Finite Group Databasebeta

← #480133

Subgroups — #480133

Order 512.

1188 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#34200C211yesmaximal, characteristic
2#17353C211yesmaximal, characteristic, complemented
3#17351C211yesmaximal, characteristic, complemented
4#11474C214yesmaximal, complemented
5#11471C214yesmaximal, complemented
6#11454C212yesmaximal, complemented
7#11425C212yesmaximal, complemented
8C2×#525C411yescharacteristic
9C2×#520C2211yescharacteristic
10C23×C4⋊C4C411yescharacteristic
11#2982—21no—
12#2972—21no—
13#2361—21no—
14#2361—21no—
15#2360—21no—
16#2360—21no—
17#2352—22no—
18D4×C4⋊C4—22no—
19#2322C412yes—
20#2322C2212yes—
21#2317C2211yescharacteristic
22#2311C411yescharacteristic
23C4×#120C412yes—
24C4×#120C2212yes—
25#2280C2211yescharacteristic
26#2274C411yescharacteristic
27#1461—21no—
28#1460—21no—
29#1460C2214yes—
30#1459C2214yes—
31#1458—22no—
32#1458C2212yes—
33#1458C2212yes—
34C2×#324—22no—
35#1326C2218yes—
36#1317C2214yes—
37#1316C2214yes—
38C22×#41C2×C411yescharacteristic
39C24×C4C2×C411yescharacteristic
40C24×C4C2×C412yes—
41#525—24no—
42#525C818yescomplemented
43C2×#123C2×C412yes—
44C2×C22⋊Q8—21no—
45C2×#121—21no—
46C2×#120C2×C412yes—
47#520—44no—
48C2×C4×Q8D411yescharacteristic
49C2×C4×Q8D411yescharacteristic
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4C2×C412yes—
54C2×C4×D4D411yescharacteristic
55C2×C4×D4D411yescharacteristic
56C2×C4×D4C2312yes—
57C2×#117C2×C411yescharacteristic
58C2×#117C2×C411yescharacteristic
59C2×#117D412yes—
60C2×#117C2311yescharacteristic
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—44no—
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4D412yes—
69C8×Q8—44no—
70C8×D4—44no—
71#431—22no—
72#431—24no—
73#431—22no—
74C2×C22⋊C8—24no—
75#391—21no—
76#390—21no—
77#389—22no—
78#389—22no—
79#389—21no—
80#389—21no—
81#388—22no—
82#388—22no—
83#387—22no—
84#387—22no—
85#386—22no—
86#386—22no—
87#386C2×C412yes—
88#386C2×C412yes—
89#385—22no—
90#385—22no—
91#385—22no—
92#385—22no—
93#385C2×C412yes—
94#385C2312yes—
95#384—21no—
96#382—21no—
97#381C2×C411yescharacteristic
98#381C2311yescharacteristic
99#380—22no—
100#380—21no—
101#380—21no—
102#379C2×C411yescharacteristic
103#379C2311yescharacteristic
104C4×C4⋊C4—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
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—24no—
115#324—22no—
116#323—24no—
117#323C2314yes—
118#323C2314yes—
119C25C22×C411yescharacteristic
120C2×#41—42no—
121C2×#41—42no—
122C2×#41C2×C814yes—
123C2×#41M1614yes—
124C22×Q8C22⋊C411yescharacteristic
125C22×D4—22no—
126C22×D4C22⋊C411yescharacteristic
127C22×D4C22×C412yes—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—44no—
136C23×C4—84no—
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C412yes—
139C23×C4C2×C814yescomplemented
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C412yes—
143C23×C4C22×C412yes—
144C23×C4C2×D412yes—
145C22×C8—44no—
146#123—24no—
147#123C2×C818yescomplemented
148C22⋊Q8—28no—
149#121—28no—
150#120—24no—
151#120C2×C818yescomplemented
152C4×Q8—24no—
153C4×Q8—42no—
154C4×D4—216no—
155C4×D4—24no—
156C4×D4—42no—
157C4×D4—44no—
158C4×D4—44no—
159C4×D4—84no—
160C4×D4—84no—
161#117—22no—
162#117—42no—
163#117—44no—
164#117—42no—
165#117C2×C814yescomplemented
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—48no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22×C412yes—
199C2×C22⋊C4C22×C411yescharacteristic
200C2×C22⋊C4C22×C411yescharacteristic
201C2×C22⋊C4C22×C411yescharacteristic
202C2×C22⋊C4C2×D412yes—
203C2×C22⋊C4C2411yescharacteristic
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42C22⋊C412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C4⋊C8—42no—
213C4⋊C8—44no—
214C4⋊C8—42no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C22⋊C8—44no—
218C4×C8—42no—
219C4×C8—44no—
220C4×C8—42no—
221#95—21no—
222#95—21no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95—21no—
230#95—22no—
231C24—21no—
232C24—21no—
233C24—41no—
234C24—41no—
235C24—42no—
236C24—42no—
237C24C2×C22⋊C412yes—
238C24C22×C812yes—
239C24C23×C411yescharacteristic
240#41—48no—
241#41—48no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8C22⋊C814yes—
245C2×D4—28no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4C22⋊C814yes—
253C2×D4C22×C814yes—
254C2×D4C2×M1614yes—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—48no—
273C22×C4—48no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—88no—
290C22×C4—88no—
291C22×C4C22⋊C818yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C22×C812yes—
300C22×C4C22×C812yes—
301C22×C4C2×M1612yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C2×C8—44no—
305C2×C8—44no—
306C2×C8—164no—
307C4⋊C4—24no—
308C4⋊C4—28no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—24no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4#9914yes—
323C22⋊C4—22no—
324C22⋊C4—28no—
325C22⋊C4—28no—
326C22⋊C4—24no—
327C22⋊C4—41no—
328C22⋊C4—42no—
329C22⋊C4—41no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4C22×C814yes—
339C42—24no—
340C42—22no—
341C42—24no—
342C42—22no—
343C42—42no—
344C42—42no—
345C42—82no—
346C42—82no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—21no—
352C23—21no—
353C23—22no—
354C23—42no—
355C23—41no—
356C23—44no—
357C23—41no—
358C23—42no—
359C23—41no—
360C23—41no—
361C23—44no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—48no—
368C23—42no—
369C23—41no—
370C23—41no—
371C23C2×C22⋊C814yes—
372C23C22×C22⋊C411yescharacteristic
373C23C23×C811yescharacteristic
374C23C22×M1611yescharacteristic
375Q8—28no—
376D4—28no—
377D4—44no—
378D4—44no—
379D4—416no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—42no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—48no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—84no—
418C2×C4—84no—
419C2×C4—88no—
420C2×C4—88no—
421C2×C4—84no—
422C2×C4—162no—
423C2×C4C2×C22⋊C812yes—
424C2×C4C2×C22⋊C814yes—
425C2×C4C2×C22⋊C812yes—
426C2×C4C2×#9914yes—
427C2×C4C2×#9912yes—
428C2×C4#41312yes—
429C8—84no—
430C8—84no—
431C22—21no—
432C22—21no—
433C22—24no—
434C22—21no—
435C22—21no—
436C22—24no—
437C22—44no—
438C22—41no—
439C22—41no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—44no—
445C22—44no—
446C22—44no—
447C22—44no—
448C22—41no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—41no—
454C22—42no—
455C22—82no—
456C22C22×C22⋊C811yescharacteristic
457C22C22×#9911yescharacteristic
458C22C2×#41311yescharacteristic
459C4—24no—
460C4—22no—
461C4—24no—
462C4—44no—
463C4—41no—
464C4—42no—
465C4—41no—
466C4—41no—
467C4—41no—
468C4—42no—
469C4—81no—
470C4—81no—
471C4—82no—
472C4#145914yes—
473C2—21no—
474C2—21no—
475C2—44no—
476C2—41no—
477C2—41no—
478C2—44no—
479C2C2×#145911yescharacteristic
480C2#3334811yescharacteristic
481C2#3334911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC23C23×C8
Frattini, cosocleC2×C22⋊C4C24
prime radicalC2×#525C4
LCSG ▷ C23 ▷ 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. #17353C2semidirect
3. #17351C2semidirect
4. #11474C2semidirect
5. #11471C2semidirect
6. #11454C2semidirect
7. #11425C2semidirect
42. #525C8semidirect
139. C23×C4C2×C8semidirect
147. #123C2×C8semidirect
151. #120C2×C8semidirect
165. #117C2×C8semidirect

Maximal subgroups

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

subgroupindex
#342002
#173532
#173512
#114742
#114712
#114542
#114252