Finite Group Databasebeta

← #200615

Subgroups — #200615

Order 512.

1660 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34946C211yesmaximal, characteristic
2C4×#518C211yesmaximal, characteristic
3#17545C212yesmaximal, complemented
4#17364C211yesmaximal, characteristic
5#9407C218yesmaximal, complemented
6#8048C212yesmaximal, complemented
7C2×#533C2211yescharacteristic
8C2×#518—21no—
9C22×C4×D4C2211yescharacteristic
10#2920C2212yes—
11#2563C412yescomplemented
12#2562—22no—
13#2561C2214yes—
14#2547C412yescomplemented
15#2503—21no—
16#2435—21no—
17#2398C414yescomplemented
18C4×#124—22no—
19#2280—22no—
20#2275—24no—
21#2274—21no—
22#2273C2214yes—
23C42×D4—24no—
24C4×#117C2211yescharacteristic
25C4×#117C2212yes—
26C2×C4×C22⋊C4—22no—
27#1508C2218yes—
28#1506—24no—
29#1503—24no—
30#1466—22no—
31#1462C2214yes—
32#1460—22no—
33#1297C2218yes—
34C23×D4D411yescharacteristic
35C24×C4D412yes—
36#533—22no—
37#533—22no—
38#533—22no—
39#533—22no—
40C2×C4⋊Q8C2×C411yescharacteristic, complemented
41C2×C4⋊Q8C2×C411yescharacteristic, complemented
42C2×#125—21no—
43C2×#124—22no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8C2312yes—
46#518—216no—
47C2×C4×Q8—21no—
48C2×C4×Q8C2×C412yescomplemented
49C2×C4×D4—28no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4C2×C412yes—
55C2×C4×D4C2×C412yes—
56C2×C4×D4D412yes—
57C2×C4×D4D412yes—
58C2×#117—22no—
59C2×#117—21no—
60C2×#117—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C22⋊C4—24no—
63C22×C22⋊C4C2312yes—
64C22×C42—24no—
65C22×C42C2311yescharacteristic
66#432—22no—
67#432—22no—
68#431—22no—
69#431—22no—
70#423—22no—
71#407C2314yes—
72C4×M16—22no—
73#400—24no—
74#397C2×C418yescomplemented
75#394—24no—
76#393C2×C418yescomplemented
77#391—22no—
78#390—22no—
79#388—28no—
80#382—22no—
81#381—24no—
82#380—24no—
83#380—24no—
84#379—24no—
85C4×C4⋊C4—28no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4C2×C412yescomplemented
89C4×C4⋊C4D414yes—
90C4×C22⋊C4—24no—
91C4×C22⋊C4—24no—
92C4×C22⋊C4—28no—
93C4×C22⋊C4—21no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4—216no—
97C4×C22⋊C4C2312yes—
98C4×C22⋊C4C2314yes—
99#376C2×C412yescomplemented
100#376D414yes—
101C2×#95—24no—
102C43—22no—
103C43—21no—
104#328—216no—
105#323—28no—
106C25C2×D412yes—
107C22×Q8—21no—
108C22×Q8C22×C412yes—
109C22×D4—28no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4C22⋊C412yes—
113C22×D4C22⋊C412yes—
114C23×C4—28no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—48no—
120C23×C4C22⋊C414yes—
121C23×C4C22⋊C414yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2411yescharacteristic
126C2×M16—24no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131#125—42no—
132#125—42no—
133#124—48no—
134C22⋊Q8—24no—
135C22⋊Q8—24no—
136C22⋊Q8—42no—
137C22⋊Q8—42no—
138C4×Q8—24no—
139C4×Q8—24no—
140C4×Q8—42no—
141C4×Q8—42no—
142C4×D4—264no—
143C4×D4—24no—
144C4×D4—24no—
145C4×D4—24no—
146C4×D4—24no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—24no—
151#117—216no—
152#117—44no—
153#117—41no—
154#117—41no—
155#117—41no—
156#117—41no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—232no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4C22×C414yes—
179C2×C22⋊C4C2×D414yes—
180C2×C42—24no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—24no—
185C2×C42—28no—
186C2×C42—24no—
187C2×C42—22no—
188C2×C42—88no—
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42#4112yes—
196C2×C42#4112yes—
197C4⋊C8—42no—
198C4⋊C8—42no—
199C22⋊C8—24no—
200C22⋊C8—24no—
201#97—42no—
202C4×C8—42no—
203#95—28no—
204#95—22no—
205#95—28no—
206#95—28no—
207#95—24no—
208#95C22×C414yes—
209#95C2×D418yes—
210C24—22no—
211C24—22no—
212C24—28no—
213C24—44no—
214C24—82no—
215C24C2×C22⋊C414yes—
216C24C22×D411yescharacteristic
217C2×Q8—24no—
218C2×Q8—42no—
219C2×Q8—44no—
220C2×Q8—82no—
221C2×D4—216no—
222C2×D4—216no—
223C2×D4—28no—
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4#10418yes—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—28no—
232C22×C4—232no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—48no—
242C22×C4—48no—
243C22×C4—48no—
244C22×C4—44no—
245C22×C4—84no—
246C22×C4—84no—
247C22×C4—84no—
248C22×C4—81no—
249C22×C4—88no—
250C22×C4—88no—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C414yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4#12011yescharacteristic
257C22×C4#12012yes—
258C22×C4#12011yescharacteristic
259C22×C4#12312yes—
260C22×C4#12311yescharacteristic
261C22×C4#12311yescharacteristic
262C22×C4C23×C411yescharacteristic
263C22×C4C22×D412yes—
264C22×C4C22×D411yescharacteristic
265C22×C4C2×#4112yes—
266M16—84no—
267C2×C8—44no—
268C4⋊C4—216no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—44no—
279C4⋊C4—42no—
280C22⋊C4—216no—
281C22⋊C4—232no—
282C22⋊C4—28no—
283C22⋊C4—28no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—416no—
288C22⋊C4—88no—
289C42—216no—
290C42—44no—
291C42—42no—
292C42—42no—
293C42—44no—
294C42—48no—
295C42—84no—
296C42—82no—
297C42—88no—
298C42C4×D412yes—
299C42C4×D412yes—
300C42C4×D412yes—
301C42C4×D412yes—
302C42#12112yes—
303C42#12112yes—
304C42#12112yes—
305C42#12112yes—
306C23—28no—
307C23—28no—
308C23—22no—
309C23—22no—
310C23—24no—
311C23—24no—
312C23—41no—
313C23—44no—
314C23—44no—
315C23—44no—
316C23—44no—
317C23—82no—
318C23—82no—
319C23—816no—
320C23C2×#10418yes—
321C23C22×C22⋊C411yescharacteristic
322C23C2×#12011yescharacteristic
323C23C2×#12311yescharacteristic
324Q8—84no—
325Q8—84no—
326D4—216no—
327D4—216no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—216no—
337C2×C4—24no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—48no—
343C2×C4—48no—
344C2×C4—44no—
345C2×C4—48no—
346C2×C4—48no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4—82no—
358C2×C4—832no—
359C2×C4#38611yescharacteristic
360C2×C4#38612yes—
361C2×C4#38611yescharacteristic
362C2×C4#38611yescharacteristic
363C2×C4#38612yes—
364C2×C4#38611yescharacteristic
365C2×C4C2×#10414yes—
366C2×C4#45514yes—
367C2×C4C2×C4×D412yes—
368C2×C4C2×#12112yes—
369C8—84no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—28no—
378C22—44no—
379C22—48no—
380C22—416no—
381C22—44no—
382C22—44no—
383C22—84no—
384C22—88no—
385C22C2×#38611yescharacteristic
386C22C22×#10412yes—
387C22C2×#45512yes—
388C4—22no—
389C4—22no—
390C4—21no—
391C4—21no—
392C4—22no—
393C4—416no—
394C4—44no—
395C4—84no—
396C4—82no—
397C4—82no—
398C4#188512yes—
399C4#188512yes—
400C4#189614yes—
401C2—22no—
402C2—22no—
403C2—41no—
404C2—48no—
405C2C2×#188511yescharacteristic
406C2C2×#189611yescharacteristic
407C2#2101411yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17545C2semidirect
5. #9407C2semidirect
6. #8048C2semidirect
11. #2563C4semidirect
14. #2547C4semidirect
17. #2398C4semidirect
40. C2×C4⋊Q8C2×C4semidirect
41. C2×C4⋊Q8C2×C4semidirect
48. C2×C4×Q8C2×C4semidirect
74. #397C2×C4semidirect
76. #393C2×C4semidirect
88. C4×C4⋊C4C2×C4semidirect
99. #376C2×C4semidirect

Maximal subgroups

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

subgroupindex
#349462
C4×#5182
#175452
#173642
#94072
#80482