Finite Group Databasebeta

← #480324

Subgroups — #480324

Order 512.

1240 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2#17361C212yesmaximal, complemented
3#11526C214yesmaximal, complemented
4#11525C214yesmaximal, complemented
5#11452C212yesmaximal, complemented
6#11425C212yesmaximal, complemented
7C2×#525C411yescharacteristic
8C2×C4×#41C2211yescharacteristic
9C23×C42C411yescharacteristic
10#2920—22no—
11C4×C22⋊Q8—22no—
12C4×#123C412yes—
13C4×#123C2212yes—
14C4×#121—22no—
15C4×#120C412yes—
16C4×#120C2212yes—
17#2297C411yescharacteristic
18#2296C2211yescharacteristic
19#2290—22no—
20#2280C2211yescharacteristic
21C42×D4—22no—
22C4×#117C411yescharacteristic
23#1467—22no—
24#1466C2214yes—
25#1465—22no—
26#1465C2214yes—
27#1458—22no—
28#1458C2214yes—
29#1326C2218yes—
30#1317C2214yes—
31#1316C2214yes—
32C22×#41C2×C411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C24×C4C2×C412yes—
35#525—24no—
36#525—24no—
37C2×#123C2×C412yes—
38C2×C22⋊Q8—21no—
39C2×#121—21no—
40C2×#120C2×C412yes—
41C4×#41—42no—
42C4×#41—41no—
43C4×#41—41no—
44C2×C4×Q8D412yes—
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4—22no—
48C2×C4×D4C2×C412yes—
49C2×C4×D4D412yes—
50C2×C4×D4C2312yes—
51C2×#117C2×C411yescharacteristic
52C2×#117C2×C411yescharacteristic
53C2×#117D412yes—
54C2×#117C2311yescharacteristic
55C22×C42—21no—
56C22×C42—22no—
57C22×C42—21no—
58C22×C42—41no—
59C22×C42—41no—
60C22×C42—42no—
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C412yes—
64C22×C42D412yes—
65#432—24no—
66#432—42no—
67#431—24no—
68#431—42no—
69#423—42no—
70#407—24no—
71C4×M16—42no—
72#391—21no—
73#390—21no—
74#389—22no—
75#388—22no—
76#388—22no—
77#387—22no—
78#387—22no—
79#386—22no—
80#386—22no—
81#386C2×C412yes—
82#386C2×C412yes—
83#385—22no—
84#385—22no—
85#385C2×C412yes—
86#385C2312yes—
87#381C2×C411yescharacteristic
88#381C2311yescharacteristic
89#380—22no—
90#379C2×C411yescharacteristic
91#379C2311yescharacteristic
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4C2×C412yes—
103C4×C22⋊C4C2×C412yes—
104#376—21no—
105C43—21no—
106#324—24no—
107#324—24no—
108#323—24no—
109#323C2318yes—
110C25C22×C411yescharacteristic
111C2×#41—22no—
112C2×#41—22no—
113C2×#41—41no—
114C2×#41—41no—
115C2×#41—42no—
116C22×Q8C22⋊C411yescharacteristic
117C22×D4—22no—
118C22×D4C22⋊C411yescharacteristic
119C22×D4C22×C412yes—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—42no—
126C23×C4—48no—
127C23×C4—41no—
128C23×C4—41no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C412yes—
133C23×C4M1614yescomplemented
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C412yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C412yes—
138C23×C4C2×D412yes—
139C2×M16—44no—
140#123—24no—
141#123—24no—
142C22⋊Q8—28no—
143#121—28no—
144#120—24no—
145#120—24no—
146C4×Q8—42no—
147C4×Q8—42no—
148C4×D4—216no—
149C4×D4—42no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—42no—
155C4×D4—48no—
156C4×D4—48no—
157#117—22no—
158#117—42no—
159#117—42no—
160#117—41no—
161#117—41no—
162#117—42no—
163#117M1614yescomplemented
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4C2×D412yes—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4C22⋊C412yes—
179C2×C22⋊C4C22×C412yes—
180C2×C22⋊C4C22×C411yescharacteristic
181C2×C22⋊C4C22×C411yescharacteristic
182C2×C22⋊C4C22×C411yescharacteristic
183C2×C22⋊C4C2×D412yes—
184C2×C22⋊C4C2411yescharacteristic
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—42no—
201C2×C42—44no—
202C2×C42—44no—
203C2×C42—42no—
204C2×C42—41no—
205C2×C42—41no—
206C2×C42—42no—
207C2×C42—41no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—41no—
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C412yes—
213C2×C42C22⋊C412yes—
214C2×C42C22⋊C411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C4⋊C8—44no—
218C4⋊C8—44no—
219C22⋊C8—28no—
220C22⋊C8—44no—
221#97—44no—
222C4×C8—44no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231C24—21no—
232C24—21no—
233C24—41no—
234C24—44no—
235C24—41no—
236C24C2×C22⋊C412yes—
237C24C2×M1612yes—
238C24C23×C411yescharacteristic
239#41—44no—
240#41—44no—
241#41—44no—
242#41—44no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×D4—28no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—41no—
251C2×D4—42no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—44no—
255C2×D4—44no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—28no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—48no—
293C22×C4—48no—
294C22×C4—48no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—48no—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×M1612yes—
311C22×C4C2×M1612yes—
312C22×C4C2×M1612yes—
313C22×C4C22×D412yes—
314M16—88no—
315C2×C8—48no—
316C4⋊C4—28no—
317C4⋊C4—24no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C22⋊C4—28no—
325C22⋊C4—28no—
326C22⋊C4—22no—
327C22⋊C4—42no—
328C22⋊C4—41no—
329C22⋊C4—42no—
330C22⋊C4—41no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4C2×M1614yes—
338C42—24no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—21no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—21no—
349C42—42no—
350C42—44no—
351C42—44no—
352C42—44no—
353C42—42no—
354C42—42no—
355C42—41no—
356C42—41no—
357C42—42no—
358C42#9911yescharacteristic
359C42#9911yescharacteristic
360C42#9912yes—
361C23—24no—
362C23—22no—
363C23—21no—
364C23—21no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—42no—
369C23—42no—
370C23—44no—
371C23—42no—
372C23—42no—
373C23—44no—
374C23—42no—
375C23—42no—
376C23—44no—
377C23—44no—
378C23—48no—
379C23—41no—
380C23—41no—
381C23C22×C22⋊C411yescharacteristic
382C23C22×M1611yescharacteristic
383C23C22×M1611yescharacteristic
384Q8—44no—
385D4—44no—
386D4—416no—
387D4—44no—
388D4—44no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—48no—
406C2×C4—44no—
407C2×C4—48no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—416no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—48no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4#40712yes—
437C2×C4#40712yes—
438C2×C4C2×#9912yes—
439C2×C4C2×#9912yes—
440C2×C4C2×#9911yescharacteristic
441C2×C4C2×#9911yescharacteristic
442C2×C4#41312yes—
443C8—88no—
444C22—22no—
445C22—24no—
446C22—21no—
447C22—21no—
448C22—41no—
449C22—41no—
450C22—42no—
451C22—44no—
452C22—42no—
453C22—44no—
454C22—44no—
455C22—44no—
456C22—48no—
457C22—44no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—82no—
464C22C2×#40711yescharacteristic
465C22C22×#9911yescharacteristic
466C22C2×#41311yescharacteristic
467C4—22no—
468C4—22no—
469C4—44no—
470C4—42no—
471C4—44no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—42no—
476C4—81no—
477C4—81no—
478C4#146512yes—
479C4#146512yes—
480C4#212014yes—
481C2—21no—
482C2—21no—
483C2—44no—
484C2—41no—
485C2—41no—
486C2—82no—
487C2C2×#146511yescharacteristic
488C2C2×#212012yes—
subgroupG/H
centerC2×C4C2×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
socleC22C22×#99
prime radicalC2×#525C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17361C2semidirect
3. #11526C2semidirect
4. #11525C2semidirect
5. #11452C2semidirect
6. #11425C2semidirect
133. C23×C4M16semidirect
163. #117M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
#173612
#115262
#115252
#114522
#114252