Finite Group Databasebeta

← #192158

Subgroups — #192158

Order 512.

1516 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34181C211yesmaximal, characteristic
2#21938C212yesmaximal, complemented
3C2×#1459C211yesmaximal, characteristic
4#13960C211yesmaximal, characteristic
5#8857C214yesmaximal, complemented
6#8854C214yesmaximal, complemented
7C2×#1294C212yesmaximal, complemented
8C2×#518—21no—
9C22×C4×D4C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11#2349—24no—
12D4×C4⋊C4—24no—
13#2316C2212yes—
14#2315—21no—
15#2315C2212yes—
16C2×#386—21no—
17#2285—21no—
18C2×#380—21no—
19C2×#380C2212yes—
20C2×#380C2211yescharacteristic
21C22×#95C2211yescharacteristic
22#1853—24no—
23#1853C2218yes—
24#1807—24no—
25#1459C418yescomplemented
26C2×#324C2212yes—
27C2×#323C2212yes—
28#1441C2212yes—
29#1439C2212yes—
30#1434—21no—
31C22⋊#95—21no—
32C22⋊#95—21no—
33#1294C2218yes—
34C23×D4D411yescharacteristic
35C24×C4—21no—
36C24×C4D411yescharacteristic
37C24×C4C2311yescharacteristic
38C23×C8—22no—
39#518—44no—
40#518—44no—
41C2×C4×D4—22no—
42C2×C4×D4—22no—
43C2×C4×D4—24no—
44C2×C4×D4—28no—
45C2×C4×D4C2×C414yes—
46C2×C4×D4Q814yes—
47C2×#117—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4D414yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4D411yescharacteristic
57C22×C22⋊C4C2311yescharacteristic
58C22×C42D411yescharacteristic
59C2×#106—28no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—24no—
62C2×C22⋊C8C2×C418yescomplemented
63C2×C22⋊C8C2314yes—
64#389—28no—
65#386—44no—
66#385—28no—
67#385—28no—
68#384—22no—
69#382—22no—
70#380—22no—
71#380—22no—
72#380—22no—
73#380—22no—
74#380—22no—
75#380—22no—
76#380—22no—
77#380—22no—
78#380D414yes—
79#380C2314yes—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—24no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95C2312yes—
91#340—28no—
92#324C2×C418yescomplemented
93#323C2×C418yescomplemented
94C25C2×D411yescharacteristic
95C25C2×D411yescharacteristic
96C22×D4—22no—
97C22×D4—22no—
98C22×D4—22no—
99C22×D4—22no—
100C22×D4C4⋊C414yes—
101C22×D4SD1614yes—
102C22×D4SD1614yes—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—24no—
108C23×C4—24no—
109C23×C4—22no—
110C23×C4—44no—
111C23×C4—42no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—82no—
115C23×C4—82no—
116C23×C4—82no—
117C23×C4C4⋊C414yes—
118C23×C4C22×C412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2×Q812yes—
126C23×C4#4114yes—
127C23×C4C2411yescharacteristic
128C22×C8—24no—
129C22×C8—28no—
130C22×C8—84no—
131C4×D4—28no—
132C4×D4—28no—
133C4×D4—28no—
134C4×D4—28no—
135C4×D4—432no—
136#117—44no—
137#117—44no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—82no—
146C2×C4⋊C4—84no—
147C2×C4⋊C4—84no—
148C2×C4⋊C4C22×C412yes—
149C2×C4⋊C4C2×D418yes—
150C2×C4⋊C4C2×Q812yes—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—44no—
159C2×C22⋊C4—44no—
160C2×C22⋊C4—44no—
161C2×C22⋊C4—44no—
162C2×C22⋊C4—44no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4—82no—
168C2×C22⋊C4C4⋊C414yes—
169C2×C22⋊C4C22×C412yes—
170C2×C22⋊C4C2×Q812yes—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—24no—
175C2×C42C4⋊C414yes—
176#106—88no—
177C22⋊C8—28no—
178C22⋊C8—24no—
179C22⋊C8C22×C418yes—
180#95—24no—
181#95—24no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—44no—
186#95—41no—
187#95—41no—
188#95—42no—
189#95—44no—
190#95—82no—
191C24—21no—
192C24—21no—
193C24—21no—
194C24—21no—
195C24—42no—
196C24—42no—
197C24—42no—
198C24—42no—
199C24C2×C4⋊C412yes—
200C24C2×C4⋊C412yes—
201C24#12012yes—
202C24#12312yes—
203C24C2×SD1612yes—
204C24C2×SD1612yes—
205C24C2×SD1612yes—
206C24C2×SD1612yes—
207C24AGL(1,ℤ/8)12yes—
208C24#13712yes—
209C24C22×D411yescharacteristic
210C2×D4—28no—
211C2×D4—28no—
212C2×D4—28no—
213C2×D4—28no—
214C2×D4—28no—
215C2×D4—28no—
216C2×D4#106116yescomplemented
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—28no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—48no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—48no—
244C22×C4—48no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—84no—
256C22×C4—84no—
257C22×C4—84no—
258C22×C4—84no—
259C22×C4—84no—
260C22×C4—84no—
261C22×C4—84no—
262C22×C4—88no—
263C22×C4—88no—
264C22×C4—84no—
265C22×C4—84no—
266C22×C4C2×C4⋊C412yes—
267C22×C4C2×C4⋊C412yes—
268C22×C4C2×C4⋊C412yes—
269C22×C4C2×C4⋊C412yes—
270C22×C4C4×D414yes—
271C22×C4C4×D414yes—
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4C22⋊Q814yes—
275C22×C4C22⋊Q814yes—
276C22×C4#12312yes—
277C22×C4C2×SD1612yes—
278C22×C4C2×SD1612yes—
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D412yes—
281C22×C4C22×Q811yescharacteristic
282C22×C4C2×#4112yes—
283C2×C8—28no—
284C2×C8—48no—
285C2×C8—88no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—48no—
289C4⋊C4—48no—
290C4⋊C4—48no—
291C4⋊C4—48no—
292C4⋊C4—84no—
293C4⋊C4—84no—
294C22⋊C4—28no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—48no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C22⋊C4—48no—
301C22⋊C4—88no—
302C22⋊C4—84no—
303C42—24no—
304C42—24no—
305C42—48no—
306C23—22no—
307C23—24no—
308C23—22no—
309C23—24no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—44no—
320C23—44no—
321C23—42no—
322C23—44no—
323C23—44no—
324C23—44no—
325C23—42no—
326C23—42no—
327C23—44no—
328C23—42no—
329C23—42no—
330C23—81no—
331C23—81no—
332C23—82no—
333C23—82no—
334C23#38514yes—
335C23C2×#10614yes—
336C23C2×#10614yes—
337C23#42814yes—
338C23#44814yes—
339C23#45014yes—
340C23#48114yes—
341C23#48314yes—
342C23C22×C4⋊C411yescharacteristic
343C23C2×#12011yescharacteristic
344C23C2×#12311yescharacteristic
345C23C22×SD1611yescharacteristic
346C23C22×SD1611yescharacteristic
347C23C2×AGL(1,ℤ/8)11yescharacteristic
348C23C2×#13711yescharacteristic
349D4—232no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—48no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—48no—
381C2×C4—84no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—816no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—168no—
397C2×C4—168no—
398C2×C4#38512yes—
399C2×C4#38512yes—
400C2×C4C2×#10614yes—
401C2×C4C2×C4×D412yes—
402C2×C4C2×C22⋊Q812yes—
403C8—48no—
404C22—21no—
405C22—21no—
406C22—24no—
407C22—21no—
408C22—21no—
409C22—24no—
410C22—22no—
411C22—28no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—28no—
416C22—24no—
417C22—44no—
418C22—44no—
419C22—44no—
420C22—44no—
421C22—44no—
422C22—42no—
423C22—42no—
424C22—44no—
425C22—44no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22#185318yes—
430C22C2×#38511yescharacteristic
431C22C22×#10611yescharacteristic
432C22C2×#42811yescharacteristic
433C22C2×#45011yescharacteristic
434C22C2×#44811yescharacteristic
435C22#301812yes—
436C22#302612yes—
437C22C2×#48311yescharacteristic
438C22C2×#48111yescharacteristic
439C4—24no—
440C4—22no—
441C4—22no—
442C4—22no—
443C4—22no—
444C4—42no—
445C4—42no—
446C4—84no—
447C4—88no—
448C4—162no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—28no—
454C2—44no—
455C2#1838112yes—
456C2#1838212yes—
457C2C2×#185311yescharacteristic
458C2#2085511yescharacteristic
459C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×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
2. #21938C2semidirect
5. #8857C2semidirect
6. #8854C2semidirect
7. C2×#1294C2semidirect
25. #1459C4semidirect
62. C2×C22⋊C8C2×C4semidirect
92. #324C2×C4semidirect
93. #323C2×C4semidirect
216. C2×D4#106semidirect

Maximal subgroups

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

subgroupindex
#341812
#219382
C2×#14592
#139602
#88572
#88542
C2×#12942