Finite Group Databasebeta

← #480359

Subgroups — #480359

Order 512.

1038 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34201C211yesmaximal, characteristic
2#17372C211yesmaximal, characteristic, complemented
3#17370C211yesmaximal, characteristic, complemented
4#11527C214yesmaximal, complemented
5#11526C214yesmaximal, complemented
6#11454C212yesmaximal, complemented
7#11425C212yesmaximal, complemented
8C2×#525C411yescharacteristic
9C2×#520C2211yescharacteristic
10C22×#117C411yescharacteristic
11#2983—21no—
12#2975—21no—
13#2363—21no—
14#2363—21no—
15#2362—21no—
16#2362—21no—
17#2354—22no—
18#2352—22no—
19#2321C412yes—
20#2321C2212yes—
21#2317C2211yescharacteristic
22#2311C411yescharacteristic
23C4×#123C412yes—
24C4×#123C2212yes—
25#2280C2211yescharacteristic
26#2274C411yescharacteristic
27#1466C2214yes—
28#1465C2214yes—
29#1464—22no—
30#1463—22no—
31#1461—21no—
32#1460—21no—
33#1458C2212yes—
34#1458C2212yes—
35#1326C2218yes—
36#1317C2214yes—
37#1316C2214yes—
38C22×#41C2×C411yescharacteristic
39C24×C4C2×C411yescharacteristic
40#525—24no—
41#525—24no—
42C2×#123C2×C412yes—
43C2×C22⋊Q8—21no—
44C2×#121—21no—
45C2×#120C2×C412yes—
46#520—44no—
47C2×C4×Q8D411yescharacteristic
48C2×C4×Q8D411yescharacteristic
49C2×C4×D4—22no—
50C2×C4×D4C2×C412yes—
51C2×C4×D4D411yescharacteristic
52C2×C4×D4D411yescharacteristic
53C2×C4×D4C2312yes—
54C2×#117—22no—
55C2×#117—21no—
56C2×#117—21no—
57C2×#117—44no—
58C2×#117C2×C411yescharacteristic
59C2×#117C2×C411yescharacteristic
60C2×#117D412yes—
61C2×#117C2311yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4D412yes—
65C22×C22⋊C4C2×C412yes—
66C22×C42C2×C412yes—
67#446—44no—
68#436—44no—
69#432—24no—
70#431—22no—
71#431—22no—
72#407—24no—
73#391—21no—
74#390—21no—
75#390—22no—
76#390—22no—
77#389—21no—
78#389—21no—
79#388—22no—
80#388—22no—
81#388—22no—
82#388—22no—
83#388—22no—
84#388—22no—
85#387—22no—
86#387—22no—
87#386—22no—
88#386—22no—
89#386C2×C412yes—
90#386C2×C412yes—
91#385C2×C412yes—
92#385C2312yes—
93#384—21no—
94#382—21no—
95#381C2×C411yescharacteristic
96#381C2311yescharacteristic
97#380—22no—
98#380—21no—
99#380—21no—
100#379C2×C411yescharacteristic
101#379C2311yescharacteristic
102C4×C4⋊C4—22no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
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×C411yescharacteristic
120C2×#41—22no—
121C2×#41—22no—
122C2×#41—42no—
123C2×#41—42no—
124C22×Q8C22⋊C411yescharacteristic
125C22×D4—22no—
126C22×D4C22⋊C411yescharacteristic
127C22×D4C22×C412yes—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C412yes—
135C23×C4M1614yescomplemented
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C412yes—
139C23×C4C22×C412yes—
140C23×C4C2×D412yes—
141C2×M16—44no—
142#123—24no—
143#123—24no—
144C22⋊Q8—28no—
145#121—28no—
146#120—24no—
147#120—24no—
148C4×Q8—24no—
149C4×Q8—42no—
150C4×D4—216no—
151C4×D4—24no—
152C4×D4—42no—
153C4×D4—44no—
154C4×D4—44no—
155#117—22no—
156#117—42no—
157#117—44no—
158#117—44no—
159#117—48no—
160#117—44no—
161#117—42no—
162#117M1614yescomplemented
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C2×D411yescharacteristic
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4C22⋊C412yes—
192C2×C22⋊C4C22×C412yes—
193C2×C22⋊C4C22×C411yescharacteristic
194C2×C22⋊C4C22×C411yescharacteristic
195C2×C22⋊C4C22×C411yescharacteristic
196C2×C22⋊C4C2×D412yes—
197C2×C22⋊C4C2411yescharacteristic
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—44no—
205C2×C42C22⋊C412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C4⋊C8—42no—
210C4⋊C8—42no—
211C4⋊C8—44no—
212C22⋊C8—24no—
213C22⋊C8—24no—
214C22⋊C8—44no—
215#97—44no—
216C4×C8—42no—
217C4×C8—42no—
218#95—21no—
219#95—21no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—21no—
226#95—21no—
227#95—22no—
228C24—21no—
229C24—21no—
230C24—41no—
231C24—41no—
232C24—81no—
233C24—81no—
234C24C2×C22⋊C412yes—
235C24C2×M1612yes—
236C24C23×C411yescharacteristic
237#41—48no—
238#41—48no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—28no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—82no—
251C2×D4—82no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—48no—
271C22×C4—48no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×M1612yes—
289C22×C4C2×M1612yes—
290C22×C4C2×M1612yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293M16—164no—
294C2×C8—44no—
295C2×C8—44no—
296C4⋊C4—28no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4#9914yes—
310C22⋊C4—28no—
311C22⋊C4—28no—
312C22⋊C4—22no—
313C22⋊C4—24no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—81no—
322C22⋊C4—81no—
323C22⋊C4—81no—
324C22⋊C4—81no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C22⋊C4C2×M1614yes—
328C42—24no—
329C42—22no—
330C42—22no—
331C42—24no—
332C42—42no—
333C42—44no—
334C42—44no—
335C42—42no—
336C42—82no—
337C42—82no—
338C23—21no—
339C23—21no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23—82no—
357C23—82no—
358C23—82no—
359C23—82no—
360C23—84no—
361C23C22×C22⋊C411yescharacteristic
362C23C22×M1611yescharacteristic
363C23C22×M1611yescharacteristic
364Q8—44no—
365D4—44no—
366D4—44no—
367D4—44no—
368D4—88no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—42no—
384C2×C4—48no—
385C2×C4—48no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—48no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4—162no—
411C2×C4#40712yes—
412C2×C4#40712yes—
413C2×C4C2×#9914yes—
414C2×C4C2×#9912yes—
415C2×C4#41312yes—
416C8—84no—
417C8—84no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—41no—
423C22—41no—
424C22—42no—
425C22—42no—
426C22—44no—
427C22—42no—
428C22—41no—
429C22—41no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—82no—
435C22—81no—
436C22—81no—
437C22—84no—
438C22—84no—
439C22—81no—
440C22—81no—
441C22C2×#40711yescharacteristic
442C22C22×#9911yescharacteristic
443C22C2×#41311yescharacteristic
444C4—24no—
445C4—22no—
446C4—44no—
447C4—41no—
448C4—42no—
449C4—41no—
450C4—41no—
451C4—41no—
452C4—42no—
453C4—42no—
454C4—81no—
455C4—81no—
456C4—82no—
457C4#146514yes—
458C2—21no—
459C2—21no—
460C2—41no—
461C2—41no—
462C2—82no—
463C2—82no—
464C2C2×#146511yescharacteristic
465C2#3335011yescharacteristic
466C2#3335111yescharacteristic
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. #17370C2semidirect
4. #11527C2semidirect
5. #11526C2semidirect
6. #11454C2semidirect
7. #11425C2semidirect
135. C23×C4M16semidirect
162. #117M16semidirect

Maximal subgroups

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

subgroupindex
#342012
#173722
#173702
#115272
#115262
#114542
#114252