Finite Group Databasebeta

← #237833

Subgroups — #237833

Order 512.

1410 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic, complemented
2#21758C212yesmaximal
3#14152C211yesmaximal, characteristic
4#10342C214yesmaximal, complemented
5#10340C214yesmaximal, complemented
6C2×#1294C212yesmaximal, complemented
7#7907C211yesmaximal, characteristic, complemented
8C2×#543—21no—
9C22×#121C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11#2857—22no—
12#2659C2214yes—
13#2549—24no—
14#2440—24no—
15#2315—21no—
16C2×#386—21no—
17C2×#386—21no—
18C2×#380C2212yes—
19C2×C4×C22⋊C4C2211yescharacteristic
20C22×#95C2211yescharacteristic
21#1893—24no—
22#1874—24no—
23#1823C2214yes—
24#1786C2214yes—
25#1445C2212yes—
26#1438C2212yes—
27#1437—22no—
28#1294C2218yes—
29#1280C2214yes—
30C23×D4—21no—
31C23×D4C2311yescharacteristic
32C24×C4—21no—
33C24×C4C2311yescharacteristic
34C23×C8—22no—
35#543—44no—
36#543—44no—
37C2×C4⋊Q8—22no—
38C2×C22⋊Q8—28no—
39C2×C22⋊Q8—24no—
40C2×#121—22no—
41C2×#121—22no—
42C2×#121—22no—
43C2×#121—24no—
44C2×#121Q814yescomplemented
45C22×C4⋊C4D414yes—
46C22×C4⋊C4C2311yescharacteristic
47C22×C22⋊C4—22no—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4D411yescharacteristic
52C22×C22⋊C4D411yescharacteristic
53C22×C42D412yes—
54C2×#102—28no—
55C2×C22⋊C8—24no—
56C2×C22⋊C8—24no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8C2314yes—
59#398—28no—
60#395—28no—
61#391—24no—
62#389—24no—
63#389—24no—
64#386—44no—
65#386—44no—
66#385—28no—
67#384—22no—
68#380—24no—
69#380—22no—
70#380—22no—
71#380D414yes—
72#380C2314yes—
73C4×C22⋊C4—22no—
74C4×C22⋊C4—22no—
75C2×#95—22no—
76C2×#95—24no—
77C2×#95—21no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95C2312yes—
81C2×#95C2312yes—
82#340—28no—
83C25—21no—
84C25C2×D411yescharacteristic
85C22×Q8—24no—
86C22×D4—24no—
87C22×D4—22no—
88C22×D4—22no—
89C22×D4—21no—
90C22×D4—42no—
91C22×D4—42no—
92C22×D4—42no—
93C22×D4—42no—
94C22×D4—42no—
95C22×D4—42no—
96C22×D4—82no—
97C22×D4C2×Q812yes—
98C23×C4—22no—
99C23×C4—21no—
100C23×C4—24no—
101C23×C4—44no—
102C23×C4—42no—
103C23×C4—42no—
104C23×C4—42no—
105C23×C4—82no—
106C23×C4—82no—
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×Q812yes—
113C23×C4#4114yes—
114C23×C4#4112yes—
115C23×C4C2411yescharacteristic
116C22×C8—24no—
117C22×C8—44no—
118C22×C8—84no—
119C4⋊Q8—84no—
120C22⋊Q8—432no—
121C22⋊Q8—48no—
122C22⋊Q8—48no—
123#121—28no—
124#121—28no—
125#121—44no—
126#121—44no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—84no—
136C2×C4⋊C4C2×D418yes—
137C2×C4⋊C4C2×Q812yes—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—44no—
146C2×C22⋊C4—44no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4—44no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4—82no—
155C2×C22⋊C4—82no—
156C2×C22⋊C4D814yes—
157C2×C22⋊C4SD1614yes—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—82no—
161#102—88no—
162C22⋊C8—24no—
163C22⋊C8—24no—
164C22⋊C8—44no—
165#95—28no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—24no—
170#95—42no—
171#95—44no—
172#95—44no—
173C24—22no—
174C24—42no—
175C24—42no—
176C24—42no—
177C24—42no—
178C24—42no—
179C24—42no—
180C24—41no—
181C24—41no—
182C24—41no—
183C24—41no—
184C24—41no—
185C24—41no—
186C24#12012yes—
187C24C22⋊Q812yes—
188C24#12412yes—
189C24C2×D812yes—
190C24C2×SD1612yes—
191C24AGL(1,ℤ/8)12yes—
192C24#13712yes—
193C24C22×D411yescharacteristic
194C2×Q8—48no—
195C2×Q8—84no—
196C2×Q8—84no—
197C2×D4—24no—
198C2×D4—48no—
199C2×D4—48no—
200C2×D4—48no—
201C2×D4—48no—
202C2×D4—42no—
203C2×D4—82no—
204C2×D4—84no—
205C2×D4—82no—
206C2×D4—84no—
207C2×D4—84no—
208C2×D4—84no—
209C2×D4—84no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—28no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—48no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—48no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4—84no—
242C22×C4—84no—
243C22×C4—84no—
244C22×C4—88no—
245C22×C4—88no—
246C22×C4—84no—
247C22×C4—84no—
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4C22⋊Q814yes—
251C22×C4C22⋊Q814yes—
252C22×C4C22⋊Q812yes—
253C22×C4#12414yes—
254C22×C4#12414yes—
255C22×C4#12412yes—
256C22×C4C2×D814yes—
257C22×C4C2×SD1614yes—
258C22×C4C22×D412yes—
259C22×C4C22×Q811yescharacteristic
260C22×C4C2×#4112yes—
261C22×C4C2×#4111yescharacteristic
262C2×C8—48no—
263C2×C8—44no—
264C2×C8—88no—
265C4⋊C4—24no—
266C4⋊C4—48no—
267C4⋊C4—48no—
268C4⋊C4—48no—
269C4⋊C4—416no—
270C4⋊C4—42no—
271C4⋊C4—84no—
272C4⋊C4—84no—
273C22⋊C4—28no—
274C22⋊C4—28no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—48no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C22⋊C4—88no—
281C22⋊C4—88no—
282C42—44no—
283C42—82no—
284C23—22no—
285C23—24no—
286C23—22no—
287C23—24no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—44no—
292C23—44no—
293C23—42no—
294C23—44no—
295C23—44no—
296C23—44no—
297C23—44no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—82no—
302C23—82no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—82no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23—82no—
312C23—82no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—82no—
318C23—82no—
319C23#39314yes—
320C23#44714yes—
321C23#44814yes—
322C23#47414yes—
323C23#47614yes—
324C23#48614yes—
325C23#48914yes—
326C23C2×#12011yescharacteristic
327C23C2×C22⋊Q811yescharacteristic
328C23C2×#12411yescharacteristic
329C23C22×D811yescharacteristic
330C23C22×SD1611yescharacteristic
331C23C2×AGL(1,ℤ/8)11yescharacteristic
332C23C2×#13711yescharacteristic
333Q8—168no—
334D4—48no—
335D4—84no—
336D4—816no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—48no—
362C2×C4—48no—
363C2×C4—48no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—816no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—168no—
378C2×C4#39312yes—
379C2×C4#39312yes—
380C2×C4C2×C22⋊Q812yes—
381C2×C4C2×#12412yes—
382C8—84no—
383C22—21no—
384C22—21no—
385C22—24no—
386C22—21no—
387C22—21no—
388C22—24no—
389C22—22no—
390C22—22no—
391C22—28no—
392C22—22no—
393C22—22no—
394C22—28no—
395C22—44no—
396C22—44no—
397C22—44no—
398C22—44no—
399C22—84no—
400C22—84no—
401C22—82no—
402C22—82no—
403C22—82no—
404C22—161no—
405C22—161no—
406C22—161no—
407C22—161no—
408C22—162no—
409C22—162no—
410C22#202318yes—
411C22C2×#39311yescharacteristic
412C22C2×#44711yescharacteristic
413C22C2×#44811yescharacteristic
414C22#301812yes—
415C22#302612yes—
416C22C2×#47411yescharacteristic
417C22C2×#47611yescharacteristic
418C22C2×#48611yescharacteristic
419C22C2×#48911yescharacteristic
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—44no—
425C4—44no—
426C4—84no—
427C4—82no—
428C4—88no—
429C2—21no—
430C2—21no—
431C2—21no—
432C2—21no—
433C2—28no—
434C2—162no—
435C2#1838012yes—
436C2#1838212yes—
437C2C2×#202311yescharacteristic
438C2#2346111yescharacteristic
439C2#2346811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35002C2semidirect
4. #10342C2semidirect
5. #10340C2semidirect
6. C2×#1294C2semidirect
7. #7907C2semidirect
44. C2×#121Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
#217582
#141522
#103422
#103402
C2×#12942
#79072