Finite Group Databasebeta

← #480332

Subgroups — #480332

Order 512.

1155 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#34265C211yesmaximal, characteristic
2#17372C211yesmaximal, characteristic, complemented
3#17370C211yesmaximal, characteristic, complemented
4#11527C214yesmaximal, complemented
5#11525C214yesmaximal, complemented
6#11452C212yesmaximal, complemented
7#11427C212yesmaximal, complemented
8C2×#535C411yescharacteristic
9C2×#520C2211yescharacteristic
10C22×C4×D4C411yescharacteristic
11#2983—21no—
12#2975—21no—
13#2367—21no—
14#2367—21no—
15#2366—21no—
16#2366—21no—
17#2363—22no—
18#2361—22no—
19#2321C412yes—
20#2321C2212yes—
21#2309C2211yescharacteristic
22C4×#120C412yes—
23C4×#120C2212yes—
24#2297C411yescharacteristic
25#2296C2211yescharacteristic
26#2274C411yescharacteristic
27#1466C2214yes—
28#1465C2214yes—
29#1464—22no—
30#1463—22no—
31#1461—21no—
32#1460—21no—
33#1458C2212yes—
34#1458C2212yes—
35#1326C2214yes—
36#1326C2214yes—
37#1316C2218yes—
38C22×#41C2×C411yescharacteristic
39C23×D4C2×C411yescharacteristic
40C24×C4C2×C412yes—
41#535—24no—
42#535—24no—
43C2×#127—21no—
44C2×#124—21no—
45C2×#121—22no—
46C2×#120C2×C412yes—
47C2×#120C2×C412yes—
48#520—44no—
49C2×C4×Q8D411yescharacteristic
50C2×C4×Q8D411yescharacteristic
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—21no—
55C2×C4×D4—21no—
56C2×C4×D4—44no—
57C2×C4×D4C2×C412yes—
58C2×C4×D4D411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4C2312yes—
61C2×#117C2×C411yescharacteristic
62C2×#117C2×C411yescharacteristic
63C2×#117D412yes—
64C2×#117C2311yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4D412yes—
67C22×C22⋊C4C2×C412yes—
68C22×C42C2×C411yescharacteristic
69#446—44no—
70#436—44no—
71#432—24no—
72#431—22no—
73#431—22no—
74#407—24no—
75#391—21no—
76#390—21no—
77#390—21no—
78#390—21no—
79#390—22no—
80#390—22no—
81#390—21no—
82#390—21no—
83#389—21no—
84#389—21no—
85#388—22no—
86#388—22no—
87#388—22no—
88#388—22no—
89#386—22no—
90#386—22no—
91#386—22no—
92#386—22no—
93#386C2×C412yes—
94#386C2×C412yes—
95#385—22no—
96#385—22no—
97#385C2×C411yescharacteristic
98#385C2×C411yescharacteristic
99#385C2311yescharacteristic
100#385C2311yescharacteristic
101#379C2×C412yes—
102#379C2312yes—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4C2×C412yes—
112C4×C22⋊C4C2×C412yes—
113#324—22no—
114#324—22no—
115#324—24no—
116#323—24no—
117#323C2314yes—
118#323C2314yes—
119C25C22×C412yes—
120C2×#41—22no—
121C2×#41—22no—
122C2×#41—42no—
123C2×#41—42no—
124C22×Q8C22⋊C411yescharacteristic
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—42no—
131C22×D4—42no—
132C22×D4C22⋊C411yescharacteristic
133C22×D4M1614yescomplemented
134C22×D4C22×C412yes—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—44no—
138C23×C4—84no—
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C412yes—
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D412yes—
146C2×M16—44no—
147#127—28no—
148#124—28no—
149#121—216no—
150#120—24no—
151#120—24no—
152#120—24no—
153#120—24no—
154C4×Q8—24no—
155C4×Q8—42no—
156C4×D4—24no—
157C4×D4—42no—
158C4×D4—44no—
159C4×D4—44no—
160C4×D4—48no—
161C4×D4—44no—
162C4×D4—44no—
163C4×D4—84no—
164C4×D4—84no—
165#117—22no—
166#117—42no—
167#117—44no—
168#117—42no—
169#117M1614yescomplemented
170C2×C4⋊C4—21no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
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—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22×C411yescharacteristic
198C2×C22⋊C4C22×C411yescharacteristic
199C2×C22⋊C4C22×C412yes—
200C2×C22⋊C4C22×C411yescharacteristic
201C2×C22⋊C4C2×D412yes—
202C2×C22⋊C4C2411yescharacteristic
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—42no—
210C2×C42C22⋊C412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C4⋊C8—42no—
215C4⋊C8—42no—
216C4⋊C8—44no—
217C22⋊C8—24no—
218C22⋊C8—24no—
219C22⋊C8—44no—
220#97—44no—
221C4×C8—42no—
222C4×C8—42no—
223#95—22no—
224#95—21no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95—21no—
230#95—21no—
231C24—22no—
232C24—22no—
233C24—42no—
234C24—42no—
235C24—82no—
236C24—82no—
237C24C2×C22⋊C412yes—
238C24C2×M1614yes—
239C24C23×C411yescharacteristic
240#41—48no—
241#41—48no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—28no—
248C2×D4—28no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—48no—
259C2×D4—48no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—82no—
263C2×D4—82no—
264C2×D4—82no—
265C2×D4—82no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—41no—
284C22×C4—44no—
285C22×C4—41no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—81no—
291C22×C4—82no—
292C22×C4—88no—
293C22×C4—88no—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×M1612yes—
302C22×C4C2×M1612yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305M16—164no—
306C2×C8—44no—
307C2×C8—44no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—24no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—81no—
319C4⋊C4—81no—
320C4⋊C4#9914yes—
321C22⋊C4—28no—
322C22⋊C4—28no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—24no—
326C22⋊C4—41no—
327C22⋊C4—42no—
328C22⋊C4—41no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—42no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4C2×M1612yes—
339C22⋊C4C2×M1612yes—
340C42—24no—
341C42—22no—
342C42—22no—
343C42—24no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—82no—
349C42—82no—
350C23—21no—
351C23—21no—
352C23—24no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—41no—
357C23—41no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—44no—
365C23—44no—
366C23—42no—
367C23—42no—
368C23—84no—
369C23—84no—
370C23—84no—
371C23—84no—
372C23—88no—
373C23C22×C22⋊C411yescharacteristic
374C23C22×M1611yescharacteristic
375C23C22×M1611yescharacteristic
376Q8—44no—
377D4—48no—
378D4—44no—
379D4—44no—
380D4—44no—
381D4—48no—
382D4—88no—
383D4—88no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—42no—
400C2×C4—48no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—84no—
421C2×C4—84no—
422C2×C4—88no—
423C2×C4—88no—
424C2×C4—82no—
425C2×C4—162no—
426C2×C4#40712yes—
427C2×C4#40712yes—
428C2×C4C2×#9914yes—
429C2×C4C2×#9912yes—
430C2×C4#41312yes—
431C8—84no—
432C8—84no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—41no—
438C22—41no—
439C22—42no—
440C22—44no—
441C22—44no—
442C22—44no—
443C22—41no—
444C22—41no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—44no—
449C22—82no—
450C22—82no—
451C22—82no—
452C22—88no—
453C22—88no—
454C22—82no—
455C22—82no—
456C22C2×#40711yescharacteristic
457C22C22×#9911yescharacteristic
458C22C2×#41311yescharacteristic
459C4—24no—
460C4—22no—
461C4—42no—
462C4—42no—
463C4—41no—
464C4—42no—
465C4—42no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—42no—
470C4—82no—
471C4—81no—
472C4#146514yes—
473C2—21no—
474C2—21no—
475C2—41no—
476C2—41no—
477C2—82no—
478C2—84no—
479C2C2×#146511yescharacteristic
480C2#3334811yescharacteristic
481C2#3335111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC2×#535C4
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. #17370C2semidirect
4. #11527C2semidirect
5. #11525C2semidirect
6. #11452C2semidirect
7. #11427C2semidirect
133. C22×D4M16semidirect
169. #117M16semidirect

Maximal subgroups

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

subgroupindex
#342652
#173722
#173702
#115272
#115252
#114522
#114272