Finite Group Databasebeta

← #188527

Subgroups — #188527

Order 512.

1266 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34206C211yesmaximal, characteristic
2#21951C211yesmaximal, characteristic, complemented
3C2×#1517C211yesmaximal, characteristic
4C2×#1502C211yesmaximal, characteristic
5#13955C211yesmaximal, characteristic, complemented
6#8575C214yesmaximal, complemented
7#8573C214yesmaximal, complemented
8#8026C211yesmaximal, characteristic, complemented
9#8014C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C2×#520—21no—
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2367—22no—
15#2366C412yescomplemented
16#2365C412yescomplemented
17#2354C412yescomplemented
18#2352—22no—
19D4×C4⋊C4C412yescomplemented
20#2325—22no—
21#2311—21no—
22C4×#121C2214yes—
23#2285C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1939—22no—
28#1939—22no—
29#1797C2214yes—
30#1517—24no—
31#1502—24no—
32C2×#327C2212yes—
33C2×#327C2212yes—
34C2×#327C2214yes—
35#1442—22no—
36C4⋊#95—21no—
37C4⋊#95—21no—
38#1434C2212yes—
39#1434C2212yes—
40#1294C2212yes—
41#1294C2212yes—
42#1294C2214yes—
43C22×#41—21no—
44#532—22no—
45#532—22no—
46#532D414yes—
47#532Q814yes—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×#127C2×C411yescharacteristic
50C2×#124—21no—
51C2×#121—21no—
52C2×#121C2312yes—
53#520—44no—
54#520—44no—
55C2×C4×Q8—22no—
56C2×C4×D4—24no—
57C2×C4×D4—22no—
58C2×C4×D4—21no—
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2×C414yes—
61C2×#117—21no—
62C2×#117—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C42—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71#431—22no—
72#431—22no—
73#431—22no—
74#431—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—24no—
78C2×C22⋊C8C2312yes—
79C2×C22⋊C8C2312yes—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82#391—22no—
83#390—22no—
84#390C2×C414yes—
85#389C2×C414yes—
86#388—24no—
87#388—24no—
88#387—24no—
89#386—24no—
90#386—24no—
91#385—24no—
92#384C2×C411yescharacteristic
93#384C2×C411yescharacteristic
94#382—21no—
95#382—22no—
96#382—22no—
97#380—21no—
98#380—22no—
99#380—22no—
100#380C2312yes—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4C2×C414yes—
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4C2×C412yes—
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C4×C22⋊C4—24no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95D412yes—
117C2×#95C2311yescharacteristic
118C2×#95C2311yescharacteristic
119C2×#95C2312yes—
120#336—24no—
121#327—24no—
122#327—24no—
123#327—28no—
124C2×#41—24no—
125C2×#41—24no—
126C2×#41—42no—
127C2×#41—42no—
128C2×#41—42no—
129C2×#41—42no—
130C22×Q8—21no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4C22×C412yes—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2411yescharacteristic
145C22×C8—22no—
146C22×C8—22no—
147C4⋊Q8—22no—
148C4⋊Q8C4⋊C414yes—
149#127—22no—
150#127C4⋊C414yes—
151#124—24no—
152#124—24no—
153#121—24no—
154#121—24no—
155#121—22no—
156#121—22no—
157#121C2×D414yes—
158#121C2×Q814yes—
159C4×Q8—48no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—48no—
164C4×D4—48no—
165C4×D4—48no—
166C4×D4—84no—
167C4×D4C4⋊C418yes—
168#117—48no—
169#117—42no—
170#117—42no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C412yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C42—24no—
202C2×C42—22no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—82no—
207C2×C42C22×C414yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C2×D414yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×Q812yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217C4⋊C8—42no—
218C4⋊C8—42no—
219C4⋊C8—84no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C4×C8—42no—
225C4×C8—42no—
226#95—24no—
227#95—22no—
228#95—24no—
229#95—22no—
230#95—21no—
231#95—21no—
232#95—22no—
233#95—21no—
234#95—21no—
235#95—22no—
236#95—44no—
237#95—44no—
238#95—42no—
239#95—82no—
240#95—82no—
241#95C22⋊C414yes—
242#95C22⋊C414yes—
243C24—22no—
244C24C22×D411yescharacteristic
245#41—48no—
246#41—48no—
247#41—48no—
248#41—48no—
249C2×Q8—24no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—81no—
266C2×D4—81no—
267C2×D4C2×C4⋊C414yes—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—28no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—48no—
287C22×C4—44no—
288C22×C4—44no—
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—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—84no—
309C22×C4—84no—
310C22×C4C2×C22⋊C414yes—
311C22×C4C2×C22⋊C414yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4#12112yes—
317C22×C4#12112yes—
318C22×C4C22⋊Q812yes—
319C22×C4C22⋊Q812yes—
320C22×C4#13512yes—
321C22×C4#13512yes—
322C22×C4AGL(1,ℤ/8)12yes—
323C22×C4AGL(1,ℤ/8)12yes—
324C22×C4C23×C411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×Q811yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—48no—
345C4⋊C4—48no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—81no—
349C4⋊C4—81no—
350C4⋊C4—84no—
351C4⋊C4C2×C4⋊C414yes—
352C22⋊C4—24no—
353C22⋊C4—24no—
354C22⋊C4—28no—
355C22⋊C4—48no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—84no—
359C22⋊C4—84no—
360C42—44no—
361C42—44no—
362C42—48no—
363C42—44no—
364C42—84no—
365C42C2×C4⋊C412yes—
366C42C2×C4⋊C412yes—
367C42#11712yes—
368C42#11712yes—
369C42#12112yes—
370C42#12112yes—
371C42C22⋊Q812yes—
372C42C22⋊Q812yes—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—24no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23C22×C22⋊C411yescharacteristic
385C23C2×#12111yescharacteristic
386C23C2×C22⋊Q811yescharacteristic
387C23C2×#13511yescharacteristic
388C23C2×#13511yescharacteristic
389C23C2×AGL(1,ℤ/8)11yescharacteristic
390C23C2×AGL(1,ℤ/8)11yescharacteristic
391Q8—48no—
392D4—44no—
393D4—44no—
394D4—44no—
395D4—44no—
396D4—44no—
397D4—44no—
398D4—48no—
399D4—84no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—416no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—168no—
456C2×C4#38012yes—
457C2×C4#38012yes—
458C2×C4#38012yes—
459C2×C4#38012yes—
460C2×C4#41612yes—
461C2×C4#41612yes—
462C2×C4#41812yes—
463C2×C4#41812yes—
464C2×C4#46314yes—
465C2×C4#46314yes—
466C2×C4#47814yes—
467C2×C4#47814yes—
468C2×C4C22×C4⋊C411yescharacteristic
469C2×C4C2×#11711yescharacteristic
470C2×C4C2×#12111yescharacteristic
471C2×C4C2×C22⋊Q811yescharacteristic
472C8—82no—
473C8—82no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—44no—
487C22—44no—
488C22—44no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22C2×#38011yescharacteristic
494C22#288311yescharacteristic
495C22#288411yescharacteristic
496C22C2×#41611yescharacteristic
497C22C2×#41811yescharacteristic
498C22#290111yescharacteristic
499C22#290111yescharacteristic
500C22C2×#46311yescharacteristic
501C22C2×#46311yescharacteristic
502C22C2×#47811yescharacteristic
503C22C2×#47811yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—22no—
509C4—22no—
510C4—44no—
511C4—44no—
512C4—44no—
513C4—82no—
514C4—82no—
515C4—82no—
516C4—82no—
517C4—162no—
518C4#180414yes—
519C4#180414yes—
520C2—21no—
521C2—21no—
522C2—21no—
523C2—21no—
524C2—84no—
525C2#1761611yescharacteristic
526C2#1761711yescharacteristic
527C2#1762611yescharacteristic
528C2#1763111yescharacteristic
529C2#2063411yescharacteristic
530C2C2×#180411yescharacteristic
531C2#2065511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21951C2semidirect
5. #13955C2semidirect
6. #8575C2semidirect
7. #8573C2semidirect
8. #8026C2semidirect
9. #8014C2semidirect
15. #2366C4semidirect
16. #2365C4semidirect
17. #2354C4semidirect
19. D4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#342062
#219512
C2×#15172
C2×#15022
#139552
#85752
#85732
#80262
#80142