Finite Group Databasebeta

← #197946

Subgroups — #197946

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3101C211yesmaximal, characteristic
2C2×#2975C211yesmaximal, characteristic
3#21835C211yesmaximal, characteristic, complemented
4#20738C211yesmaximal, characteristic
5C2×#1554C211yesmaximal, characteristic, complemented
6#9225C214yesmaximal, complemented
7#9223C214yesmaximal, complemented
8#8038C211yesmaximal, characteristic, complemented
9#7878C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12#3101—24no—
13#3101C418yescomplemented
14C2×#483—21no—
15C2×#476C2212yes—
16C2×#461C2212yes—
17C2×#450—21no—
18#2975—24no—
19#2975—24no—
20C2×C4×SD16—22no—
21C2×#436—22no—
22C2×#436C2214yes—
23C2×#432C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#407—22no—
28#2285C2211yescharacteristic
29#1990C2212yescomplemented
30#1990C2212yescomplemented
31#1989—22no—
32#1825—22no—
33#1824—21no—
34#1802C2214yes—
35#1786—21no—
36#1762—22no—
37#1554—24no—
38C2×#333C2212yescomplemented
39C2×#332C2212yescomplemented
40#1296C2214yescomplemented
41#1277C2214yescomplemented
42C23×D4D411yescharacteristic
43C24×C4—21no—
44C22×SD16—22no—
45C22×M16—24no—
46#533—22no—
47#533—22no—
48#533—22no—
49#533C2×C414yes—
50C2×C4⋊Q8C2311yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×D4—21no—
58C2×C4×D4—22no—
59C2×C4×D4—21no—
60C2×C4×D4—22no—
61C2×C4×D4C2×C414yes—
62C2×C4×D4D414yes—
63C2×C4×D4D412yes—
64C2×C4×D4C2312yes—
65C2×#117—21no—
66C2×#117—21no—
67C22×C4⋊C4D411yescharacteristic
68C22×C22⋊C4—21no—
69C22×C42C2311yescharacteristic
70#483—28no—
71#476—24no—
72#476C2×C418yes—
73#461—24no—
74#461C2×C418yes—
75#450—28no—
76C4×SD16—216no—
77#436—216no—
78#436—28no—
79#436—28no—
80#432—22no—
81#432—22no—
82#431—22no—
83#431—22no—
84#431—22no—
85#431—22no—
86#431C2×C414yes—
87#431D414yes—
88C2×#106—22no—
89C2×C4⋊C8D412yes—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93C2×#103—22no—
94C2×#102—22no—
95C2×#102—24no—
96#407—216no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8D412yes—
102C2×#97—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2312yes—
106#384C2312yes—
107#382—21no—
108#380—21no—
109#380—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112#340—24no—
113#340—24no—
114#340—24no—
115#333—24no—
116#332—24no—
117C25—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×D4—22no—
121C22×D4—21no—
122C22×D4—44no—
123C22×D4C22⋊C414yes—
124C22×D4C2×D412yes—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4—84no—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4#4111yescharacteristic
134C2×SD16—28no—
135C2×SD16—48no—
136C2×M16—216no—
137C2×M16—88no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8C2×D414yes—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8C22×C412yes—
149C4⋊Q8C22×C412yes—
150#125—24no—
151#125—42no—
152C22⋊Q8—24no—
153C22⋊Q8—24no—
154C22⋊Q8—42no—
155C22⋊Q8—42no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—42no—
159C4×Q8—42no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4C22⋊C418yes—
168C4×D4C22×C414yes—
169C4×D4C2×D414yes—
170#117—24no—
171#117—42no—
172#117—82no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4C22⋊C414yes—
185C2×C4⋊C4C2×D412yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C42—41no—
193C2×C42—41no—
194C2×C42C22×C412yes—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D412yes—
199C2×C42#4111yescharacteristic
200C2×C42#4112yes—
201C2×C42C2411yescharacteristic
202#106—28no—
203C4⋊C8—24no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8C22×C414yes—
209C4⋊C8C2×D414yes—
210#103—28no—
211#102—28no—
212#102—48no—
213C22⋊C8—216no—
214C22⋊C8—24no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C22⋊C8—42no—
218#97—42no—
219C4×C8—24no—
220C4×C8—24no—
221C4×C8—22no—
222C4×C8—22no—
223#95—41no—
224#95—41no—
225C24—22no—
226C24—24no—
227C24—44no—
228C24—82no—
229C24—81no—
230C24—81no—
231C24#12111yescharacteristic
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—44no—
237C2×Q8—81no—
238C2×Q8—81no—
239C2×D4—22no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—48no—
244C2×D4—44no—
245C2×D4—48no—
246C2×D4C2×C22⋊C414yes—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—28no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—48no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—88no—
261C22×C4—88no—
262C22×C4—84no—
263C22×C4—82no—
264C22×C4C2×C22⋊C412yes—
265C22×C4C4×D412yes—
266C22×C4C4×D412yes—
267C22×C4#12012yes—
268C22×C4#12112yes—
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4#12312yes—
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C2×#4111yescharacteristic
276SD16—416no—
277M16—816no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—28no—
282C2×C8—44no—
283C2×C8—44no—
284C2×C8—42no—
285C2×C8—44no—
286C4⋊C4—22no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—82no—
301C4⋊C4C2×C22⋊C414yes—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—28no—
305C22⋊C4—44no—
306C22⋊C4—82no—
307C42—44no—
308C42—82no—
309C42C2×C22⋊C412yes—
310C42C4×D412yes—
311C42C4×D412yes—
312C42#12012yes—
313C42#12112yes—
314C42#12312yes—
315C42C23×C411yescharacteristic
316C42C22×D411yescharacteristic
317C42C22×D411yescharacteristic
318C42C2×#4111yescharacteristic
319C23—22no—
320C23—22no—
321C23—21no—
322C23—21no—
323C23—24no—
324C23—48no—
325C23—44no—
326C23—44no—
327C23—42no—
328C23—42no—
329C23—84no—
330C23—84no—
331C23—88no—
332C23—88no—
333C23#38612yes—
334C23C2×#12111yescharacteristic
335Q8—44no—
336Q8—82no—
337Q8—82no—
338Q8—82no—
339D4—24no—
340D4—24no—
341D4—48no—
342D4—44no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—48no—
358C2×C4—48no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—88no—
362C2×C4—84no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4#38612yes—
370C2×C4#38612yes—
371C2×C4#38612yes—
372C2×C4C22×C22⋊C411yescharacteristic
373C2×C4C2×C4×D411yescharacteristic
374C2×C4C2×C4×D411yescharacteristic
375C2×C4C2×#12011yescharacteristic
376C2×C4C2×#12111yescharacteristic
377C2×C4C2×#12311yescharacteristic
378C2×C4#57514yes—
379C2×C4#57714yes—
380C8—44no—
381C8—44no—
382C8—44no—
383C8—82no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—42no—
388C22—42no—
389C22—44no—
390C22—44no—
391C22—44no—
392C22—816no—
393C22—84no—
394C22—84no—
395C22C2×#38611yescharacteristic
396C22#304212yes—
397C22#305412yes—
398C22#306811yescharacteristic
399C22#306911yescharacteristic
400C4—21no—
401C4—21no—
402C4—22no—
403C4—22no—
404C4—22no—
405C4—44no—
406C4—82no—
407C4—82no—
408C4—162no—
409C4#289612yes—
410C4#294912yes—
411C4#300912yes—
412C4#305412yes—
413C2—42no—
414C2—84no—
415C2#1943012yes—
416C2#1943112yes—
417C2#2093311yescharacteristic
418C2#2097011yescharacteristic
419C2#2100411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21835C2semidirect
5. C2×#1554C2semidirect
6. #9225C2semidirect
7. #9223C2semidirect
8. #8038C2semidirect
9. #7878C2semidirect
13. #3101C4semidirect
29. #1990C22semidirect
30. #1990C22semidirect
38. C2×#333C22semidirect
39. C2×#332C22semidirect
40. #1296C22semidirect
41. #1277C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31012
C2×#29752
#218352
#207382
C2×#15542
#92252
#92232
#80382
#78782