Finite Group Databasebeta

← #493750

Subgroups — #493750

Order 512.

1315 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#535C211yesmaximal, characteristic
2C2×#2134C212yesmaximal
3#20637C212yesmaximal, complemented
4#12803C214yesmaximal, complemented
5#12801C214yesmaximal, complemented
6#12587C212yesmaximal, complemented
7C2×#535C2211yescharacteristic
8C2×C4×#41C2211yescharacteristic
9C22×C4×D4C411yescharacteristic
10C2×#423—22no—
11C2×#413C2214yes—
12#2328C412yes—
13#2326C2212yes—
14#2324—22no—
15#2321—22no—
16#2317C411yescharacteristic
17#2311C2211yescharacteristic
18#2309C411yescharacteristic
19C4×#127C412yes—
20C4×#124C2212yes—
21C4×#121—22no—
22C4×#120—22no—
23C4×#117C411yescharacteristic
24#2134—28no—
25C2×#355C2218yes—
26#1801C2214yescomplemented
27#1801C2214yescomplemented
28#1800—22no—
29#1800—22no—
30#1742—22no—
31#1394C2218yescomplemented
32C22×#41C2311yescharacteristic
33C23×D4C2×C411yescharacteristic
34C24×C4—21no—
35C22×M16—24no—
36#535—24no—
37#535D414yes—
38#535Q814yes—
39C2×#127C2×C412yes—
40C2×#124C2312yes—
41C2×#121—22no—
42C2×#120—22no—
43C4×#41—42no—
44C4×#41—42no—
45C2×C4×Q8C2311yescharacteristic
46C2×C4×Q8C2311yescharacteristic
47C2×C4×D4—24no—
48C2×C4×D4—21no—
49C2×C4×D4—21no—
50C2×C4×D4—42no—
51C2×C4×D4—42no—
52C2×C4×D4C2×C412yes—
53C2×C4×D4C2×C412yes—
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4C2×C411yescharacteristic
56C2×C4×D4D412yes—
57C2×C4×D4D412yes—
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#423—24no—
67#423—24no—
68#423—42no—
69#423—42no—
70C2×C4⋊C8—42no—
71C2×C4⋊C8—42no—
72#413—28no—
73#413—28no—
74C2×#101C2318yes—
75C2×#100—24no—
76#391C2312yes—
77#390—24no—
78#390—22no—
79#390C2×C412yes—
80#390C2×C412yes—
81#390C2×C412yes—
82#389—22no—
83#388—24no—
84#388—24no—
85#386—22no—
86#386—24no—
87#386—24no—
88#385—24no—
89#385—22no—
90#384C2×C412yes—
91#382C2×C412yes—
92#380—22no—
93#379—22no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4—24no—
99#376C2×C412yes—
100C43C2×C412yes—
101#355—216no—
102#343—24no—
103#343—24no—
104C25—21no—
105C2×#41—42no—
106C2×#41—42no—
107C2×#41C2×D412yes—
108C2×#41C2×Q812yes—
109C2×#41#4114yes—
110C22×Q8C2411yescharacteristic
111C22×D4—24no—
112C22×D4—21no—
113C22×D4—22no—
114C22×D4—42no—
115C22×D4—42no—
116C22×D4C4⋊C414yes—
117C22×D4C22×C412yes—
118C22×D4C22×C411yescharacteristic
119C22×D4C2×D412yes—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—84no—
126C23×C4C22⋊C412yes—
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22⋊C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C2×D412yes—
132C2×M16—28no—
133C2×M16—44no—
134C2×M16—48no—
135C2×M16—48no—
136C22×C8—44no—
137#127—24no—
138#127C4⋊C418yes—
139#124—24no—
140#124C2×D414yes—
141#124C2×Q814yes—
142#121—28no—
143#121—28no—
144#120—28no—
145#120—28no—
146C4×Q8—42no—
147C4×Q8—41no—
148C4×Q8—41no—
149C4×D4—41no—
150C4×D4—42no—
151C4×D4—41no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—42no—
155C4×D4—42no—
156C4×D4—44no—
157C4×D4—42no—
158C4×D4—42no—
159C4×D4—44no—
160C4×D4—42no—
161C4×D4—42no—
162C4×D4—88no—
163#117—22no—
164#117—42no—
165#117—42no—
166#117—42no—
167#117—42no—
168#117C4⋊C414yes—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C4⋊C4C22×C411yescharacteristic
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D412yes—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C22⋊C412yes—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42C22⋊C412yes—
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C412yes—
205C2×C42C22×C412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C4⋊C8—44no—
209C4⋊C8—44no—
210C4⋊C8—44no—
211C4⋊C8—44no—
212#101—28no—
213#101—28no—
214#100—48no—
215#95—22no—
216#95—22no—
217#95—24no—
218#95—24no—
219C24—22no—
220C24—24no—
221C24—42no—
222C24—42no—
223C24—82no—
224C24—82no—
225C24C2×C22⋊C411yescharacteristic
226C24C2×C22⋊C412yes—
227#41—44no—
228#41—44no—
229#41—44no—
230#41—44no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8C22×D411yescharacteristic
234C2×Q8C22×Q811yescharacteristic
235C2×Q8C2×#4112yes—
236C2×D4—216no—
237C2×D4—22no—
238C2×D4—24no—
239C2×D4—41no—
240C2×D4—42no—
241C2×D4—41no—
242C2×D4—42no—
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×D4C2×C4⋊C412yes—
253C2×D4#11712yes—
254C2×D4#12114yes—
255C2×D4C22⋊Q814yes—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—48no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—82no—
279C22×C4—84no—
280C22×C4—82no—
281C22×C4—81no—
282C22×C4—88no—
283C22×C4—88no—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C4⋊C412yes—
292C22×C4#11712yes—
293C22×C4#12114yes—
294C22×C4C22⋊Q814yes—
295C22×C4C23×C411yescharacteristic
296C22×C4C22×D412yes—
297M16—48no—
298M16—48no—
299M16—88no—
300C2×C8—48no—
301C2×C8—88no—
302C4⋊C4—24no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—84no—
311C22⋊C4—24no—
312C22⋊C4—28no—
313C22⋊C4—28no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—44no—
318C22⋊C4—42no—
319C22⋊C4—44no—
320C22⋊C4—84no—
321C22⋊C4—84no—
322C22⋊C4—84no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—21no—
332C42—21no—
333C42—22no—
334C42—42no—
335C42—44no—
336C42—82no—
337C42#9912yes—
338C42#9912yes—
339C42C2×C4⋊C414yes—
340C23—21no—
341C23—22no—
342C23—21no—
343C23—22no—
344C23—24no—
345C23—41no—
346C23—41no—
347C23—42no—
348C23—42no—
349C23—44no—
350C23—44no—
351C23—48no—
352C23—42no—
353C23—42no—
354C23—88no—
355C23—84no—
356C23—88no—
357C23—84no—
358C23#38012yes—
359C23#38014yes—
360C23C22×C22⋊C411yescharacteristic
361Q8—41no—
362Q8—41no—
363Q8—42no—
364D4—42no—
365D4—42no—
366D4—44no—
367D4—44no—
368D4—44no—
369D4—44no—
370D4—48no—
371D4—816no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—84no—
408C2×C4—88no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4#38012yes—
412C2×C4C2×#9912yes—
413C2×C4C2×#9911yescharacteristic
414C2×C4C2×#9912yes—
415C2×C4C2×#9911yescharacteristic
416C2×C4#41012yes—
417C2×C4C22×C4⋊C411yescharacteristic
418C2×C4C2×#11711yescharacteristic
419C2×C4C2×#12112yes—
420C2×C4C2×C22⋊Q812yes—
421C8—88no—
422C22—21no—
423C22—21no—
424C22—22no—
425C22—24no—
426C22—44no—
427C22—42no—
428C22—42no—
429C22—48no—
430C22—41no—
431C22—42no—
432C22—42no—
433C22—41no—
434C22—44no—
435C22—82no—
436C22—84no—
437C22—816no—
438C22—82no—
439C22—82no—
440C22C2×#38011yescharacteristic
441C22C22×#9911yescharacteristic
442C22C2×#41011yescharacteristic
443C4—21no—
444C4—22no—
445C4—22no—
446C4—21no—
447C4—21no—
448C4—22no—
449C4—44no—
450C4—41no—
451C4—42no—
452C4—42no—
453C4—41no—
454C4—82no—
455C4—82no—
456C4#179914yes—
457C4#213412yescomplemented
458C2—21no—
459C2—21no—
460C2—42no—
461C2—44no—
462C2—84no—
463C2C2×#179911yescharacteristic
464C2C2×#213411yescharacteristic
465C2#3336911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20637C2semidirect
4. #12803C2semidirect
5. #12801C2semidirect
6. #12587C2semidirect
26. #1801C22semidirect
27. #1801C22semidirect
31. #1394C22semidirect
457. C4#2134semidirect

Maximal subgroups

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

subgroupindex
C4×#5352
C2×#21342
#206372
#128032
#128012
#125872