Finite Group Databasebeta

← #164814

Subgroups — #164814

Order 512.

901 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36406C211yesmaximal, characteristic
2#22012C212yesmaximal, complemented
3#7727C214yesmaximal, complemented
4#7682C212yesmaximal, complemented
5#7645C212yesmaximal, complemented
6#7635C211yesmaximal, characteristic, complemented
7#7597C212yesmaximal, complemented
8#7567C211yesmaximal, characteristic, complemented
9C2×#556C411yescharacteristic
10C2×#547C411yescharacteristic
11#2862C2211yescharacteristic
12#2857C2212yes—
13#2553—24no—
14#2546C411yescharacteristic
15#2542C411yescharacteristic
16#2499C412yes—
17#2432C412yes—
18#2427—22no—
19#2360—22no—
20C4×#124C2211yescharacteristic
21#2297C2212yes—
22C42×D4C2211yescharacteristic
23#1954—22no—
24#1939—22no—
25#1924—24no—
26#1817C2214yes—
27#1797C2214yes—
28#1767C2212yes—
29#1744C2212yes—
30#1272C2214yes—
31#1272C2214yes—
32#1272C2212yes—
33#1272C2212yes—
34#1270C2211yescharacteristic
35#1270C2211yescharacteristic
36#1270C2212yes—
37C22×#41C2×C412yes—
38#556—44no—
39#547—44no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×C4⋊Q8C2×C412yes—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#126—22no—
44C2×#124—21no—
45C2×#124C2×C411yescharacteristic
46C2×#123—22no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8C2×C412yes—
49C2×C22⋊Q8C2×C412yes—
50C2×#121—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4D412yes—
56C2×#117C2×C412yes—
57C22×C4⋊C4—21no—
58C22×C42C2312yes—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—24no—
61C2×C22⋊C8—24no—
62C2×C22⋊C8—24no—
63C2×#97—22no—
64C2×C4×C8—22no—
65#399—22no—
66#398—22no—
67#397—24no—
68#394—24no—
69#391—22no—
70#391C2×C412yes—
71#391C2×C412yes—
72#391C2311yescharacteristic
73#391C2312yes—
74#390C2×C412yes—
75#390C2311yescharacteristic
76#389—22no—
77#389—24no—
78#388—22no—
79#387—24no—
80#386—24no—
81#385—24no—
82#384C2×C412yes—
83#382—22no—
84#382C2×C411yescharacteristic
85#380—24no—
86C4×C4⋊C4C2×C411yescharacteristic
87C4×C4⋊C4D412yes—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—24no—
90C4×C22⋊C4C2×C412yes—
91C4×C22⋊C4D414yes—
92C43C2311yescharacteristic
93#336—24no—
94#336—22no—
95#336C2312yes—
96#336C2314yes—
97#336C2312yes—
98C2×#41—24no—
99C2×#41—42no—
100C2×#41—44no—
101C2×#41—82no—
102C22×Q8—21no—
103C22×Q8C22×C411yescharacteristic
104C22×Q8C22×C412yes—
105C22×D4—22no—
106C22×D4C22⋊C411yescharacteristic
107C22×D4C22×C411yescharacteristic
108C23×C4—21no—
109C23×C4—22no—
110C23×C4C22×C412yes—
111C23×C4C2×D414yes—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—24no—
115C4⋊Q8—42no—
116C4⋊Q8—42no—
117C4⋊Q8—82no—
118#126—48no—
119#124—22no—
120#124—41no—
121#124—44no—
122#123—48no—
123C22⋊Q8—48no—
124C22⋊Q8—44no—
125C22⋊Q8—44no—
126#121—44no—
127C4×Q8—42no—
128C4×Q8—44no—
129C4×D4—24no—
130C4×D4—44no—
131C4×D4—42no—
132#117—44no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22⋊C411yescharacteristic
144C2×C4⋊C4C22×C411yescharacteristic
145C2×C4⋊C4C22×C412yes—
146C2×C4⋊C4C22×C411yescharacteristic
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4C22⋊C412yes—
155C2×C22⋊C4C22⋊C414yes—
156C2×C42—22no—
157C2×C42—21no—
158C2×C42—82no—
159C2×C42C22×C411yescharacteristic
160C2×C42C22×C411yescharacteristic
161C2×C42C22×C412yes—
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D414yes—
165C2×C42C2411yescharacteristic
166C4⋊C8—44no—
167C4⋊C8—84no—
168C22⋊C8—84no—
169C22⋊C8—84no—
170#97—82no—
171C4×C8—44no—
172#95—22no—
173#95—24no—
174#95—24no—
175#95—22no—
176#95—24no—
177#95C22⋊C414yes—
178#95C22⋊C412yes—
179C24—21no—
180C24C2×C22⋊C412yes—
181#41—88no—
182#41—88no—
183C2×Q8—21no—
184C2×Q8—22no—
185C2×Q8—21no—
186C2×Q8—22no—
187C2×Q8—41no—
188C2×Q8—42no—
189C2×Q8—41no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—44no—
193C2×D4—21no—
194C2×D4—22no—
195C2×D4—24no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—44no—
199C2×D4—44no—
200C2×D4—82no—
201C2×D4#13112yes—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—42no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—82no—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C414yes—
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C23×C411yescharacteristic
226C22×C4C22×D412yes—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—44no—
231C2×C8—44no—
232C2×C8—42no—
233C2×C8—44no—
234C2×C8—42no—
235C2×C8—42no—
236C4⋊C4—24no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—48no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—84no—
248C4⋊C4#10014yes—
249C22⋊C4—24no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—42no—
253C22⋊C4—48no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—44no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C42—21no—
260C42—21no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—24no—
265C42—44no—
266C42—41no—
267C42—42no—
268C42—44no—
269C42—42no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—24no—
275C23—24no—
276C23—41no—
277C23—41no—
278C23—42no—
279C23—42no—
280C23C22×C22⋊C411yescharacteristic
281Q8—81no—
282Q8—81no—
283Q8—82no—
284Q8—82no—
285Q8—84no—
286D4—42no—
287D4—42no—
288D4—44no—
289D4—88no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—84no—
319C2×C4—84no—
320C2×C4—88no—
321C2×C4—84no—
322C2×C4C2×#10014yes—
323C2×C4C2×#10012yes—
324C2×C4#42114yes—
325C2×C4C2×#13111yescharacteristic
326C2×C4#56811yescharacteristic
327C2×C4#57712yes—
328C2×C4#57812yes—
329C8—82no—
330C8—84no—
331C8—82no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—44no—
336C22—44no—
337C22—44no—
338C22—42no—
339C22—81no—
340C22—82no—
341C22—81no—
342C22#288011yescharacteristic
343C22#288311yescharacteristic
344C22C22×#10011yescharacteristic
345C22#289812yes—
346C22C2×#42112yes—
347C4—21no—
348C4—21no—
349C4—24no—
350C4—22no—
351C4—22no—
352C4—41no—
353C4—42no—
354C4—44no—
355C4—82no—
356C4—81no—
357C4—81no—
358C4—81no—
359C4—82no—
360C4#153614yes—
361C2—82no—
362C2—82no—
363C2#1742111yescharacteristic
364C2#1752312yes—
365C2#1760012yes—
366C2C2×#153611yescharacteristic
367C2#1777311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#547C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22012C2semidirect
3. #7727C2semidirect
4. #7682C2semidirect
5. #7645C2semidirect
6. #7635C2semidirect
7. #7597C2semidirect
8. #7567C2semidirect

Maximal subgroups

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

subgroupindex
#364062
#220122
#77272
#76822
#76452
#76352
#75972
#75672