Finite Group Databasebeta

← #480731

Subgroups — #480731

Order 512.

1350 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1350 conjugacy classes of subgroups — too many to draw. Only its 202 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2544C211yesmaximal, characteristic
2#17362C212yesmaximal, complemented
3#11566C218yesmaximal, complemented
4#11506C212yesmaximal, complemented
5#11505C212yesmaximal, complemented
6C22×#127C411yescharacteristic
7C22×#124C411yescharacteristic
8C22×#124C411yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C2×#407—22no—
11#2544—22no—
12#2544—22no—
13#2544C414yes—
14#2544C2214yes—
15C2×#394—22no—
16C2×#391C2211yescharacteristic
17C2×#390C2211yescharacteristic
18C2×#386—22no—
19C2×C4×C4⋊C4C411yescharacteristic
20#1465—22no—
21#1465C2214yescomplemented
22#1463C2214yes—
23#1462—22no—
24#1459—22no—
25#1459C2214yescomplemented
26#1319C2218yescomplemented
27#1318C2218yescomplemented
28C23×Q8C2×C411yescharacteristic
29C23×D4—22no—
30C23×D4C2×C411yescharacteristic
31C23×D4C2×C411yescharacteristic
32C24×C4D412yes—
33C22×M16—42no—
34C2×#127—21no—
35C2×#127—21no—
36C2×#127—44no—
37C2×#127C2×C414yes—
38C2×#124—21no—
39C2×#124—21no—
40C2×#124—22no—
41C2×#124—42no—
42C2×#124—41no—
43C2×#124—41no—
44C2×#124—44no—
45C2×#124C2×C414yes—
46C2×#124C2×C414yes—
47C2×C4×D4—42no—
48C2×C4×D4—41no—
49C2×C4×D4—41no—
50C2×C4×D4C2×C412yes—
51C2×C4×D4D412yes—
52C2×C4×D4D412yes—
53C2×C4×D4C2312yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4D412yes—
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64#407—24no—
65#407—24no—
66#407—42no—
67#407—42no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8—42no—
70#394—24no—
71#394—24no—
72#394—24no—
73#394—24no—
74#391—22no—
75#391C2×C412yes—
76#391C2312yes—
77#390—22no—
78#390C2×C412yes—
79#390C2312yes—
80#386—24no—
81#386—24no—
82#386—24no—
83#386—24no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4C2×C414yes—
86C2×#95—21no—
87C2×#95—22no—
88#324—24no—
89#324C2318yes—
90#323—24no—
91#323—24no—
92C25—21no—
93C25C22⋊C412yes—
94C22×Q8—21no—
95C22×Q8—42no—
96C22×Q8—42no—
97C22×Q8—41no—
98C22×Q8—41no—
99C22×Q8C22×C412yes—
100C22×D4—24no—
101C22×D4—24no—
102C22×D4—24no—
103C22×D4—21no—
104C22×D4—42no—
105C22×D4—42no—
106C22×D4—41no—
107C22×D4—41no—
108C22×D4—41no—
109C22×D4—41no—
110C22×D4—48no—
111C22×D4—42no—
112C22×D4—48no—
113C22×D4—84no—
114C22×D4C22⋊C411yescharacteristic
115C22×D4C22⋊C411yescharacteristic
116C22×D4C22×C412yes—
117C22×D4C22×C412yes—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—84no—
126C23×C4C22⋊C411yescharacteristic
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C2×M16—44no—
135C2×M16—44no—
136C2×M16—88no—
137C22×C8—44no—
138#127—48no—
139#127—44no—
140#127—44no—
141#124—44no—
142#124—44no—
143#124—44no—
144#124—44no—
145#124—44no—
146#124—44no—
147#124—48no—
148C4×D4—42no—
149C4×D4—42no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—42no—
155C4×D4—42no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4C22×C411yescharacteristic
161C2×C4⋊C4C2×D412yes—
162C2×C4⋊C4C2411yescharacteristic
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—48no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—84no—
178C2×C22⋊C4C22⋊C412yes—
179C2×C22⋊C4C22⋊C412yes—
180C2×C22⋊C4C2×D412yes—
181C2×C22⋊C4C2×D412yes—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42C22⋊C412yes—
190C2×C42C22×C412yes—
191C2×C42C22×C412yes—
192C2×C42C22×C412yes—
193C22⋊C8—28no—
194C22⋊C8—44no—
195C22⋊C8—44no—
196C22⋊C8—44no—
197#95—24no—
198#95—24no—
199#95—24no—
200C24—24no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24—42no—
205C24—44no—
206C24—44no—
207C24—44no—
208C24—42no—
209C24—42no—
210C24—44no—
211C24—41no—
212C24—41no—
213C24C2×C22⋊C411yescharacteristic
214C24C2×C22⋊C412yes—
215C24C2×C22⋊C412yes—
216C2×Q8—24no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—84no—
223C2×Q8—84no—
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—22no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—42no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—48no—
234C2×D4—48no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—48no—
240C2×D4—48no—
241C2×D4—48no—
242C2×D4—48no—
243C2×D4—42no—
244C2×D4—84no—
245C2×D4—84no—
246C2×D4—84no—
247C2×D4—84no—
248C2×D4—832no—
249C2×D4#10014yes—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—48no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—84no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—84no—
289C22×C4—84no—
290C22×C4#9912yes—
291C22×C4#9911yescharacteristic
292C22×C4#9911yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D412yes—
299M16—88no—
300M16—88no—
301C2×C8—48no—
302C2×C8—88no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—48no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—48no—
312C22⋊C4—48no—
313C22⋊C4—48no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—84no—
317C22⋊C4—84no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—42no—
323C42—42no—
324C42—42no—
325C42—42no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—44no—
336C23—44no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—416no—
342C23—44no—
343C23—44no—
344C23—44no—
345C23—44no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—82no—
350C23—82no—
351C23—88no—
352C23—88no—
353C23—82no—
354C23—82no—
355C23—88no—
356C23—82no—
357C23—82no—
358C23—82no—
359C23—82no—
360C23C2×#9911yescharacteristic
361C23C2×#9911yescharacteristic
362C23C2×#10014yes—
363C23C22×C22⋊C411yescharacteristic
364Q8—84no—
365Q8—84no—
366D4—44no—
367D4—48no—
368D4—44no—
369D4—84no—
370D4—84no—
371D4—816no—
372D4—816no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—48no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—84no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4C2×#9912yes—
415C2×C4C2×#9912yes—
416C2×C4C2×#10012yes—
417C2×C4#56812yes—
418C8—88no—
419C22—24no—
420C22—24no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—24no—
428C22—44no—
429C22—44no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—82no—
436C22—88no—
437C22—88no—
438C22—82no—
439C22—84no—
440C22—84no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22—164no—
445C22—164no—
446C22#212712yes—
447C22#213212yes—
448C22#287911yescharacteristic
449C22C22×#9911yescharacteristic
450C22C22×#10011yescharacteristic
451C4—22no—
452C4—22no—
453C4—22no—
454C4—22no—
455C4—44no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—82no—
460C4—82no—
461C2—21no—
462C2—21no—
463C2—22no—
464C2—22no—
465C2—44no—
466C2—82no—
467C2—162no—
468C2#1739611yescharacteristic
469C2C2×#212711yescharacteristic
470C2C2×#213211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#127C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17362C2semidirect
3. #11566C2semidirect
4. #11506C2semidirect
5. #11505C2semidirect
21. #1465C22semidirect
25. #1459C22semidirect
26. #1319C22semidirect
27. #1318C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25442
#173622
#115662
#115062
#115052