Finite Group Databasebeta

← #207249

Subgroups — #207249

Order 512.

930 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2#21951C211yesmaximal, characteristic
3#13804C211yesmaximal, characteristic
4#9913C214yesmaximal
5#9910C214yesmaximal
6#8026C211yesmaximal, characteristic
7#8026C211yesmaximal, characteristic
8#8014C211yesmaximal, characteristic
9#8014C211yesmaximal, characteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2851C2214yes—
13#2646—22no—
14#2571C412yescomplemented
15#2571C412yescomplemented
16#2570—22no—
17#2570—22no—
18#2552C412yescomplemented
19#2552C412yescomplemented
20#2317—21no—
21#2317—21no—
22#2285C2211yescharacteristic
23#2285C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1939—22no—
29#1939—22no—
30#1797C2214yes—
31#1436—22no—
32C4⋊#95—21no—
33#1434C2212yes—
34C4×#95C2212yes—
35#1294C2212yes—
36#1294C2212yes—
37#1294C2212yes—
38#1294C2212yes—
39#1294C2214yes—
40#1294C2214yes—
41C2×#125C2×C414yes—
42C2×#117—21no—
43C2×#117—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4D411yescharacteristic
48C22×C42—21no—
49C22×C42—22no—
50C22×C42C2312yes—
51C22×C42C2311yescharacteristic
52#431—22no—
53#431—22no—
54#431—22no—
55#431—22no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—24no—
59C2×C22⋊C8C2312yes—
60C2×C22⋊C8C2312yes—
61C2×C4×C8—21no—
62C2×C4×C8—21no—
63#400—24no—
64#400—24no—
65#398—24no—
66#398—24no—
67#389—22no—
68#389—22no—
69#389—22no—
70#389—22no—
71#389C2×C414yes—
72#389C2×C414yes—
73#387—24no—
74#387—24no—
75#385—24no—
76#385—24no—
77#384—22no—
78#384—22no—
79#384C2×C411yescharacteristic
80#384C2×C411yescharacteristic
81#384C2×C411yescharacteristic
82#384C2×C411yescharacteristic
83#382—22no—
84#382—22no—
85#382—21no—
86#382—21no—
87#382—22no—
88#382C2×C414yes—
89#382Q814yes—
90#380—21no—
91#380—21no—
92#380C2312yes—
93#380C2312yes—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—44no—
98C4×C4⋊C4C2×C412yes—
99C4×C4⋊C4C2×C412yes—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4C2×C412yes—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4D412yes—
104C4×C4⋊C4Q814yes—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95D412yes—
110C2×#95C2311yescharacteristic
111C2×#95C2312yes—
112C2×#95C2311yescharacteristic
113#336—24no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2411yescharacteristic
122C22×C8—22no—
123C22×C8—22no—
124#125—84no—
125#117—42no—
126#117—42no—
127#117—42no—
128#117—42no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C22×C412yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—24no—
164C2×C42—42no—
165C2×C42—42no—
166C2×C42—44no—
167C2×C42—44no—
168C2×C42—82no—
169C2×C42—82no—
170C2×C42C22×C414yes—
171C2×C42C22×C411yescharacteristic
172C2×C42C22×C411yescharacteristic
173C2×C42C2×D414yes—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×Q812yes—
177C2×C42C2×Q812yes—
178C2×C42#4112yes—
179C2×C42#4112yes—
180C4⋊C8—42no—
181C4⋊C8—42no—
182C4⋊C8—84no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C4×C8—42no—
188C4×C8—42no—
189#95—22no—
190#95—22no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95—22no—
197#95—22no—
198#95—21no—
199#95—21no—
200#95—22no—
201#95—44no—
202#95—44no—
203#95—82no—
204#95C22⋊C414yes—
205#95C22⋊C414yes—
206#95C2×Q812yes—
207#95C2×Q812yes—
208#95C2×Q814yes—
209C24C22×D411yescharacteristic
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4C2×C22⋊C414yes—
248C22×C4C2×C22⋊C414yes—
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C22⋊Q812yes—
254C22×C4C22⋊Q812yes—
255C22×C4C22⋊Q812yes—
256C22×C4C22⋊Q812yes—
257C22×C4#13512yes—
258C22×C4#13512yes—
259C22×C4AGL(1,ℤ/8)12yes—
260C22×C4#13712yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×Q811yescharacteristic
264C22×C4C22×Q811yescharacteristic
265C22×C4C2×#4111yescharacteristic
266C22×C4C2×#4111yescharacteristic
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—48no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—48no—
287C4⋊C4—81no—
288C4⋊C4—81no—
289C4⋊C4—81no—
290C4⋊C4—81no—
291C4⋊C4—84no—
292C4⋊C4—84no—
293C4⋊C4—84no—
294C4⋊C4—84no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C42—44no—
300C42—44no—
301C42—44no—
302C42—44no—
303C42—48no—
304C42—88no—
305C42C4×Q812yes—
306C42C4×Q812yes—
307C42C4×Q812yes—
308C42C4×Q812yes—
309C42#12412yes—
310C42#12412yes—
311C42C4⋊Q812yes—
312C42C4⋊Q812yes—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23C22×C22⋊C411yescharacteristic
318C23C2×C22⋊Q811yescharacteristic
319C23C2×C22⋊Q811yescharacteristic
320C23C2×#13511yescharacteristic
321C23C2×#13511yescharacteristic
322C23C2×AGL(1,ℤ/8)11yescharacteristic
323C23C2×#13711yescharacteristic
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4#39112yes—
382C2×C4#39112yes—
383C2×C4#39112yes—
384C2×C4#39112yes—
385C2×C4#41612yes—
386C2×C4#41612yes—
387C2×C4#41712yes—
388C2×C4#41712yes—
389C2×C4#47814yes—
390C2×C4#47814yes—
391C2×C4#47914yes—
392C2×C4#47914yes—
393C2×C4C2×C4×Q811yescharacteristic
394C2×C4C2×C4×Q811yescharacteristic
395C2×C4C2×#12411yescharacteristic
396C2×C4C2×C4⋊Q811yescharacteristic
397C8—82no—
398C8—82no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22C2×#39111yescharacteristic
412C22#288311yescharacteristic
413C22#288411yescharacteristic
414C22C2×#41611yescharacteristic
415C22C2×#41711yescharacteristic
416C22#290111yescharacteristic
417C22#290111yescharacteristic
418C22C2×#47811yescharacteristic
419C22C2×#47811yescharacteristic
420C22C2×#47911yescharacteristic
421C22C2×#47911yescharacteristic
422C4—22no—
423C4—22no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—44no—
429C4—44no—
430C4—44no—
431C4—82no—
432C4—84no—
433C4—82no—
434C4—82no—
435C4—82no—
436C4—82no—
437C4#198914yes—
438C4#198914yes—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2#1761611yescharacteristic
444C2#1761711yescharacteristic
445C2#1762811yescharacteristic
446C2#1763011yescharacteristic
447C2C2×#198911yescharacteristic
448C2#2133011yescharacteristic
449C2#2133411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2571C4semidirect
15. #2571C4semidirect
18. #2552C4semidirect
19. #2552C4semidirect

Maximal subgroups

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

subgroupindex
#351812
#219512
#138042
#99132
#99102
#80262
#80142