Finite Group Databasebeta

← #194872

Subgroups — #194872

Order 512.

1426 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2611C211yesmaximal, characteristic
2#34052C211yesmaximal, characteristic
3#21838C212yesmaximal, complemented
4C2×#1464C211yesmaximal, characteristic
5#9027C212yesmaximal, complemented
6#9025C212yesmaximal
7#9023C212yesmaximal, complemented
8#9021C212yesmaximal
9#8022C212yesmaximal, complemented
10C22×#125C2211yescharacteristic
11C22×#117—21no—
12C22×#117C2211yescharacteristic
13C2×#432C2212yes—
14#2611—24no—
15#2611C418yescomplemented
16C2×#400C2211yescharacteristic
17C2×#399C2211yescharacteristic
18C2×#398C2211yescharacteristic
19C2×#397C2211yescharacteristic
20#2317—24no—
21#2316—21no—
22#2316C2212yes—
23#2315—21no—
24#2315C2212yes—
25#2296—24no—
26C2×#389—21no—
27C2×#388—21no—
28C2×#387—21no—
29C2×#385—21no—
30#2285—21no—
31#2285C2212yes—
32C2×#376C2211yescharacteristic
33#1992—24no—
34#1989—24no—
35#1824C2214yes—
36#1823C2214yes—
37#1464—24no—
38C2×#324C2212yes—
39C2×#323C2212yes—
40#1294C2214yes—
41#1294C2214yes—
42C24×C4D411yescharacteristic
43C2×#125—22no—
44C2×#125—22no—
45C2×#125—22no—
46C2×#125—24no—
47C2×#125C2×C414yescomplemented
48C2×#117—22no—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—24no—
55C2×#117—44no—
56C2×#117—44no—
57C2×#117C2×C414yes—
58C2×#117D414yes—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4D411yescharacteristic
72C22×C22⋊C4C2311yescharacteristic
73C22×C42—22no—
74C22×C42—21no—
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77#432—24no—
78#432—24no—
79C2×C4⋊C8—22no—
80C2×C22⋊C8C2314yes—
81C2×#97—22no—
82#400—22no—
83#400C2×C414yescomplemented
84#399—22no—
85#399C2×C414yes—
86#398—22no—
87#398C2×C414yescomplemented
88#397—22no—
89#397C2×C414yes—
90#389—28no—
91#389—28no—
92#389—28no—
93#388—28no—
94#387—28no—
95#385—28no—
96#385—28no—
97#385—28no—
98#384—22no—
99#384D414yes—
100#382—22no—
101#382—22no—
102#380—22no—
103#380—22no—
104#380—22no—
105#380C2314yes—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4D414yes—
110#376—22no—
111#376C2×C414yescomplemented
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95C2311yescharacteristic
118C2×#95C2311yescharacteristic
119#340—28no—
120#340—28no—
121#324—24no—
122#323—24no—
123C25C2×D411yescharacteristic
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—24no—
136C23×C4C22⋊C414yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—24no—
144#125—28no—
145#125—28no—
146#125—44no—
147#125—44no—
148#117—28no—
149#117—28no—
150#117—28no—
151#117—28no—
152#117—48no—
153#117—48no—
154#117—44no—
155#117—44no—
156#117—44no—
157#117—44no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—48no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4C22⋊C414yes—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D418yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4C22⋊C414yes—
206C2×C22⋊C4C22×C412yes—
207C2×C22⋊C4C2×D412yes—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—48no—
215C2×C42—44no—
216C2×C42C22⋊C414yes—
217C2×C42C22×C412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42#4114yes—
222C4⋊C8—44no—
223C22⋊C8—24no—
224C22⋊C8—24no—
225#97—44no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—21no—
235#95—21no—
236#95—24no—
237#95C22×C412yes—
238#95C22×C412yes—
239C24—21no—
240C24—21no—
241C24—21no—
242C24—21no—
243C24—42no—
244C24—42no—
245C24C2×C22⋊C412yes—
246C24AGL(1,ℤ/8)12yes—
247C24#13712yes—
248C24C22×D411yescharacteristic
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—28no—
252C22×C4—28no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—28no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—28no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—48no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—84no—
300C22×C4—84no—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4#12012yes—
307C22×C4#12011yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4#12312yes—
310C22×C4#12312yes—
311C22×C4AGL(1,ℤ/8)12yes—
312C22×C4#13712yes—
313C22×C4C23×C411yescharacteristic
314C22×C4C22×D412yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C2×#4112yes—
317C2×C8—48no—
318C2×C8—44no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—28no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—48no—
325C4⋊C4—48no—
326C4⋊C4—48no—
327C4⋊C4—48no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—48no—
333C4⋊C4—48no—
334C4⋊C4—84no—
335C4⋊C4—84no—
336C4⋊C4—84no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—28no—
340C22⋊C4—48no—
341C22⋊C4—48no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—48no—
345C42—24no—
346C42—24no—
347C42—48no—
348C42—48no—
349C42—48no—
350C42C4×D414yes—
351C42C4×D414yes—
352C42#12114yes—
353C42#12114yes—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—42no—
363C23—42no—
364C23—44no—
365C23—44no—
366C23—42no—
367C23—42no—
368C23—82no—
369C23—82no—
370C23—81no—
371C23—81no—
372C23#41714yes—
373C23C22×C22⋊C411yescharacteristic
374C23C2×#12011yescharacteristic
375C23C2×#12311yescharacteristic
376C23C2×AGL(1,ℤ/8)11yescharacteristic
377C23C2×AGL(1,ℤ/8)11yescharacteristic
378C23C2×#13711yescharacteristic
379C23C2×#13711yescharacteristic
380C2×C4—216no—
381C2×C4—216no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—48no—
422C2×C4—48no—
423C2×C4—48no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—816no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—82no—
437C2×C4#38612yes—
438C2×C4#38612yes—
439C2×C4#38614yes—
440C2×C4#41714yes—
441C2×C4C2×C4×D412yes—
442C2×C4C2×#12112yes—
443C8—84no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—24no—
453C22—44no—
454C22—44no—
455C22—44no—
456C22—42no—
457C22—42no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22C2×#38611yescharacteristic
462C22C2×#41711yescharacteristic
463C22C2×#41711yescharacteristic
464C22#300311yescharacteristic
465C22#300411yescharacteristic
466C22#301812yes—
467C22#302812yes—
468C22#309211yescharacteristic
469C22#309211yescharacteristic
470C4—22no—
471C4—22no—
472C4—24no—
473C4—24no—
474C4—22no—
475C4—22no—
476C4—44no—
477C4—42no—
478C4—42no—
479C4—82no—
480C4—88no—
481C4—82no—
482C4—82no—
483C2—21no—
484C2—21no—
485C2—21no—
486C2—21no—
487C2—44no—
488C2#1833912yes—
489C2#1834012yes—
490C2#2090911yescharacteristic
491C2#2095211yescharacteristic
492C2#2096211yescharacteristic
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
3. #21838C2semidirect
5. #9027C2semidirect
7. #9023C2semidirect
9. #8022C2semidirect
15. #2611C4semidirect
47. C2×#125C2×C4semidirect
83. #400C2×C4semidirect
87. #398C2×C4semidirect
111. #376C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26112
#340522
#218382
C2×#14642
#90272
#90252
#90232
#90212
#80222