—
93C2×#41—24no—
94C2×#41—24no—
95C2×#41—24no—
96C2×#41—24no—
97C2×#41—22no—
98C2×#41—22no—
99C2×#41—22no—
100C2×#41—44no—
101C2×#41—44no—
102C2×#41—42no—
103C2×#41—42no—
104C2×#41—48no—
105C2×#41—48no—
106C2×#41—44no—
107C2×#41—44no—
108C2×#41M1614yes—
109C22×Q8—28no—
110C22×Q8—24no—
111C22×Q8—24no—
112C22×Q8—22no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8—42no—
116C22×Q8—44no—
117C22×Q8—42no—
118C22×Q8—44no—
119C22×Q8—44no—
120C22×Q8—44no—
121C22×Q8C22⋊C411yescharacteristic
122C22×Q8M1614yes—
123C22×Q8C22×C411yescharacteristic
124C22×Q8C22×C412yes—
125C22×D4—22no—
126C22×D4—21no—
127C22×D4C22⋊C412yes—
128C22×D4C22⋊C411yescharacteristic
129C22×D4C22×C411yescharacteristic
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4C22⋊C412yes—
137C23×C4C22⋊C414yes—
138C23×C4C22×C412yes—
139C23×C4C22×C411yescharacteristic
140C4×Q8—28no—
141C4×Q8—44no—
142C4×D4—44no—
143C4×D4—48no—
144C4×D4—48no—
145C4×D4—44no—
146#117—42no—
147#117—44no—
148#117—42no—
149#117—44no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4C22⋊C414yes—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C2411yescharacteristic
159C2×C22⋊C4—21no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4C2×D412yes—
165C2×C22⋊C4C2×D414yes—
166C2×C42—22no—
167C2×C42—21no—
168C2×C42—44no—
169C2×C42—42no—
170C2×C42C22×C412yes—
171C2×C42C22×C411yescharacteristic
172C2×C42C22×C411yescharacteristic
173C2×C42C2×D412yes—
174C2×C42C2×D414yes—
175C4⋊C8—44no—
176C4⋊C8—48no—
177C22⋊C8—28no—
178#97—48no—
179C4×C8—44no—
180#95—22no—
181#95—24no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—22no—
186#95—24no—
187C24—21no—
188C24C2×C22⋊C411yescharacteristic
189C24C2×C22⋊C412yes—
190#41—216no—
191#41—216no—
192#41—416no—
193#41—416no—
194#41—48no—
195#41—48no—
196#41—464no—
197#41—432no—
198C2×Q8—28no—
199C2×Q8—21no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—24no—
203C2×Q8—22no—
204C2×Q8—48no—
205C2×Q8—44no—
206C2×Q8—416no—
207C2×Q8—416no—
208C2×Q8—48no—
209C2×Q8—42no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—41no—
213C2×Q8—432no—
214C2×Q8—44no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8#9914yes—
218C2×Q8C2×M1612yes—
219C2×Q8C2×M1614yes—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—28no—
223C2×D4—24no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—24no—
227C2×D4—22no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—42no—
231C2×D4—44no—
232C2×D4—42no—
233C2×D4—44no—
234C2×D4#10014yes—
235C2×D4C2×M1612yes—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—28no—
243C22×C4—28no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—82no—
266C22×C4—81no—
267C22×C4—84no—
268C22×C4—82no—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C414yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×M1612yes—
274C22×C4C2×M1612yes—
275C22×C4C23×C411yescharacteristic
276C22×C4C22×D412yes—
277C2×C8—48no—
278C4⋊C4—28no—
279C4⋊C4—24no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—44no—
284C4⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—42no—
289C22⋊C4—44no—
290C22⋊C4—82no—
291C22⋊C4—84no—
292C42—24no—
293C42—28no—
294C42—41no—
295C42—42no—
296C42—41no—
297C42—44no—
298C42—42no—
299C23—22no—
300C23—21no—
301C23—22no—
302C23—21no—
303C23—22no—
304C23—42no—
305C23—41no—
306C23—42no—
307C23—41no—
308C23—42no—
309C23—42no—
310C23—41no—
311C23—41no—
312C23#40712yes—
313C23C2×#10014yes—
314C23C22×C22⋊C411yescharacteristic
315Q8—28no—
316Q8—44no—
317Q8—42no—
318Q8—42no—
319Q8—44no—
320Q8—416no—
321Q8—48no—
322D4—216no—
323D4—28no—
324D4—42no—
325D4—42no—
326D4—48no—
327D4—416no—
328C2×C4—28no—
329C2×C4—28no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—416no—
343C2×C4—48no—
344C2×C4—416no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—48no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—88no—
362C2×C4—84no—
363C2×C4—164no—
364C2×C4—162no—
365C2×C4#40712yes—
366C2×C4C2×#9914yes—
367C2×C4C2×#9912yes—
368C2×C4C2×#10012yes—
369C2×C4C22×M1611yescharacteristic
370C2×C4C22×M1611yescharacteristic
371C8—88no—
372C22—21no—
373C22—21no—
374C22—22no—
375C22—24no—
376C22—44no—
377C22—42no—
378C22—42no—
379C22—44no—
380C22—44no—
381C22—41no—
382C22—42no—
383C22—41no—
384C22—42no—
385C22—42no—
386C22—82no—
387C22—81no—
388C22—81no—
389C22C2×#40711yescharacteristic
390C22C22×#9911yescharacteristic
391C22C22×#10011yescharacteristic
392C4—21no—
393C4—21no—
394C4—24no—
395C4—42no—
396C4—42no—
397C4—42no—
398C4—44no—
399C4—44no—
400C4—42no—
401C4—82no—
402C4—81no—
403C4#146214yes—
404C2—21no—
405C2—21no—
406C2—42no—
407C2—44no—
408C2—82no—
409C2C2×#146211yescharacteristic
410C2#3336011yescharacteristic
411C2#3337111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×2−1+4C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17372C2semidirect
3. #11510C2semidirect
4. #11507C2semidirect
5. #11434C2semidirect
6. #11431C2semidirect

Maximal subgroups

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

subgroupindex
#342592
#173722
#115102
#115072
#114342
#114312