Finite Group Databasebeta

← #493747

Subgroups — #493747

Order 512.

1355 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#535C211yesmaximal, characteristic
2C2×#2128C212yesmaximal
3#20632C212yesmaximal, complemented
4#12799C214yesmaximal, complemented
5#12797C214yesmaximal, complemented
6#12583C212yesmaximal, complemented
7C2×#535C2211yescharacteristic
8C2×C4×#41C2211yescharacteristic
9C22×C4×D4C411yescharacteristic, complemented
10C2×#410C2214yes—
11#2328C2212yes—
12#2326C412yescomplemented
13#2324—22no—
14#2321—22no—
15#2317C411yescharacteristic, complemented
16#2311C2211yescharacteristic
17#2309C411yescharacteristic, complemented
18C4×#127C2212yes—
19C4×#124C412yescomplemented
20C4×#121—22no—
21C4×#120—22no—
22#2285—22no—
23C4×#117C411yescharacteristic, complemented
24#2128—28no—
25C2×#353C2218yes—
26#1799—22no—
27#1799—22no—
28#1799C2214yescomplemented
29#1799C2214yescomplemented
30#1740—22no—
31#1393C2218yescomplemented
32C22×#41C2311yescharacteristic
33C23×D4C2×C411yescharacteristic
34C24×C4—21no—
35#535—24no—
36#535D414yes—
37#535Q814yes—
38C2×#127C2312yes—
39C2×#124C2×C412yes—
40C2×#121—22no—
41C2×#120—22no—
42C4×#41—42no—
43C4×#41—42no—
44C2×C4×Q8C2×C411yescharacteristic
45C2×C4×Q8C2×C411yescharacteristic
46C2×C4×D4—24no—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×C4×D4—42no—
50C2×C4×D4—42no—
51C2×C4×D4C2×C412yes—
52C2×C4×D4C2×C412yes—
53C2×C4×D4D412yes—
54C2×C4×D4D412yes—
55C2×C4×D4C2311yescharacteristic
56C2×C4×D4C2311yescharacteristic
57C2×#117—24no—
58C2×#117C2×C411yescharacteristic
59C2×#117C2×C411yescharacteristic
60C2×#117C2×C411yescharacteristic
61C2×#117D412yes—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C22⋊C4—21no—
64C22×C42C2×C411yescharacteristic
65C22×C42D412yes—
66#410—28no—
67#410—28no—
68C2×#99—24no—
69C2×#99C2318yes—
70#391C2×C412yes—
71#390—24no—
72#390—22no—
73#390C2×C412yes—
74#390C2×C412yes—
75#390C2312yes—
76#389—22no—
77#388—24no—
78#388—24no—
79#386—22no—
80#386—24no—
81#386—24no—
82#385—24no—
83#385—22no—
84#384—42no—
85#384C2×C412yes—
86#382—42no—
87#382C2×C412yes—
88#380—22no—
89#380—24no—
90#380—24no—
91#379—22no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—42no—
95C4×C4⋊C4—42no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4—24no—
99#376C2×C412yes—
100C43C2×C412yes—
101#353—216no—
102#342—24no—
103#342—24no—
104C25—21no—
105C2×#41—42no—
106C2×#41—42no—
107C2×#41C2×D412yes—
108C2×#41C2×Q812yes—
109C2×#41#4114yes—
110C22×Q8C22×C411yescharacteristic
111C22×D4—24no—
112C22×D4—21no—
113C22×D4—22no—
114C22×D4—42no—
115C22×D4—42no—
116C22×D4C4⋊C414yescomplemented
117C22×D4C22×C412yes—
118C22×D4C2×D412yes—
119C22×D4C2411yescharacteristic
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—84no—
126C23×C4C22⋊C411yescharacteristic
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132#127—24no—
133#127C2×D414yes—
134#127C2×Q814yes—
135#124—24no—
136#124C4⋊C418yescomplemented
137#121—28no—
138#121—28no—
139#120—28no—
140#120—28no—
141C4×Q8—42no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×D4—42no—
145C4×D4—42no—
146C4×D4—42no—
147C4×D4—42no—
148C4×D4—41no—
149C4×D4—42no—
150C4×D4—41no—
151C4×D4—44no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—44no—
155C4×D4—42no—
156C4×D4—42no—
157C4×D4—88no—
158#117—22no—
159#117—42no—
160#117—42no—
161#117—42no—
162#117—42no—
163#117—48no—
164#117—48no—
165#117C4⋊C414yescomplemented
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C22⋊C4—21no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C2×D412yes—
191C2×C22⋊C4C2×D412yes—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—44no—
201C2×C42C22⋊C412yes—
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C412yes—
204C2×C42C22⋊C412yes—
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C412yes—
209C2×C42C22×C412yes—
210#99—28no—
211#99—28no—
212#99—48no—
213#95—22no—
214#95—22no—
215#95—24no—
216#95—24no—
217#95—44no—
218#95—44no—
219C24—22no—
220C24—24no—
221C24—42no—
222C24—42no—
223C24—82no—
224C24—82no—
225C24C2×C22⋊C411yescharacteristic
226C24C22×D412yes—
227#41—44no—
228#41—44no—
229#41—44no—
230#41—44no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8C2×C4⋊C412yes—
234C2×Q8#11712yes—
235C2×D4—28no—
236C2×D4—28no—
237C2×D4—22no—
238C2×D4—24no—
239C2×D4—42no—
240C2×D4—41no—
241C2×D4—42no—
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—48no—
246C2×D4—48no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—84no—
250C2×D4—84no—
251C2×D4C2×C4⋊C414yes—
252C2×D4#12114yes—
253C2×D4C22⋊Q814yes—
254C2×D4C22×D411yescharacteristic
255C2×D4C22×Q811yescharacteristic
256C2×D4C2×#4112yes—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—48no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—48no—
280C22×C4—82no—
281C22×C4—84no—
282C22×C4—82no—
283C22×C4—81no—
284C22×C4—88no—
285C22×C4—88no—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C4⋊C412yes—
295C22×C4#11712yes—
296C22×C4#12114yes—
297C22×C4C22⋊Q814yes—
298C22×C4C23×C411yescharacteristic
299C4⋊C4—24no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—88no—
308C4⋊C4—84no—
309C4⋊C4—88no—
310C4⋊C4—88no—
311C22⋊C4—24no—
312C22⋊C4—216no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—44no—
317C22⋊C4—42no—
318C22⋊C4—44no—
319C22⋊C4—48no—
320C22⋊C4—48no—
321C22⋊C4—84no—
322C22⋊C4—84no—
323C22⋊C4—84no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—21no—
333C42—21no—
334C42—22no—
335C42—42no—
336C42—44no—
337C42—88no—
338C42—82no—
339C42#9912yes—
340C42#9912yes—
341C42C2×C4⋊C414yes—
342C23—22no—
343C23—21no—
344C23—21no—
345C23—22no—
346C23—24no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—44no—
352C23—44no—
353C23—48no—
354C23—42no—
355C23—42no—
356C23—88no—
357C23—84no—
358C23—88no—
359C23—84no—
360C23#38012yes—
361C23C22×C22⋊C411yescharacteristic
362C23C2×#12112yes—
363C23C2×C22⋊Q812yes—
364Q8—42no—
365Q8—42no—
366D4—44no—
367D4—44no—
368D4—41no—
369D4—41no—
370D4—42no—
371D4—44no—
372D4—44no—
373D4—48no—
374D4—88no—
375D4—88no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—41no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—84no—
412C2×C4—88no—
413C2×C4—84no—
414C2×C4—84no—
415C2×C4—88no—
416C2×C4—88no—
417C2×C4—88no—
418C2×C4#38014yes—
419C2×C4#38012yes—
420C2×C4C2×#9912yes—
421C2×C4C2×#9911yescharacteristic
422C2×C4C2×#9912yes—
423C2×C4C2×#9911yescharacteristic
424C2×C4#41012yes—
425C2×C4C22×C4⋊C411yescharacteristic
426C2×C4C2×#11711yescharacteristic
427C22—21no—
428C22—21no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—48no—
437C22—41no—
438C22—42no—
439C22—42no—
440C22—41no—
441C22—44no—
442C22—82no—
443C22—84no—
444C22—816no—
445C22—82no—
446C22—82no—
447C22C2×#38011yescharacteristic
448C22C22×#9911yescharacteristic
449C22C2×#41011yescharacteristic
450C4—21no—
451C4—22no—
452C4—21no—
453C4—21no—
454C4—24no—
455C4—44no—
456C4—41no—
457C4—42no—
458C4—42no—
459C4—41no—
460C4—82no—
461C4—82no—
462C4—168no—
463C4#179914yes—
464C4#212812yescomplemented
465C2—21no—
466C2—21no—
467C2—42no—
468C2—42no—
469C2—42no—
470C2—84no—
471C2C2×#179911yescharacteristic
472C2C2×#212811yescharacteristic
473C2#3335811yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20632C2semidirect
4. #12799C2semidirect
5. #12797C2semidirect
6. #12583C2semidirect
9. C22×C4×D4C4semidirect
12. #2326C4semidirect
15. #2317C4semidirect
17. #2309C4semidirect
19. C4×#124C4semidirect
23. C4×#117C4semidirect
28. #1799C22semidirect
29. #1799C22semidirect
31. #1393C22semidirect
116. C22×D4C4⋊C4semidirect
136. #124C4⋊C4semidirect
165. #117C4⋊C4semidirect
464. C4#2128semidirect

Maximal subgroups

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

subgroupindex
C4×#5352
C2×#21282
#206322
#127992
#127972
#125832