Finite Group Databasebeta

← #162899

Subgroups — #162899

Order 512.

1255 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2C4×#407C212yesmaximal, complemented
3#7675C214yesmaximal, complemented
4#7674C214yesmaximal, complemented
5#7632C212yesmaximal, complemented
6#7631C212yesmaximal, complemented
7C2×#525C411yescharacteristic
8C23×C42C2211yescharacteristic
9C2×#407—22no—
10C2×C4×M16—22no—
11#2855C412yes—
12#2855C2212yes—
13#2854C412yes—
14#2854C2212yes—
15#2853—22no—
16#2849—22no—
17C4×#123—22no—
18C4×#123—22no—
19#2297C411yescharacteristic
20#2297C2211yescharacteristic
21#2296C411yescharacteristic
22#2296C2211yescharacteristic
23#2280C411yescharacteristic
24#1749—22no—
25#1749C2214yes—
26C4×C22⋊C8—22no—
27C4×C22⋊C8C2214yes—
28#1728C2214yes—
29#1272C2214yescomplemented
30#1271C2218yescomplemented
31#1270C2214yescomplemented
32C22×#41C2×C411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C24×C4D412yes—
35C22×M16—24no—
36#525—44no—
37C2×#123—22no—
38C2×#123—21no—
39C2×#123—22no—
40C2×C22⋊Q8C2×C412yes—
41C2×#121C2×C412yes—
42C2×#120—21no—
43C2×C4×D4—22no—
44C2×#117C2×C411yescharacteristic
45C2×#117C2×C411yescharacteristic
46C2×#117C2×C411yescharacteristic
47C22×C42—44no—
48C22×C42C2×C412yes—
49C22×C42C2×C411yescharacteristic
50C22×C42D412yes—
51C22×C42D412yes—
52C22×C42D412yes—
53C22×C42C2312yes—
54C22×C42C2311yescharacteristic
55#407—216no—
56C2×C22⋊C8—24no—
57C2×C22⋊C8—24no—
58C4×M16—48no—
59C2×#97—24no—
60C2×C4×C8—24no—
61#391C2×C411yescharacteristic
62#391C2×C412yes—
63#391C2311yescharacteristic
64#390C2×C411yescharacteristic
65#390C2×C412yes—
66#390C2311yescharacteristic
67#389C2×C412yes—
68#389C2×C412yes—
69#389C2×C412yes—
70#389C2312yes—
71#388—22no—
72#388—22no—
73#388—22no—
74#388—22no—
75#387—22no—
76#387—22no—
77#387—22no—
78#387—22no—
79#386—22no—
80#386—22no—
81#386—22no—
82#386—22no—
83#385—22no—
84#385—22no—
85#385—22no—
86#385—22no—
87#381—22no—
88#381—21no—
89#381—22no—
90#380C2×C412yes—
91#380C2×C412yes—
92#379—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4—21no—
99C4×C22⋊C4—22no—
100#336—24no—
101#336C2318yes—
102C25C22⋊C411yescharacteristic
103C2×#41—42no—
104C2×#41—42no—
105C2×#41—42no—
106C22×Q8C22×C411yescharacteristic
107C22×D4—22no—
108C22×D4—21no—
109C22×D4C22×C411yescharacteristic
110C23×C4—24no—
111C23×C4—28no—
112C23×C4—21no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—44no—
116C23×C4—44no—
117C23×C4C22⋊C412yes—
118C23×C4C22⋊C412yes—
119C23×C4C22⋊C412yes—
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22⋊C412yes—
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C2×M16—216no—
128C2×M16—48no—
129C2×M16—48no—
130C22×C8—28no—
131#123—48no—
132#123—44no—
133#123—48no—
134C22⋊Q8—44no—
135#121—44no—
136#120—44no—
137C4×D4—48no—
138#117—42no—
139#117—42no—
140#117—42no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4C22×C412yes—
147C2×C4⋊C4C22×C412yes—
148C2×C4⋊C4C22×C411yescharacteristic
149C2×C4⋊C4C22×C411yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C42—28no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—44no—
166C2×C42—44no—
167C2×C42—44no—
168C2×C42—44no—
169C2×C42—42no—
170C2×C42—42no—
171C2×C42C22⋊C412yes—
172C2×C42C22⋊C412yes—
173C2×C42C22⋊C411yescharacteristic
174C2×C42C22⋊C411yescharacteristic
175C2×C42C22×C412yes—
176C2×C42C22×C411yescharacteristic
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2×D412yes—
180C2×C42C2411yescharacteristic
181C22⋊C8—216no—
182C22⋊C8—216no—
183#97—48no—
184C4×C8—48no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195C24—22no—
196C24—24no—
197C24—21no—
198C24—41no—
199C24—41no—
200C24—42no—
201C24C2×C22⋊C411yescharacteristic
202C24C2×C22⋊C412yes—
203#41—88no—
204C2×Q8—22no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×D4—22no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—44no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—432no—
250C22×C4—416no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4#9914yes—
255C22×C4#10418yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D412yes—
263M16—432no—
264C2×C8—216no—
265C2×C8—48no—
266C2×C8—48no—
267C4⋊C4—24no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C22⋊C4—42no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C42—24no—
288C42—24no—
289C42—24no—
290C42—24no—
291C42—41no—
292C42—41no—
293C42—41no—
294C42—42no—
295C42—42no—
296C42—41no—
297C42—42no—
298C42—42no—
299C42—42no—
300C23—21no—
301C23—22no—
302C23—22no—
303C23—21no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—48no—
311C23—44no—
312C23—44no—
313C23—48no—
314C23—44no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—42no—
319C23—42no—
320C23C2×#9912yes—
321C23C2×#10414yes—
322C23C22×C22⋊C411yescharacteristic
323Q8—82no—
324D4—82no—
325D4—88no—
326D4—84no—
327C2×C4—216no—
328C2×C4—28no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—44no—
345C2×C4—48no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—48no—
349C2×C4—48no—
350C2×C4—48no—
351C2×C4—48no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—84no—
366C2×C4C2×#9914yes—
367C2×C4#41312yes—
368C2×C4C2×#10414yes—
369C2×C4C2×#10414yes—
370C2×C4#56812yes—
371C2×C4#57514yes—
372C8—416no—
373C22—24no—
374C22—24no—
375C22—24no—
376C22—24no—
377C22—22no—
378C22—416no—
379C22—41no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—41no—
385C22—42no—
386C22—81no—
387C22—82no—
388C22—82no—
389C22—81no—
390C22—81no—
391C22#287911yescharacteristic
392C22C22×#9911yescharacteristic
393C22C2×#41311yescharacteristic
394C22#289612yes—
395C22C22×#10412yes—
396C4—24no—
397C4—24no—
398C4—24no—
399C4—22no—
400C4—22no—
401C4—24no—
402C4—42no—
403C4—81no—
404C4—82no—
405C4—82no—
406C4—81no—
407C4—81no—
408C2—44no—
409C2—42no—
410C2—161no—
411C2#1739211yescharacteristic
412C2#1750312yes—
413C2#1754212yes—
414C2#1763811yescharacteristic
415C2#1775711yescharacteristic
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. C4×#407C2semidirect
3. #7675C2semidirect
4. #7674C2semidirect
5. #7632C2semidirect
6. #7631C2semidirect
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
C4×#4072
#76752
#76742
#76322
#76312