Finite Group Databasebeta

← #239881

Subgroups — #239881

Order 512.

1032 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35281C211yesmaximal, characteristic
2#21885C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic
4#21169C211yesmaximal, characteristic
5C22×#332C211yesmaximal, characteristic
6C2×#1481C211yesmaximal, characteristic
7#10419C214yesmaximal, complemented
8#10418C214yesmaximal, complemented
9#7891C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#427—21no—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16#2868—21no—
17#2862—21no—
18#2660—22no—
19#2567—22no—
20#2546C2214yes—
21#2502—22no—
22#2434—22no—
23#2317—21no—
24#2317—21no—
25#2285C2211yescharacteristic
26C4×#117C2211yescharacteristic
27#1990C2214yes—
28#1987—22no—
29#1987—21no—
30#1981—21no—
31#1949—22no—
32#1948C2212yes—
33#1947C2212yes—
34#1892—22no—
35#1824—22no—
36#1803C2214yes—
37#1735—21no—
38C2×#332—24no—
39C2×#332C2212yes—
40C2×#332C2212yes—
41#1481—24no—
42C2×Q8⋊C8C2212yes—
43C2×Q8⋊C8C2212yes—
44#1279C2214yes—
45C23×Q8—21no—
46C23×C8—21no—
47C23×C8—21no—
48#533—22no—
49#533—22no—
50#533—22no—
51#533—22no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D412yes—
62C2×C4⋊Q8C2312yes—
63C2×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8D412yes—
66C2×C4×Q8—21no—
67C2×C4×Q8C2312yes—
68C2×#117—21no—
69C2×#117—21no—
70C2×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4D412yes—
74C22×C42C2311yescharacteristic
75#431—22no—
76#431—22no—
77#427—44no—
78C2×#107—22no—
79C2×#106—22no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8D414yes—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#103—24no—
91C2×C22⋊C8—21no—
92C2×C4×C8C2312yes—
93#398—28no—
94#395—28no—
95#391—24no—
96#391—24no—
97#389—24no—
98#389—24no—
99#385—24no—
100#385—24no—
101#384—24no—
102#384—22no—
103#384—22no—
104#384C2312yes—
105#382—22no—
106#382—22no—
107#382—21no—
108#380—21no—
109#380—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4C2312yes—
114C4×C22⋊C4—21no—
115C4×C22⋊C4—21no—
116#376—21no—
117C43—21no—
118#340—24no—
119#340—28no—
120#340—22no—
121#340—22no—
122#332—24no—
123#332—24no—
124Q8⋊C8—24no—
125Q8⋊C8—24no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—82no—
132C22×Q8—82no—
133C22×Q8C2×D412yes—
134C23×C4—22no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C22×C8—24no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—42no—
144C22×C8—42no—
145C22×C8—42no—
146C22×C8—82no—
147C4⋊Q8—24no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—24no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161C4⋊Q8—84no—
162C4⋊Q8—84no—
163#125—42no—
164#125—42no—
165C22⋊Q8—24no—
166C22⋊Q8—24no—
167C22⋊Q8—42no—
168C22⋊Q8—42no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—42no—
174C4×Q8—42no—
175#117—22no—
176#117—22no—
177#117—42no—
178#117—42no—
179#117—42no—
180#117—42no—
181#117SD1614yes—
182#117Q1614yes—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—41no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—41no—
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42#4112yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221C2×C42C2411yescharacteristic
222#107—44no—
223#106—44no—
224C4⋊C8—24no—
225C4⋊C8—22no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—22no—
229C4⋊C8—42no—
230C4⋊C8—42no—
231C4⋊C8C2×D412yes—
232C4⋊C8C2×D412yes—
233#103—84no—
234C22⋊C8—42no—
235C4×C8—22no—
236C4×C8—22no—
237#95—28no—
238#95—28no—
239#95—22no—
240#95—22no—
241#95—41no—
242#95—41no—
243C24#12011yescharacteristic
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—44no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—44no—
252C2×Q8—44no—
253C2×Q8—42no—
254C2×Q8—44no—
255C2×Q8—84no—
256C2×Q8—88no—
257C2×Q8—84no—
258C2×Q8—84no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—84no—
270C22×C4#12011yescharacteristic
271C22×C4#12011yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4#12112yes—
274C22×C4#12112yes—
275C22×C4#12112yes—
276C22×C4#12312yes—
277C22×C4#12312yes—
278C22×C4#12412yes—
279C22×C4C2×SD1612yes—
280C22×C4C2×Q1612yes—
281C22×C4#13712yes—
282C22×C4#13712yes—
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—48no—
293C2×C8—44no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—44no—
297C2×C8—84no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4—22no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—88no—
320C4⋊C4—88no—
321C4⋊C4C2×SD1614yes—
322C4⋊C4C2×Q1614yes—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—44no—
326C22⋊C4—82no—
327C42—24no—
328C42—24no—
329C42—44no—
330C42—42no—
331C42—42no—
332C42—42no—
333C42—42no—
334C42—82no—
335C42#12112yes—
336C42#12112yes—
337C42#12112yes—
338C42#12312yes—
339C42#12312yes—
340C42#12412yes—
341C42C22×D411yescharacteristic
342C42C2×#4111yescharacteristic
343C42C2×#4111yescharacteristic
344C42C2×#4111yescharacteristic
345C23—22no—
346C23—22no—
347C23#39412yes—
348C23#44812yes—
349C23#45112yes—
350C23C2×#12011yescharacteristic
351Q8—44no—
352Q8—42no—
353Q8—42no—
354Q8—84no—
355Q8—88no—
356Q8—84no—
357Q8—82no—
358Q8—82no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—48no—
378C2×C4—44no—
379C2×C4—88no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4#39412yes—
384C2×C4#39412yes—
385C2×C4#39412yes—
386C2×C4#44812yes—
387C2×C4#45112yes—
388C2×C4#46114yes—
389C2×C4#46214yes—
390C2×C4#48714yes—
391C2×C4#49114yes—
392C2×C4C2×#12111yescharacteristic
393C2×C4C2×#12111yescharacteristic
394C2×C4C2×#12111yescharacteristic
395C2×C4C2×#12311yescharacteristic
396C2×C4C2×#12311yescharacteristic
397C2×C4C2×#12411yescharacteristic
398C2×C4C22×SD1611yescharacteristic
399C2×C4C22×Q1611yescharacteristic
400C2×C4C2×#13711yescharacteristic
401C2×C4C2×#13711yescharacteristic
402C2×C4#57612yes—
403C2×C4#57712yes—
404C8—42no—
405C8—42no—
406C8—82no—
407C8—82no—
408C22—22no—
409C22—24no—
410C22—24no—
411C22—22no—
412C22#204314yes—
413C22C2×#39411yescharacteristic
414C22C2×#44811yescharacteristic
415C22C2×#45111yescharacteristic
416C22#301212yes—
417C22#302012yes—
418C4—22no—
419C4—21no—
420C4—21no—
421C4—22no—
422C4—44no—
423C4—44no—
424C4—84no—
425C4—82no—
426C4—82no—
427C4—162no—
428C4#204314yes—
429C4C2×#46111yescharacteristic
430C4C2×#46211yescharacteristic
431C4#305711yescharacteristic
432C4#305811yescharacteristic
433C4#309612yes—
434C4C2×#48711yescharacteristic
435C4C2×#49111yescharacteristic
436C2—24no—
437C2#1888412yes—
438C2#1888512yes—
439C2#2348611yescharacteristic
440C2#2349411yescharacteristic
441C2C2×#204311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×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
7. #10419C2semidirect
8. #10418C2semidirect

Maximal subgroups

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

subgroupindex
#352812
#218852
#218342
#211692
C22×#3322
C2×#14812
#104192
#104182
#78912