Finite Group Databasebeta

← #482107

Subgroups — #482107

Order 512.

1116 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36009C211yesmaximal, characteristic
2#20638C211yesmaximal, characteristic, complemented
3#17350C211yesmaximal, characteristic, complemented
4#11884C214yesmaximal, complemented
5#11881C214yesmaximal, complemented
6#11700C212yesmaximal, complemented
7#11697C212yesmaximal, complemented
8C2×#547C411yescharacteristic
9C2×#546—21no—
10C2×#518C2211yescharacteristic
11#2972—21no—
12#2905—21no—
13C22×#100—21no—
14#2621C2211yescharacteristic
15#2617C412yes—
16#2547C411yescharacteristic
17#2543C2212yes—
18#2535C411yescharacteristic
19#2486C2212yes—
20#2427—21no—
21#2425—22no—
22#2415—21no—
23#2412—22no—
24#2392C2211yescharacteristic
25C2×#397C412yes—
26#2349C411yescharacteristic
27#2339—21no—
28#1800—22no—
29#1800C2214yes—
30#1742C2212yes—
31#1460—21no—
32#1459C2214yes—
33#1458C2212yes—
34C2×#323—22no—
35#1346C2214yescomplemented
36#1346C2218yescomplemented
37#1345C2214yescomplemented
38C22×#41—21no—
39C23×D4D411yescharacteristic
40C22×M16—22no—
41#547—44no—
42#546—44no—
43#546—44no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#124—21no—
46C2×#123—22no—
47C2×#123—22no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×#121—21no—
51C2×#121—22no—
52C2×#121C2×C412yes—
53C2×#120—22no—
54#518—44no—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4D412yes—
62C2×C4×D4D411yescharacteristic
63C2×C4×D4C2312yes—
64C2×#117D411yescharacteristic
65C2×#117C2311yescharacteristic
66C22×C4⋊C4C2×C412yes—
67C22×C22⋊C4—22no—
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4D412yes—
70C8×D4—44no—
71#431—22no—
72#424—44no—
73#423—24no—
74C2×#100—24no—
75C2×#100—44no—
76C2×C22⋊C8—24no—
77#398—21no—
78#398C2312yes—
79#398C2311yescharacteristic
80#397—22no—
81#397—22no—
82#397—21no—
83#397C2×C412yes—
84#397C2×C412yes—
85#397C2×C411yescharacteristic
86#397C2×C412yes—
87#397C2×C412yes—
88#396—21no—
89#394—22no—
90#394—22no—
91#394—21no—
92#394C2311yescharacteristic
93#392C2×C411yescharacteristic
94#392C2×C412yes—
95#391—21no—
96#391—21no—
97#389—21no—
98#389—22no—
99#388—22no—
100#386—22no—
101#386—22no—
102#386—22no—
103#386—22no—
104#385—22no—
105#382C2×C411yescharacteristic
106#381—22no—
107#380—22no—
108#380—22no—
109#380—21no—
110#380C2×C412yes—
111C4×C4⋊C4C2×C411yescharacteristic
112C4×C22⋊C4—21no—
113C2×#95—22no—
114C2×#95C2×C412yes—
115#343—22no—
116#343C2314yes—
117#324C2314yes—
118#323—22no—
119#323—24no—
120C25C22⋊C412yes—
121C2×#41—44no—
122C2×#41—44no—
123C2×#41—44no—
124C22×Q8—21no—
125C22×D4—22no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—42no—
129C22×D4—42no—
130C22×D4C22⋊C411yescharacteristic
131C22×D4C22⋊C411yescharacteristic
132C22×D4D814yes—
133C22×D4SD1614yes—
134C22×D4C22×C411yescharacteristic
135C22×D4C2×D412yes—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22×C412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C2×M16—24no—
145C2×M16—42no—
146C2×M16—44no—
147C2×M16—44no—
148C22×C8—42no—
149C22×C8—44no—
150C4⋊Q8—42no—
151#124—44no—
152#123—48no—
153#123—44no—
154#123—44no—
155C22⋊Q8—42no—
156C22⋊Q8—42no—
157C22⋊Q8—44no—
158#121—44no—
159#121—44no—
160#121—44no—
161#121—44no—
162#120—48no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—44no—
166C4×D4—42no—
167C4×D4—42no—
168C4×D4—44no—
169C4×D4—42no—
170C4×D4—44no—
171C4×D4—42no—
172C4×D4—42no—
173#117—24no—
174#117—42no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2411yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—48no—
204C2×C22⋊C4C22⋊C412yes—
205C2×C22⋊C4C22⋊C412yes—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22×C411yescharacteristic
208C2×C22⋊C4C22×C412yes—
209C2×C22⋊C4C2×D411yescharacteristic
210C2×C22⋊C4C2×D412yes—
211C2×C42—21no—
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C4⋊C8—48no—
216C4⋊C8—42no—
217#100—48no—
218C22⋊C8—24no—
219C22⋊C8—44no—
220C4×C8—42no—
221#95—21no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—21no—
228#95—21no—
229#95—22no—
230#95C22×C411yescharacteristic
231#95C22×C412yes—
232#95C22×C411yescharacteristic
233C24—21no—
234C24—42no—
235C24—42no—
236C24—82no—
237C24—82no—
238C24#10218yes—
239C24C2×C22⋊C411yescharacteristic
240C24C2×C22⋊C412yes—
241C24C2×C22⋊C412yes—
242#41—816no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—48no—
251C2×D4—48no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—41no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—42no—
258C2×D4—44no—
259C2×D4—44no—
260C2×D4—81no—
261C2×D4—81no—
262C2×D4—81no—
263C2×D4—82no—
264C2×D4—82no—
265C2×D4#10214yes—
266C2×D4#10214yes—
267C2×D4C2×D814yes—
268C2×D4C2×SD1614yes—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×D812yes—
299C22×C4C2×SD1612yes—
300C22×C4C23×C411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303M16—44no—
304M16—44no—
305M16—84no—
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—84no—
309C2×C8—164no—
310C4⋊C4—22no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—82no—
323C4⋊C4—81no—
324C4⋊C4#9914yes—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—44no—
342C22⋊C4—81no—
343C22⋊C4—82no—
344C22⋊C4—81no—
345C42—22no—
346C42—24no—
347C42—22no—
348C42—42no—
349C23—21no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—21no—
354C23—22no—
355C23—48no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—88no—
370C23—162no—
371C23—162no—
372C23—164no—
373C23C2×#10212yes—
374C23C2×#10214yes—
375C23C2×#10214yes—
376C23C22×C22⋊C411yescharacteristic
377Q8—84no—
378D4—28no—
379D4—28no—
380D4—44no—
381D4—44no—
382D4—82no—
383D4—88no—
384D4—84no—
385D4—164no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—84no—
417C2×C4—84no—
418C2×C4—88no—
419C2×C4—84no—
420C2×C4—162no—
421C2×C4—162no—
422C2×C4—164no—
423C2×C4C2×#9914yes—
424C2×C4C2×#9912yes—
425C2×C4C2×#10212yes—
426C2×C4#42112yes—
427C2×C4C22×D811yescharacteristic
428C2×C4C22×SD1611yescharacteristic
429C8—84no—
430C8—84no—
431C22—21no—
432C22—21no—
433C22—22no—
434C22—24no—
435C22—24no—
436C22—44no—
437C22—42no—
438C22—44no—
439C22—44no—
440C22—42no—
441C22—42no—
442C22—41no—
443C22—42no—
444C22—42no—
445C22—41no—
446C22—82no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—168no—
456C22C22×#9911yescharacteristic
457C22C22×#10211yescharacteristic
458C22C2×#42111yescharacteristic
459C4—24no—
460C4—21no—
461C4—21no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—42no—
467C4—81no—
468C4—82no—
469C4—81no—
470C4—81no—
471C4—82no—
472C4#151914yes—
473C2—21no—
474C2—21no—
475C2—44no—
476C2—44no—
477C2—42no—
478C2—161no—
479C2C2×#151911yescharacteristic
480C2#3335011yescharacteristic
481C2#3337111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radical#36009C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20638C2semidirect
3. #17350C2semidirect
4. #11884C2semidirect
5. #11881C2semidirect
6. #11700C2semidirect
7. #11697C2semidirect
35. #1346C22semidirect
36. #1346C22semidirect
37. #1345C22semidirect

Maximal subgroups

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

subgroupindex
#360092
#206382
#173502
#118842
#118812
#117002
#116972