Finite Group Databasebeta

← #363996

Subgroups — #363996

Order 512.

1046 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60124C211yesmaximal, characteristic
2#20757C211yesmaximal, characteristic, complemented
3#20370C211yesmaximal, characteristic, complemented
4#11246C211yesmaximal, characteristic, complemented
5#11245C211yesmaximal, characteristic, complemented
6#11101C211yesmaximal, characteristic
7#11100C211yesmaximal, characteristic
8#10937C212yesmaximal
9#10890C212yesmaximal, complemented
10#10810C212yesmaximal, complemented
11#10716C212yesmaximal
12C22×#117C2211yescharacteristic
13#2924—24no—
14#2924—24no—
15#2921C414yescomplemented
16#2920C414yescomplemented
17C2×#430C2211yescharacteristic
18#2911C2211yescharacteristic
19#2905C2211yescharacteristic
20#2905C2211yescharacteristic
21C2×#407C2212yes—
22C2×#407C2211yescharacteristic
23C2×#407C2211yescharacteristic
24#2873C2212yes—
25#2873C2211yescharacteristic
26#2873C2211yescharacteristic
27#2280C2212yes—
28#1833—24no—
29#1747—24no—
30C2×#344C2212yes—
31C2×#344C2212yes—
32#1742C2212yes—
33#1738C2212yes—
34#1733C2212yes—
35C2×#338C2211yescharacteristic
36C2×H(3,ℤ/4)C2211yescharacteristic
37C2×H(3,ℤ/4)C2212yes—
38C2×H(3,ℤ/4)C2212yes—
39C2×H(3,ℤ/4)C2212yes—
40C2×H(3,ℤ/4)C2212yes—
41C24×C4C2311yescharacteristic
42C22×M16C2312yes—
43C22×M16C2311yescharacteristic
44C22×M16C2311yescharacteristic
45C22×M16C2311yescharacteristic
46C22×M16C2311yescharacteristic
47C22×M16C2312yes—
48C2×#117—21no—
49C2×#117—21no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117C2×C412yes—
55C2×#117C2312yes—
56C2×#117C2311yescharacteristic
57C2×#117C2312yes—
58C2×#117C2311yescharacteristic
59C22×C4⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C42C2×C412yes—
62#433—28no—
63#433—28no—
64#432—28no—
65#431—28no—
66#430—42no—
67C2×#108—22no—
68#428—42no—
69#427—22no—
70#424—22no—
71#424—22no—
72#424—22no—
73#424—22no—
74#423C2×C414yes—
75#423C2×C414yes—
76#407—22no—
77#407—22no—
78#407—22no—
79#407—22no—
80#407—42no—
81#407C2×C412yes—
82#407C2×C412yes—
83#407C2×C412yes—
84#407C2×C412yes—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—24no—
88#405—24no—
89#405—24no—
90#405—22no—
91#405—22no—
92#405D414yes—
93#405D414yes—
94C4×M16—24no—
95C4×M16C2×C414yes—
96C4×M16C2×C414yes—
97#381—22no—
98#380—42no—
99#379—22no—
100C4×C22⋊C4—24no—
101#344—24no—
102#344—42no—
103#343—42no—
104#341—24no—
105#339—24no—
106#338—22no—
107H(3,ℤ/4)—22no—
108H(3,ℤ/4)—24no—
109H(3,ℤ/4)—24no—
110H(3,ℤ/4)—24no—
111H(3,ℤ/4)—24no—
112C25#4111yescharacteristic
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4#4111yescharacteristic
125C23×C4#4111yescharacteristic
126C23×C4#4111yescharacteristic
127C23×C4C2411yescharacteristic
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—21no—
134C2×M16—21no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16C22×C412yes—
151C2×M16C22×C412yes—
152C2×M16C22×C412yes—
153C2×M16C22×C412yes—
154C2×M16C2×D412yes—
155C2×M16C2×D412yes—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8C2×D412yes—
160C22×C8C2×D412yes—
161C22×C8#4114yes—
162#117—24no—
163#117—24no—
164#117—24no—
165#117—24no—
166#117—21no—
167#117—21no—
168#117—21no—
169#117—21no—
170#117—22no—
171#117—22no—
172#117—22no—
173#117—22no—
174#117—24no—
175#117—24no—
176#117—22no—
177#117—22no—
178#117—22no—
179#117—22no—
180#117C2×D412yes—
181#117C2×D412yes—
182#117#4112yes—
183#117#4112yes—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—44no—
201C2×C42—22no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—22no—
205C2×C42C22×C412yes—
206C2×C42C22×C412yes—
207C2×C42#4114yes—
208#108—44no—
209#107—42no—
210#106—42no—
211C4⋊C8—28no—
212C4⋊C8—28no—
213C22⋊C8—24no—
214C22⋊C8—24no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C22⋊C8—44no—
218C22⋊C8—44no—
219#97—28no—
220#97—24no—
221#97C22⋊C418yescomplemented
222C4×C8—28no—
223C4×C8—24no—
224C4×C8C22⋊C418yescomplemented
225#95—44no—
226C24—21no—
227C24—21no—
228C24—22no—
229C24—22no—
230C24—82no—
231C24#11711yescharacteristic
232C24#11711yescharacteristic
233C24C2×#4111yescharacteristic
234C22×C4—24no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—84no—
271C22×C4#11711yescharacteristic
272C22×C4#11711yescharacteristic
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C2×#4111yescharacteristic
277C22×C4C2×#4111yescharacteristic
278C22×C4C2×#4112yes—
279C22×C4C2×#4112yes—
280C22×C4C2×#4111yescharacteristic
281C22×C4H(5,𝔽2)12yes—
282C22×C42−1+412yes—
283M16—24no—
284M16—24no—
285M16—22no—
286M16—22no—
287M16—24no—
288M16—24no—
289M16—24no—
290M16—42no—
291M16—42no—
292M16—41no—
293M16—41no—
294M16—42no—
295M16—42no—
296M16—42no—
297M16—42no—
298M16—42no—
299M16—42no—
300M16—42no—
301C2×C8—24no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8C2×C22⋊C414yes—
316C2×C8C2×C22⋊C414yes—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C22⋊C4—28no—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—88no—
337C42—28no—
338C42—22no—
339C42—22no—
340C42—24no—
341C42—24no—
342C42—42no—
343C42—42no—
344C42C2×C22⋊C414yes—
345C42#11714yes—
346C23—22no—
347C23—21no—
348C23—22no—
349C23—21no—
350C23—42no—
351C23—44no—
352C23—42no—
353C23—42no—
354C23—44no—
355C23—42no—
356C23—41no—
357C23—41no—
358C23—81no—
359C23—81no—
360C23—82no—
361C23—81no—
362C23—81no—
363C23—82no—
364C23C2×#11711yescharacteristic
365C23#52511yescharacteristic
366C23#52511yescharacteristic
367C23#55112yes—
368C23#55212yes—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—21no—
375C2×C4—48no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—44no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4—82no—
398C2×C4C22×C22⋊C411yescharacteristic
399C2×C4C2×#11711yescharacteristic
400C2×C4C4×#4112yes—
401C2×C4#52012yes—
402C2×C4#52511yescharacteristic
403C2×C4#52512yes—
404C2×C4#52511yescharacteristic
405C2×C4#53211yescharacteristic
406C2×C4#53211yescharacteristic
407C2×C4#53511yescharacteristic
408C2×C4#53611yescharacteristic
409C2×C4#53712yes—
410C2×C4#55112yes—
411C2×C4#55212yes—
412C2×C4#55212yes—
413C2×C4#55212yes—
414C2×C4#55512yes—
415C2×C4#55512yes—
416C8—24no—
417C8—24no—
418C8—42no—
419C8—44no—
420C8—42no—
421C8—82no—
422C22—22no—
423C22—24no—
424C22—41no—
425C22—42no—
426C22—41no—
427C22—41no—
428C22—42no—
429C22—41no—
430C22—42no—
431C22—41no—
432C22—41no—
433C22—42no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—84no—
438C22—82no—
439C22—82no—
440C22#228011yescharacteristic
441C22#234312yes—
442C22#260912yes—
443C22#261011yescharacteristic
444C22#261111yescharacteristic
445C4—44no—
446C4—44no—
447C4—41no—
448C4—41no—
449C4—84no—
450C4—162no—
451C4#229711yescharacteristic
452C4#231111yescharacteristic
453C4#234612yes—
454C4#260912yes—
455C4#261011yescharacteristic
456C4#261111yescharacteristic
457C2—21no—
458C2—22no—
459C2—21no—
460C2—22no—
461C2—21no—
462C2—42no—
463C2—81no—
464C2#1395711yescharacteristic
subgroupG/H
centerC4#2297
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC2#13957
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20757C2semidirect
3. #20370C2semidirect
4. #11246C2semidirect
5. #11245C2semidirect
9. #10890C2semidirect
10. #10810C2semidirect
15. #2921C4semidirect
16. #2920C4semidirect
221. #97C22⋊C4semidirect
224. C4×C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
#601242
#207572
#203702
#112462
#112452
#111012
#111002
#109372
#108902
#108102
#107162