Finite Group Databasebeta

← #480352

Subgroups — #480352

Order 512.

1268 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34210C211yesmaximal, characteristic
2#17369C212yesmaximal, complemented
3#11533C214yesmaximal, complemented
4#11529C214yesmaximal, complemented
5#11447C212yesmaximal, complemented
6#11433C212yesmaximal, complemented
7C22×#127C411yescharacteristic
8C2×#518C411yescharacteristic
9C2×#518C2211yescharacteristic
10#2975—22no—
11#2366—22no—
12#2366—22no—
13D4×C22⋊C4—24no—
14#2328C412yes—
15#2328C2212yes—
16#2309C411yescharacteristic
17#2309C2211yescharacteristic
18C4×#127C412yes—
19C4×#127C2212yes—
20#2285C2211yescharacteristic
21C4×#117C411yescharacteristic
22#1465C2218yescomplemented
23#1462—24no—
24#1460—22no—
25#1458C2214yes—
26#1325C2214yescomplemented
27#1324C2214yescomplemented
28#1318C2218yescomplemented
29C23×D4—22no—
30C23×D4C2×C411yescharacteristic
31C23×D4C2×C411yescharacteristic
32C2×#127—22no—
33C2×#127—24no—
34C2×#127—24no—
35C2×#127C2×C412yes—
36C2×#127C2×C412yes—
37#518—44no—
38#518—44no—
39C2×C4×D4—24no—
40C2×C4×D4—24no—
41C2×C4×D4—21no—
42C2×C4×D4—21no—
43C2×C4×D4C2×C412yes—
44C2×C4×D4C2×C412yes—
45C2×C4×D4C2×C412yes—
46C2×C4×D4D412yes—
47C2×C4×D4D412yes—
48C2×C4×D4C2312yes—
49C2×#117C2×C411yescharacteristic
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117C2311yescharacteristic
53C22×C22⋊C4—22no—
54C22×C22⋊C4D414yes—
55C22×C42C2×C411yescharacteristic
56#436—48no—
57#431—24no—
58#407—28no—
59#390—22no—
60#390—24no—
61#390—24no—
62#390—22no—
63#390C2×C412yes—
64#390C2×C412yes—
65#390C2×C412yes—
66#390C2312yes—
67#386—24no—
68#386—24no—
69#386—24no—
70#386—24no—
71#384C2×C411yescharacteristic
72#384C2311yescharacteristic
73#382C2×C411yescharacteristic
74#382C2311yescharacteristic
75#380—22no—
76#379—22no—
77#379—22no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—22no—
81C4×C22⋊C4—22no—
82#376C2×C412yes—
83C43C2×C412yes—
84#324C2318yes—
85#323—24no—
86#323—28no—
87C25—21no—
88C25C22⋊C412yes—
89C22×D4—24no—
90C22×D4—216no—
91C22×D4—24no—
92C22×D4—21no—
93C22×D4—22no—
94C22×D4—22no—
95C22×D4—22no—
96C22×D4—42no—
97C22×D4—42no—
98C22×D4—42no—
99C22×D4—42no—
100C22×D4—84no—
101C22×D4—84no—
102C22×D4C22⋊C412yes—
103C22×D4M1614yescomplemented
104C22×D4C22×C412yes—
105C22×D4C22×C412yes—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4C22⋊C414yes—
110C23×C4C22×C411yescharacteristic
111C23×C4C2×D414yes—
112C2×M16—48no—
113#127—216no—
114#127—24no—
115#127—24no—
116#127—24no—
117#127—24no—
118C4×D4—28no—
119C4×D4—44no—
120C4×D4—44no—
121C4×D4—44no—
122C4×D4—44no—
123C4×D4—44no—
124C4×D4—44no—
125C4×D4—44no—
126C4×D4—88no—
127C4×D4—88no—
128#117—24no—
129#117—42no—
130#117—42no—
131#117—42no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4C22⋊C411yescharacteristic
136C2×C4⋊C4C22⋊C411yescharacteristic
137C2×C4⋊C4C22×C411yescharacteristic
138C2×C4⋊C4C22×C411yescharacteristic
139C2×C4⋊C4C22×C411yescharacteristic
140C2×C4⋊C4C2411yescharacteristic
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—48no—
150C2×C22⋊C4—48no—
151C2×C22⋊C4C22⋊C412yes—
152C2×C22⋊C4C22⋊C414yes—
153C2×C22⋊C4C2×D412yes—
154C2×C22⋊C4C2×D414yes—
155C2×C42—21no—
156C2×C42—22no—
157C2×C42—21no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42M1614yescomplemented
161C2×C42C22×C412yes—
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C412yes—
165C2×C42C2×D412yes—
166C4⋊C8—44no—
167C22⋊C8—28no—
168C22⋊C8—48no—
169C4×C8—44no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95—22no—
174C24—22no—
175C24—24no—
176C24—24no—
177C24—42no—
178C24—42no—
179C24—42no—
180C24—42no—
181C24—82no—
182C24—82no—
183C24—82no—
184C24—82no—
185C24C2×C22⋊C414yes—
186C24C2×C22⋊C411yescharacteristic
187C2×D4—216no—
188C2×D4—216no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—24no—
193C2×D4—24no—
194C2×D4—24no—
195C2×D4—24no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—48no—
201C2×D4—48no—
202C2×D4—48no—
203C2×D4—48no—
204C2×D4—48no—
205C2×D4—48no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—832no—
209C2×D4—84no—
210C2×D4—84no—
211C2×D4C2×M1614yes—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—24no—
217C22×C4—28no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—84no—
227C22×C4—84no—
228C22×C4—84no—
229C22×C4—84no—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×M1612yes—
235C22×C4C2×M1612yes—
236C22×C4C23×C411yescharacteristic
237C22×C4C22×D412yes—
238M16—168no—
239C2×C8—48no—
240C4⋊C4—22no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4#9914yes—
249C22⋊C4—24no—
250C22⋊C4—28no—
251C22⋊C4—42no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—44no—
257C22⋊C4—84no—
258C22⋊C4—84no—
259C22⋊C4—84no—
260C22⋊C4—84no—
261C42—21no—
262C42—21no—
263C42—24no—
264C42—24no—
265C42—24no—
266C42—42no—
267C42—44no—
268C42—44no—
269C42—42no—
270C42C2×M1612yes—
271C42C2×M1612yes—
272C23—21no—
273C23—21no—
274C23—24no—
275C23—24no—
276C23—24no—
277C23—24no—
278C23—41no—
279C23—41no—
280C23—44no—
281C23—44no—
282C23—44no—
283C23—44no—
284C23—48no—
285C23—48no—
286C23—42no—
287C23—42no—
288C23—88no—
289C23—88no—
290C23—162no—
291C23—162no—
292C23—164no—
293C23—164no—
294C23—162no—
295C23—162no—
296C23#40712yes—
297C23C22×C22⋊C411yescharacteristic
298D4—44no—
299D4—44no—
300D4—44no—
301D4—44no—
302D4—432no—
303D4—48no—
304D4—48no—
305D4—816no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—42no—
318C2×C4—48no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—88no—
332C2×C4—164no—
333C2×C4—164no—
334C2×C4—164no—
335C2×C4#40712yes—
336C2×C4C2×#9914yes—
337C2×C4C2×#9912yes—
338C2×C4#41312yes—
339C2×C4C22×M1611yescharacteristic
340C2×C4C22×M1611yescharacteristic
341C8—88no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—42no—
346C22—42no—
347C22—48no—
348C22—48no—
349C22—41no—
350C22—41no—
351C22—44no—
352C22—44no—
353C22—44no—
354C22—44no—
355C22—84no—
356C22—84no—
357C22—82no—
358C22—82no—
359C22—168no—
360C22—168no—
361C22C2×#40711yescharacteristic
362C22C22×#9911yescharacteristic
363C22C2×#41311yescharacteristic
364C4—24no—
365C4—24no—
366C4—21no—
367C4—21no—
368C4—42no—
369C4—42no—
370C4—42no—
371C4—42no—
372C4—82no—
373C4—81no—
374C4—81no—
375C4#146514yes—
376C2—21no—
377C2—21no—
378C2—42no—
379C2—84no—
380C2—84no—
381C2C2×#146511yescharacteristic
382C2#3335911yescharacteristic
383C2#3337311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#127C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17369C2semidirect
3. #11533C2semidirect
4. #11529C2semidirect
5. #11447C2semidirect
6. #11433C2semidirect
22. #1465C22semidirect
26. #1325C22semidirect
27. #1324C22semidirect
28. #1318C22semidirect
103. C22×D4M16semidirect
160. C2×C42M16semidirect

Maximal subgroups

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

subgroupindex
#342102
#173692
#115332
#115292
#114472
#114332