Finite Group Databasebeta

← #480530

Subgroups — #480530

Order 512.

1043 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35633C211yesmaximal, characteristic
2#17372C211yesmaximal, characteristic, complemented
3#17351C211yesmaximal, characteristic, complemented
4#11560C214yesmaximal, complemented
5#11558C214yesmaximal, complemented
6#11428C212yesmaximal, complemented
7#11425C212yesmaximal, complemented
8C2×#536C411yescharacteristic
9C2×#525C411yescharacteristic
10C22×C22⋊Q8C411yescharacteristic
11C2×#520C2211yescharacteristic
12#2983—21no—
13#2972—21no—
14#2534C412yes—
15#2534C2212yes—
16#2486C412yes—
17#2486C2212yes—
18#2429—21no—
19#2429—21no—
20#2427—21no—
21#2427—21no—
22#2417—22no—
23#2415—22no—
24#2316C411yescharacteristic
25#2310C2211yescharacteristic
26#2280C2211yescharacteristic
27#1466C2214yes—
28#1464—22no—
29#1461—21no—
30#1460—21no—
31#1459C2214yes—
32#1458C2212yes—
33#1458C2212yes—
34C2×#324—22no—
35#1317C2214yes—
36#1317C2218yes—
37#1316C2214yes—
38C22×#41C2×C411yescharacteristic
39C23×Q8C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41#536—44no—
42C2×C4⋊Q8—21no—
43C2×#124—21no—
44#525—44no—
45C2×#123C2×C412yes—
46C2×#123C2×C412yes—
47C2×#123C2×C412yes—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—44no—
54C2×#121—21no—
55C2×#120C2×C412yes—
56#520—44no—
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×D4—22no—
60C2×C4×D4C2×C412yes—
61C2×C4×D4D411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×C4×D4C2312yes—
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117D412yes—
67C2×#117C2311yescharacteristic
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4D412yes—
71C22×C22⋊C4C2×C412yes—
72#446—44no—
73C8×D4—44no—
74#432—24no—
75#431—22no—
76#431—22no—
77C2×C22⋊C8—24no—
78#398—22no—
79#398—22no—
80#397—22no—
81#397—22no—
82#396—22no—
83#396—22no—
84#394—22no—
85#394—22no—
86#391—21no—
87#391—21no—
88#391—21no—
89#391—21no—
90#389—22no—
91#389—21no—
92#389—21no—
93#389—22no—
94#389—22no—
95#388—22no—
96#388—22no—
97#386—22no—
98#386—22no—
99#381C2×C412yes—
100#381C2×C411yescharacteristic
101#381C2312yes—
102#381C2311yescharacteristic
103#380—22no—
104#380—22no—
105#380—21no—
106#380—21no—
107#380C2×C412yes—
108#380C2×C412yes—
109#379C2×C411yescharacteristic
110#379C2311yescharacteristic
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113#324—22no—
114#324—24no—
115#324—22no—
116#323—24no—
117#323C2314yes—
118#323C2314yes—
119C25C22×C411yescharacteristic
120C2×#41—24no—
121C2×#41—24no—
122C2×#41—42no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—24no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×Q8C22⋊C411yescharacteristic
129C22×D4—22no—
130C22×D4C22⋊C411yescharacteristic
131C22×D4C22×C412yes—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—24no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C412yes—
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C412yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C412yes—
143C23×C4C2×D412yes—
144C22×C8—44no—
145C4⋊Q8—44no—
146#124—44no—
147#123—44no—
148#123—44no—
149#123—44no—
150C22⋊Q8—44no—
151C22⋊Q8—48no—
152C22⋊Q8—48no—
153C22⋊Q8—44no—
154C22⋊Q8—44no—
155#121—44no—
156#120—44no—
157C4×Q8—24no—
158C4×Q8—42no—
159C4×D4—24no—
160C4×D4—42no—
161C4×D4—44no—
162C4×D4—44no—
163C4×D4—48no—
164#117—42no—
165#117—44no—
166#117—42no—
167#117—42no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22×C412yes—
200C2×C22⋊C4C22×C411yescharacteristic
201C2×C22⋊C4C22×C411yescharacteristic
202C2×C22⋊C4C22×C411yescharacteristic
203C2×C22⋊C4C2×D412yes—
204C2×C22⋊C4C2411yescharacteristic
205C2×C42—21no—
206C2×C42—21no—
207C2×C42C22⋊C412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C4⋊C8—42no—
212C4⋊C8—42no—
213C4⋊C8—44no—
214C22⋊C8—24no—
215C22⋊C8—24no—
216C22⋊C8—44no—
217#97—44no—
218C4×C8—42no—
219C4×C8—42no—
220#95—21no—
221#95—21no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—21no—
228#95—22no—
229#95—22no—
230C24—21no—
231C24—22no—
232C24—41no—
233C24—42no—
234C24—41no—
235C24—81no—
236C24C2×C22⋊C412yes—
237C24C23×C411yescharacteristic
238#41—416no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—88no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—42no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—82no—
257C2×D4#10014yes—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—88no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4#13114yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C2×C8—44no—
298C2×C8—44no—
299C2×C8—164no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4#9914yes—
316C22⋊C4—24no—
317C22⋊C4—41no—
318C22⋊C4—42no—
319C22⋊C4—41no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—42no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—81no—
332C22⋊C4—81no—
333C42—22no—
334C42—22no—
335C42—24no—
336C42—42no—
337C42—42no—
338C42—42no—
339C23—22no—
340C23—21no—
341C23—21no—
342C23—22no—
343C23—22no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—42no—
352C23—42no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—82no—
357C23—84no—
358C23—84no—
359C23—82no—
360C23C2×#10014yes—
361C23C22×C22⋊C411yescharacteristic
362C23C2×#13112yes—
363Q8—44no—
364Q8—84no—
365Q8—84no—
366D4—28no—
367D4—48no—
368D4—88no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—28no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—42no—
381C2×C4—48no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—84no—
406C2×C4—82no—
407C2×C4—84no—
408C2×C4—162no—
409C2×C4#40812yes—
410C2×C4#40812yes—
411C2×C4C2×#9914yes—
412C2×C4C2×#9912yes—
413C2×C4C2×#10012yes—
414C8—84no—
415C8—84no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—24no—
420C22—44no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—41no—
427C22—41no—
428C22—42no—
429C22—42no—
430C22—82no—
431C22—81no—
432C22—81no—
433C22—84no—
434C22—84no—
435C22—81no—
436C22—81no—
437C22C2×#40811yescharacteristic
438C22C22×#9911yescharacteristic
439C22C22×#10011yescharacteristic
440C4—24no—
441C4—22no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—42no—
446C4—41no—
447C4—42no—
448C4—81no—
449C4—81no—
450C4—82no—
451C4—81no—
452C4—82no—
453C4#147114yes—
454C2—21no—
455C2—21no—
456C2—42no—
457C2—44no—
458C2—82no—
459C2C2×#147111yescharacteristic
460C2#3334911yescharacteristic
461C2#3335011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC2×#525C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17372C2semidirect
3. #17351C2semidirect
4. #11560C2semidirect
5. #11558C2semidirect
6. #11428C2semidirect
7. #11425C2semidirect

Maximal subgroups

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

subgroupindex
#356332
#173722
#173512
#115602
#115582
#114282
#114252