Finite Group Databasebeta

← #192131

Subgroups — #192131

Order 512.

1280 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34052C211yesmaximal, characteristic
2#21981C212yesmaximal, complemented
3C2×#1458C211yesmaximal, characteristic
4#14340C211yesmaximal, characteristic
5#8857C214yesmaximal, complemented
6#8854C214yesmaximal
7#8026C212yesmaximal, complemented
8C22×#117—21no—
9C22×#117C2211yescharacteristic
10C2×#431C2212yes—
11#2317—22no—
12#2317—22no—
13#2316—21no—
14#2316C2212yes—
15#2315—21no—
16#2315C2212yes—
17#2296—24no—
18#2285—21no—
19#2285C2212yes—
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#380—21no—
23#1948—24no—
24#1853C2218yes—
25#1844—22no—
26#1834—22no—
27#1458C418yescomplemented
28C2×#324C2212yes—
29C2×#323C2212yes—
30#1441—21no—
31#1441C2212yes—
32#1439—21no—
33#1439C2212yes—
34C4⋊#95—21no—
35#1294C2218yes—
36C24×C4D411yescharacteristic
37C2×#117—22no—
38C2×#117—22no—
39C2×#117—22no—
40C2×#117—22no—
41C2×#117—22no—
42C2×#117—22no—
43C2×#117—24no—
44C2×#117—44no—
45C2×#117—44no—
46C2×#117C2×C414yes—
47C2×#117Q814yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4D411yescharacteristic
59C22×C22⋊C4C2311yescharacteristic
60C22×C42—22no—
61C22×C42D411yescharacteristic
62C22×C42C2311yescharacteristic
63#431—24no—
64#431C2×C418yescomplemented
65C2×#106—28no—
66C2×C4⋊C8—22no—
67C2×C22⋊C8C2314yes—
68C2×C4×C8—22no—
69#389—24no—
70#389—24no—
71#389—24no—
72#389—24no—
73#385—24no—
74#385—24no—
75#385—24no—
76#385—24no—
77#384—21no—
78#384—21no—
79#384—22no—
80#384—22no—
81#384D412yes—
82#384D412yes—
83#382—22no—
84#382—22no—
85#382—24no—
86#380—22no—
87#380—22no—
88#380—22no—
89#380—44no—
90#380C2314yes—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4D414yes—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95C2312yes—
100#340—28no—
101#324C2×C418yescomplemented
102#323C2×C418yes—
103C25C2×D411yescharacteristic
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4C4⋊C414yes—
114C23×C4SD1614yes—
115C23×C4SD1614yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2411yescharacteristic
122C22×C8—24no—
123#117—28no—
124#117—28no—
125#117—28no—
126#117—28no—
127#117—48no—
128#117—44no—
129#117—48no—
130#117—44no—
131#117—44no—
132#117—44no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4C4⋊C414yes—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C2×D414yes—
163C2×C4⋊C4C2×D414yes—
164C2×C4⋊C4C2×Q812yes—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—82no—
175C2×C22⋊C4—82no—
176C2×C22⋊C4C4⋊C414yes—
177C2×C22⋊C4C22×C412yes—
178C2×C22⋊C4C2×Q812yes—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—22no—
182C2×C42—24no—
183C2×C42—24no—
184C2×C42—48no—
185C2×C42C4⋊C414yes—
186C2×C42C22×C412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2×Q812yes—
190C2×C42#4114yes—
191#106—88no—
192C4⋊C8—28no—
193C22⋊C8—24no—
194C22⋊C8C22×C418yes—
195C4×C8—28no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—42no—
202#95—44no—
203#95—82no—
204#95—82no—
205C24—21no—
206C24—21no—
207C24—21no—
208C24—21no—
209C24—41no—
210C24—41no—
211C24—41no—
212C24—41no—
213C24C2×C4⋊C412yes—
214C24C2×SD1612yes—
215C24C2×SD1612yes—
216C24C22×D411yescharacteristic
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—28no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—28no—
229C22×C4—28no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4#106116yescomplemented
265C22×C4C2×C4⋊C412yes—
266C22×C4C2×C4⋊C412yes—
267C22×C4C2×C4⋊C412yes—
268C22×C4C2×C4⋊C412yes—
269C22×C4C2×C4⋊C412yes—
270C22×C4#12012yes—
271C22×C4#12011yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4#12312yes—
274C22×C4#12312yes—
275C22×C4C2×SD1612yes—
276C22×C4C2×SD1612yes—
277C22×C4C2×SD1612yes—
278C22×C4C2×SD1612yes—
279C22×C4AGL(1,ℤ/8)12yes—
280C22×C4#13712yes—
281C22×C4C23×C411yescharacteristic
282C22×C4C22×D412yes—
283C22×C4C22×Q811yescharacteristic
284C22×C4C2×#4112yes—
285C2×C8—28no—
286C2×C8—44no—
287C2×C8—44no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—28no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—48no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—84no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—84no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—28no—
311C22⋊C4—48no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—88no—
315C42—24no—
316C42—24no—
317C42—48no—
318C42—48no—
319C42C4×D414yes—
320C42C4×D414yes—
321C42C22⋊Q814yes—
322C42C22⋊Q814yes—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23C2×#10614yes—
350C23C22×C4⋊C411yescharacteristic
351C23C2×#12011yescharacteristic
352C23C2×#12311yescharacteristic
353C23C22×SD1611yescharacteristic
354C23C22×SD1611yescharacteristic
355C23C2×AGL(1,ℤ/8)11yescharacteristic
356C23C2×#13711yescharacteristic
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—216no—
368C2×C4—216no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—88no—
391C2×C4—88no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4#38514yes—
407C2×C4#38512yes—
408C2×C4#38512yes—
409C2×C4C2×#10614yes—
410C2×C4C2×#10614yes—
411C2×C4#42814yes—
412C2×C4#44812yes—
413C2×C4#44812yes—
414C2×C4#45012yes—
415C2×C4#45012yes—
416C2×C4#48114yes—
417C2×C4#48314yes—
418C2×C4C2×C4×D412yes—
419C2×C4C2×C22⋊Q812yes—
420C8—48no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—24no—
434C22—44no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22C2×#38511yescharacteristic
445C22C22×#10611yescharacteristic
446C22C2×#42811yescharacteristic
447C22C2×#45011yescharacteristic
448C22C2×#44811yescharacteristic
449C22#302111yescharacteristic
450C22#302211yescharacteristic
451C22#302711yescharacteristic
452C22#302811yescharacteristic
453C22C2×#48311yescharacteristic
454C22C2×#48111yescharacteristic
455C4—24no—
456C4—24no—
457C4—22no—
458C4—22no—
459C4—42no—
460C4—42no—
461C4—84no—
462C4—84no—
463C4—162no—
464C4—162no—
465C4#185314yes—
466C4#185314yes—
467C2—21no—
468C2—21no—
469C2—21no—
470C2—21no—
471C2—44no—
472C2#1830111yescharacteristic
473C2#1830211yescharacteristic
474C2#1830911yescharacteristic
475C2#1831011yescharacteristic
476C2C2×#185311yescharacteristic
477C2#2085811yescharacteristic
478C2#2089011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21981C2semidirect
5. #8857C2semidirect
7. #8026C2semidirect
27. #1458C4semidirect
64. #431C2×C4semidirect
101. #324C2×C4semidirect
264. C22×C4#106semidirect

Maximal subgroups

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

subgroupindex
#340522
#219812
C2×#14582
#143402
#88572
#88542
#80262