Finite Group Databasebeta

← #163616

Subgroups — #163616

Order 512.

1119 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#36626C211yesmaximal, characteristic
2#20715C212yesmaximal, complemented
3#7701C214yesmaximal, complemented
4#7698C214yesmaximal, complemented
5#7646C212yesmaximal, complemented
6#7645C212yesmaximal, complemented
7C2×#525C411yescharacteristic
8C23×C42C2211yescharacteristic
9C2×#432—22no—
10#2855C412yes—
11#2855C2212yes—
12#2854C412yes—
13#2854C2212yes—
14#2853—22no—
15#2849—22no—
16C4×#123—22no—
17C4×#123—22no—
18#2297C411yescharacteristic
19#2297C2211yescharacteristic
20#2296C411yescharacteristic
21#2296C2211yescharacteristic
22#2280C411yescharacteristic
23#1817C2214yes—
24#1797—22no—
25#1797C2214yes—
26#1779—22no—
27#1749—22no—
28#1749C2214yes—
29#1272C2214yescomplemented
30#1271C2218yescomplemented
31#1270C2214yescomplemented
32C22×#41C2×C411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C24×C4D412yes—
35#525—44no—
36C2×#123—22no—
37C2×#123—21no—
38C2×#123—22no—
39C2×C22⋊Q8C2×C412yes—
40C2×#121C2×C412yes—
41C2×#120—21no—
42C2×C4×D4—22no—
43C2×#117C2×C411yescharacteristic
44C2×#117C2×C411yescharacteristic
45C2×#117C2×C411yescharacteristic
46C22×C42—44no—
47C22×C42C2×C412yes—
48C22×C42C2×C411yescharacteristic
49C22×C42D412yes—
50C22×C42D412yes—
51C22×C42D412yes—
52C22×C42C2312yes—
53C22×C42C2311yescharacteristic
54#432—48no—
55C2×C4⋊C8—24no—
56C2×C22⋊C8—24no—
57C2×C22⋊C8—24no—
58C2×C22⋊C8—24no—
59C2×#97—24no—
60#391C2×C411yescharacteristic
61#391C2×C412yes—
62#391C2311yescharacteristic
63#390C2×C411yescharacteristic
64#390C2×C412yes—
65#390C2311yescharacteristic
66#389C2×C412yes—
67#389C2×C412yes—
68#389C2×C412yes—
69#389C2312yes—
70#388—22no—
71#388—22no—
72#388—22no—
73#388—22no—
74#387—22no—
75#387—22no—
76#387—22no—
77#387—22no—
78#386—22no—
79#386—22no—
80#386—22no—
81#386—22no—
82#385—22no—
83#385—22no—
84#385—22no—
85#385—22no—
86#381—22no—
87#381—21no—
88#381—22no—
89#380C2×C412yes—
90#380C2×C412yes—
91#379—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4—22no—
99#336—24no—
100#336C2318yes—
101C25C22⋊C411yescharacteristic
102C2×#41—42no—
103C2×#41—42no—
104C2×#41—42no—
105C22×Q8C22×C411yescharacteristic
106C22×D4—22no—
107C22×D4—21no—
108C22×D4C22×C411yescharacteristic
109C23×C4—24no—
110C23×C4—21no—
111C23×C4—42no—
112C23×C4—42no—
113C23×C4—44no—
114C23×C4—44no—
115C23×C4—44no—
116C23×C4C22⋊C412yes—
117C23×C4C22⋊C412yes—
118C23×C4C22⋊C412yes—
119C23×C4C22⋊C412yes—
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C22×C412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C22×C8—28no—
127#123—48no—
128#123—48no—
129#123—44no—
130C22⋊Q8—44no—
131#121—44no—
132#120—44no—
133C4×D4—48no—
134#117—42no—
135#117—42no—
136#117—42no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4C22×C412yes—
143C2×C4⋊C4C22×C412yes—
144C2×C4⋊C4C22×C411yescharacteristic
145C2×C4⋊C4C22×C411yescharacteristic
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C42—28no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—44no—
162C2×C42—44no—
163C2×C42—44no—
164C2×C42—44no—
165C2×C42—42no—
166C2×C42—42no—
167C2×C42C22⋊C412yes—
168C2×C42C22⋊C412yes—
169C2×C42C22⋊C411yescharacteristic
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22×C411yescharacteristic
172C2×C42C22×C412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2×D412yes—
176C2×C42C2411yescharacteristic
177C4⋊C8—48no—
178C22⋊C8—48no—
179C22⋊C8—48no—
180C22⋊C8—48no—
181#97—48no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192C24—22no—
193C24—24no—
194C24—21no—
195C24—41no—
196C24—41no—
197C24—42no—
198C24C2×C22⋊C411yescharacteristic
199C24C2×C22⋊C412yes—
200#41—88no—
201C2×Q8—22no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×D4—22no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—44no—
208C2×D4—44no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—44no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—416no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—432no—
251C22×C4—42no—
252C22×C4#9914yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C23×C411yescharacteristic
259C22×C4C22×D412yes—
260C2×C8—48no—
261C2×C8—48no—
262C2×C8—48no—
263C4⋊C4—24no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C22⋊C4—44no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—42no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C42—24no—
284C42—24no—
285C42—24no—
286C42—24no—
287C42—41no—
288C42—41no—
289C42—41no—
290C42—42no—
291C42—42no—
292C42—41no—
293C42—42no—
294C42—42no—
295C42—42no—
296C23—21no—
297C23—22no—
298C23—22no—
299C23—21no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—48no—
307C23—44no—
308C23—44no—
309C23—48no—
310C23—44no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—42no—
315C23—42no—
316C23C2×#9912yes—
317C23#42114yes—
318C23C22×C22⋊C411yescharacteristic
319Q8—82no—
320D4—82no—
321D4—88no—
322D4—84no—
323C2×C4—28no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—44no—
342C2×C4—48no—
343C2×C4—48no—
344C2×C4—48no—
345C2×C4—48no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—48no—
349C2×C4—48no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—84no—
364C2×C4C2×#9914yes—
365C2×C4#41312yes—
366C2×C4#56812yes—
367C2×C4#57514yes—
368C8—88no—
369C22—24no—
370C22—24no—
371C22—24no—
372C22—24no—
373C22—22no—
374C22—416no—
375C22—41no—
376C22—41no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—81no—
383C22—81no—
384C22—82no—
385C22—82no—
386C22—81no—
387C22#287911yescharacteristic
388C22C22×#9911yescharacteristic
389C22C2×#41311yescharacteristic
390C22#289612yes—
391C22C2×#42112yes—
392C4—24no—
393C4—22no—
394C4—22no—
395C4—24no—
396C4—42no—
397C4—42no—
398C4—42no—
399C4—81no—
400C4—81no—
401C4—82no—
402C4—82no—
403C4—81no—
404C2—44no—
405C2—42no—
406C2—161no—
407C2#1739211yescharacteristic
408C2#1749412yes—
409C2#1758212yes—
410C2#1767811yescharacteristic
411C2#1774011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#525C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20715C2semidirect
3. #7701C2semidirect
4. #7698C2semidirect
5. #7646C2semidirect
6. #7645C2semidirect
29. #1272C22semidirect
30. #1271C22semidirect
31. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#366262
#207152
#77012
#76982
#76462
#76452