Finite Group Databasebeta

← #6433748

Subgroups — #6433748

Order 512.

1968 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#34633C211yesmaximal, characteristic
2C2×#2629C211yesmaximal, characteristic, complemented
3#30196C214yesmaximal, complemented
4#30194C214yesmaximal, complemented
5#30191C218yesmaximal, complemented
6#23747C212yesmaximal, complemented
7#23630C212yesmaximal, complemented
8#23508C212yesmaximal, complemented
9#23453C212yesmaximal, complemented
10#20971C214yesmaximal, complemented
11#20335C211yesmaximal, characteristic, complemented
12C23×M16C2212yes—
13C2×#532C2211yescharacteristic
14C22×#125C2211yescharacteristic
15C2×#525C2212yes—
16C22×#117C2211yescharacteristic
17C2×#417C2214yes—
18C2×#417C2214yes—
19#2629—24no—
20#2629C2214yes—
21#2629C2214yes—
22#2621C2212yes—
23#2620C2214yes—
24#2619C2212yes—
25#2618—22no—
26#2617—22no—
27#2392—24no—
28C2×#398—22no—
29C2×#396—22no—
30#2297C2212yes—
31C2×#389C2212yes—
32#2285C2211yescharacteristic
33C2×#384C2211yescharacteristic
34C2×#380C2212yes—
35#2101C2218yescomplemented
36#2100C2214yescomplemented
37#2099C2214yescomplemented
38#2072C2218yescomplemented
39#2071C2218yescomplemented
40#2043C2214yescomplemented
41#2042C2218yescomplemented
42#2041C2214yescomplemented
43#2026C2218yescomplemented
44#2025C2218yescomplemented
45#1895C2218yes—
46#1894C2218yes—
47#1893C2218yes—
48#1892C2218yes—
49#1736C2214yes—
50#1736C2214yes—
51#1729C2214yes—
52#1729C2214yes—
53#1728C2212yes—
54#1728C2212yes—
55C22×#41C2312yes—
56C24×C4C2311yescharacteristic
57C22×M16—28no—
58C22×M16—44no—
59C22×M16—44no—
60C22×M16C2314yes—
61C22×M16C2314yes—
62C23×C8C2314yes—
63#532—44no—
64C2×C4⋊Q8C2311yescharacteristic
65C2×#127C2311yescharacteristic
66C2×#125—22no—
67C2×#125—44no—
68C2×#125C2314yes—
69C2×#124C2312yes—
70#525—216no—
71C2×#123—22no—
72C2×C22⋊Q8C2314yes—
73C2×#121—22no—
74C2×#121C2314yes—
75C2×#120—22no—
76C2×C4×D4—24no—
77C2×C4×D4—22no—
78C2×#117—44no—
79C2×#117D414yes—
80C2×#117C2312yes—
81C2×#117C2312yes—
82C2×#117C2312yes—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4D412yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C22⋊C4—22no—
90C22×C22⋊C4D412yes—
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#417—48no—
94#417—48no—
95C2×#103C2318yes—
96C2×#103C2318yes—
97C2×#102C2318yes—
98C2×#102C2318yes—
99#398—24no—
100#398—28no—
101#398—28no—
102#397—24no—
103#396—24no—
104#396—24no—
105#396—28no—
106#396—28no—
107#394—24no—
108#394—24no—
109#394—24no—
110#392—24no—
111#391C2314yes—
112#390C2314yes—
113#389—22no—
114#389—22no—
115#389C2314yes—
116#389C2314yes—
117#386—24no—
118#384—22no—
119#384C2314yes—
120#384C2312yes—
121#382C2312yes—
122#380—22no—
123#380—24no—
124#380C2318yes—
125C4×C4⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C2×#95—22no—
129C2×#95—22no—
130#340C2318yes—
131#340C2318yes—
132#340C2318yes—
133#340C2318yes—
134#336C2318yes—
135#336C2314yes—
136#336C2314yes—
137C25C2×D411yescharacteristic
138C2×#41—28no—
139C2×#41—28no—
140C2×#41—24no—
141C2×#41—24no—
142C22×Q8C2412yes—
143C22×D4—22no—
144C22×D4—24no—
145C22×D4—22no—
146C22×D4C2412yes—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4#4118yes—
158C23×C4C2412yes—
159C23×C4C2411yescharacteristic
160C2×M16—28no—
161C2×M16—28no—
162C2×M16—28no—
163C2×M16—28no—
164C2×M16—44no—
165C2×M16—44no—
166C2×M16—432no—
167C22×C8—28no—
168C22×C8—44no—
169C22×C8—44no—
170C22×C8C2418yes—
171C4⋊Q8—42no—
172#127—42no—
173#125—44no—
174#125—44no—
175#125—48no—
176#124—44no—
177#123—48no—
178C22⋊Q8—28no—
179C22⋊Q8—28no—
180#121—28no—
181#121—28no—
182#121—48no—
183#120—48no—
184C4×D4—48no—
185C4×D4—416no—
186#117—28no—
187#117—48no—
188#117—42no—
189#117—42no—
190#117—42no—
191#117—42no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—48no—
207C2×C4⋊C4—84no—
208C2×C4⋊C4—84no—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2414yes—
212C2×C4⋊C4C2414yes—
213C2×C4⋊C4C2414yes—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—84no—
223C2×C22⋊C4C2×D414yes—
224C2×C22⋊C4C2×D414yes—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—22no—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2411yescharacteristic
231C2×C42C2412yes—
232C2×C42C2411yescharacteristic
233#103—48no—
234#103—48no—
235#102—48no—
236#102—48no—
237#95—28no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243C24—22no—
244C24—22no—
245C24—22no—
246C24—42no—
247C24—42no—
248C24#12414yes—
249C24C22×D412yes—
250C24C22×D411yescharacteristic
251#41—416no—
252#41—416no—
253C2×Q8—24no—
254C2×Q8—22no—
255C2×Q8—24no—
256C2×Q8—22no—
257C2×D4—24no—
258C2×D4—22no—
259C2×D4—24no—
260C2×D4—22no—
261C2×D4—44no—
262C2×D4—48no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—82no—
267C2×D4—82no—
268C2×D4—84no—
269C2×D4—84no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—216no—
275C22×C4—216no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4#12418yes—
300C22×C4#12414yes—
301C22×C4C22×D412yes—
302C22×C4C22×D414yes—
303C22×C4C22×D412yes—
304C22×C4C22×D412yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4114yes—
307C22×C4C2×#4114yes—
308C22×C4C2×#4114yes—
309C22×C4H(5,𝔽2)14yes—
310C22×C4H(5,𝔽2)12yes—
311C22×C42−1+412yes—
312C22×C4C2511yescharacteristic
313M16—416no—
314M16—416no—
315C2×C8—28no—
316C2×C8—28no—
317C2×C8—28no—
318C2×C8—416no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—28no—
323C4⋊C4—24no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—416no—
327C4⋊C4—44no—
328C4⋊C4—48no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—88no—
334C4⋊C4—84no—
335C4⋊C4—84no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—48no—
339C22⋊C4—44no—
340C22⋊C4—48no—
341C22⋊C4—48no—
342C22⋊C4—84no—
343C22⋊C4—84no—
344C22⋊C4—88no—
345C42—24no—
346C42—24no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—44no—
352C23—22no—
353C23—22no—
354C23—24no—
355C23—24no—
356C23—22no—
357C23—22no—
358C23—28no—
359C23—48no—
360C23—44no—
361C23—44no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—84no—
366C23—82no—
367C23—82no—
368C23C2×#12414yes—
369C23C2×#12412yes—
370C23#53412yes—
371C23#53712yes—
372C23C23×D411yescharacteristic
373Q8—44no—
374Q8—44no—
375D4—44no—
376D4—44no—
377D4—88no—
378D4—816no—
379D4—88no—
380C2×C4—28no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—216no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—48no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—84no—
411C2×C4—88no—
412C2×C4—88no—
413C2×C4—88no—
414C2×C4C2×#12414yes—
415C2×C4C2×#12412yes—
416C2×C4#53412yes—
417C2×C4#53712yes—
418C2×C4#53818yes—
419C2×C4#53918yes—
420C2×C4C22×#4112yes—
421C2×C4C2×H(5,𝔽2)11yescharacteristic
422C2×C4C2×H(5,𝔽2)12yes—
423C2×C4C2×2−1+411yescharacteristic
424C8—48no—
425C22—24no—
426C22—24no—
427C22—22no—
428C22—22no—
429C22—24no—
430C22—28no—
431C22—44no—
432C22—42no—
433C22—42no—
434C22—88no—
435C22—82no—
436C22—84no—
437C22—84no—
438C22—82no—
439C22—82no—
440C22#263914yes—
441C22C22×#12411yescharacteristic
442C22C2×#53411yescharacteristic
443C22C2×#53711yescharacteristic
444C22#358512yes—
445C22#358612yes—
446C4—24no—
447C4—44no—
448C4—42no—
449C4—42no—
450C4—82no—
451C4—84no—
452C4—84no—
453C4—84no—
454C4#263914yes—
455C4C2×#53812yes—
456C4C2×#53912yes—
457C2—24no—
458C2—44no—
459C2—162no—
460C2C2×#263911yescharacteristic
461C2#6062112yes—
462C2#6075512yes—
463C2#6092011yescharacteristic
464C2#6092111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#34633C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2629C2semidirect
3. #30196C2semidirect
4. #30194C2semidirect
5. #30191C2semidirect
6. #23747C2semidirect
7. #23630C2semidirect
8. #23508C2semidirect
9. #23453C2semidirect
10. #20971C2semidirect
11. #20335C2semidirect
35. #2101C22semidirect
36. #2100C22semidirect
37. #2099C22semidirect
38. #2072C22semidirect
39. #2071C22semidirect
40. #2043C22semidirect
41. #2042C22semidirect
42. #2041C22semidirect
43. #2026C22semidirect
44. #2025C22semidirect

Maximal subgroups

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

subgroupindex
#346332
C2×#26292
#301962
#301942
#301912
#237472
#236302
#235082
#234532
#209712
#203352