Finite Group Databasebeta

← #204124

Subgroups — #204124

Order 512.

1530 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#124C211yesmaximal, characteristic
2#20810C211yesmaximal, characteristic, complemented
3C42×M16C211yesmaximal, characteristic, complemented
4C2×#1491C212yesmaximal, complemented
5C2×#1490C212yesmaximal
6#9646C214yesmaximal, complemented
7#9639C214yesmaximal, complemented
8C22×#124C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11C2×#432C2212yes—
12C2×C4×M16—22no—
13C2×C4×M16C2212yes—
14C4×#124—22no—
15C4×#124—22no—
16C4×#124C412yes—
17C4×#124C412yes—
18C4×#124C2212yes—
19C4×#124C2212yes—
20C2×#391C411yescharacteristic
21C2×#390C411yescharacteristic
22C2×#388—22no—
23C2×#382C411yescharacteristic
24C2×C4×C22⋊C4—22no—
25C2×C43C2211yescharacteristic
26#1842C2212yescomplemented
27#1831C2212yescomplemented
28#1797—22no—
29#1767—22no—
30C4×#97—22no—
31C4×#97C2212yescomplemented
32C42×C8C2212yescomplemented
33#1491—28no—
34#1490—28no—
35C2×Q8⋊C8C2218yescomplemented
36C2×D4⋊C8C2218yes—
37C23×Q8C2×C411yescharacteristic
38C23×D4C2×C411yescharacteristic
39C24×C4—21no—
40C22×M16—22no—
41C2×#124—28no—
42C2×#124—22no—
43C2×#124C2×C412yes—
44C2×#124C2×C412yes—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—24no—
48C2×C4×Q8—24no—
49C2×C4×Q8C2×C412yes—
50C2×C4×Q8C2312yes—
51C2×C4×D4—21no—
52C2×C4×D4—21no—
53C2×C4×D4—24no—
54C2×C4×D4—24no—
55C2×C4×D4C2×C412yes—
56C2×C4×D4C2312yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C42—21no—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42D412yes—
67C22×C42D412yes—
68C22×C42C2311yescharacteristic
69#432—24no—
70#432—24no—
71C2×C4⋊C8—24no—
72C2×C4⋊C8C2314yes—
73C2×C22⋊C8—22no—
74C4×M16—216no—
75C4×M16—24no—
76C4×M16D418yes—
77C2×#97—24no—
78C2×#97—22no—
79C2×#97—22no—
80C2×C4×C8—24no—
81C2×C4×C8C2314yes—
82#391—22no—
83#391C2×C412yes—
84#391C2×C412yes—
85#390—22no—
86#390C2×C412yes—
87#390C2×C412yes—
88#388—24no—
89#388—24no—
90#388—24no—
91#388—24no—
92#382—22no—
93#382C2×C412yes—
94#382C2×C412yes—
95C4×C22⋊C4—24no—
96C4×C22⋊C4—24no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4—24no—
99C2×#95—22no—
100C2×#95—22no—
101C43C2×C411yescharacteristic
102C43C2×C411yescharacteristic
103C43D412yes—
104C43D412yes—
105C43C2311yescharacteristic
106C43C2311yescharacteristic
107#336—24no—
108Q8⋊C8—216no—
109D4⋊C8—216no—
110C25—21no—
111C22×Q8—24no—
112C22×Q8—21no—
113C22×Q8—24no—
114C22×Q8—42no—
115C22×Q8C22×C412yes—
116C22×D4—24no—
117C22×D4—21no—
118C22×D4—24no—
119C22×D4—42no—
120C22×D4C22×C412yes—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—48no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C412yes—
131C23×C4C22×C411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C2×M16—216no—
135C2×M16—28no—
136C22×C8—24no—
137C22×C8—24no—
138#124—216no—
139#124—24no—
140#124—24no—
141#124—24no—
142#124M1618yescomplemented
143C4×Q8—28no—
144C4×Q8—24no—
145C4×Q8—24no—
146C4×Q8—44no—
147C4×Q8—42no—
148C4×Q8—42no—
149C4×D4—28no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—44no—
153C4×D4—42no—
154C4×D4—42no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C22⋊C4—216no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—48no—
172C2×C22⋊C4—44no—
173C2×C42—28no—
174C2×C42—24no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—42no—
178C2×C42—44no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42C22⋊C412yes—
182C2×C42C22⋊C412yes—
183C2×C42C22⋊C412yes—
184C2×C42C22⋊C412yes—
185C2×C42C22⋊C412yes—
186C2×C42C22⋊C412yes—
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D414yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42#4114yes—
200C2×C42C2411yescharacteristic
201C4⋊C8—24no—
202C4⋊C8—24no—
203C4⋊C8—48no—
204C22⋊C8—44no—
205#97—216no—
206#97—28no—
207#97—44no—
208C4×C8—216no—
209C4×C8—24no—
210C4×C8C2×D418yes—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—24no—
215C24—24no—
216C24—22no—
217C24—44no—
218C24—42no—
219C24—42no—
220C24—44no—
221C24C2×C22⋊C411yescharacteristic
222C2×Q8—24no—
223C2×Q8—22no—
224C2×Q8—24no—
225C2×Q8—24no—
226C2×Q8—416no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8C2×M1614yes—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—22no—
233C2×D4—24no—
234C2×D4—416no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4C2×M1614yes—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—48no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—48no—
261C22×C4—44no—
262C22×C4—88no—
263C22×C4—88no—
264C22×C4#10418yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C4×D414yes—
273C22×C4#12114yes—
274C22×C4#12412yes—
275C22×C4#12712yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279M16—232no—
280C2×C8—216no—
281C2×C8—28no—
282C2×C8—44no—
283C2×C8—48no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C22⋊C4—216no—
291C22⋊C4—216no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—48no—
295C22⋊C4—48no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—24no—
300C42—24no—
301C42—24no—
302C42—24no—
303C42—24no—
304C42—24no—
305C42—24no—
306C42—42no—
307C42—42no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42—44no—
312C42C4×D414yes—
313C42#12114yes—
314C42#12412yes—
315C42#12712yes—
316C42C2×M1614yes—
317C42C22×D412yes—
318C42C2×#4112yes—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—24no—
323C23—48no—
324C23—416no—
325C23—42no—
326C23—44no—
327C23—44no—
328C23—42no—
329C23—84no—
330C23—88no—
331C23—84no—
332C23#39012yes—
333C23C2×#10414yes—
334C23C22×C22⋊C411yescharacteristic
335Q8—28no—
336Q8—48no—
337Q8—44no—
338D4—28no—
339D4—48no—
340D4—44no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—28no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—28no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—48no—
361C2×C4—48no—
362C2×C4—48no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—88no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4#39012yes—
375C2×C4#39012yes—
376C2×C4#39012yes—
377C2×C4#40714yes—
378C2×C4C2×#10414yes—
379C2×C4C2×#10414yes—
380C2×C4C2×C4×D412yes—
381C2×C4C2×#12112yes—
382C2×C4C2×#12411yescharacteristic
383C2×C4C2×#12711yescharacteristic
384C2×C4C22×M1612yes—
385C2×C4#56812yes—
386C2×C4#57514yes—
387C8—216no—
388C8—84no—
389C22—24no—
390C22—24no—
391C22—22no—
392C22—22no—
393C22—48no—
394C22—44no—
395C22—816no—
396C22—84no—
397C22—82no—
398C22—82no—
399C22#196214yes—
400C22C2×#39011yescharacteristic
401C22C2×#40711yescharacteristic
402C22#287911yescharacteristic
403C22#289612yes—
404C22C22×#10412yes—
405C4—24no—
406C4—24no—
407C4—24no—
408C4—22no—
409C4—22no—
410C4—22no—
411C4—22no—
412C4—42no—
413C4—44no—
414C4—82no—
415C4—82no—
416C4D4×M1614yes—
417C4#314012yes—
418C4#315312yes—
419C2—24no—
420C2—84no—
421C2#1744712yes—
422C2#1750212yes—
423C2#2119011yescharacteristic
424C2#2120411yescharacteristic
425C2C2×#196211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20810C2semidirect
3. C42×M16C2semidirect
4. C2×#1491C2semidirect
6. #9646C2semidirect
7. #9639C2semidirect
26. #1842C22semidirect
27. #1831C22semidirect
31. C4×#97C22semidirect
32. C42×C8C22semidirect
35. C2×Q8⋊C8C22semidirect
142. #124M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1242
#208102
C42×M162
C2×#14912
C2×#14902
#96462
#96392