Finite Group Databasebeta

← #497310

Subgroups — #497310

Order 512.

1183 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39360C211yesmaximal, characteristic
2C2×#2129C212yesmaximal, complemented
3#22406C212yesmaximal
4#13224C212yesmaximal, complemented
5#13091C214yesmaximal, complemented
6#12798C211yesmaximal, characteristic, complemented
7#12698C212yesmaximal, complemented
8C4×#354C211yesmaximal, characteristic, complemented
9C2×D4×Q8C411yescharacteristic, complemented
10C2×#547C2211yescharacteristic
11C2×#536C2212yes—
12C2×#410C2214yes—
13#2868C411yescharacteristic, complemented
14#2680—24no—
15#2621C412yescomplemented
16#2547C2211yescharacteristic
17#2546C411yescharacteristic, complemented
18#2440—22no—
19#2434C412yescomplemented
20#2432C2212yes—
21#2422—22no—
22#2344—22no—
23C4×#124C2211yescharacteristic
24#2293C411yescharacteristic, complemented
25#2129—28no—
26C2×#354C2214yes—
27C2×#354C2212yes—
28C2×#354C2211yescharacteristic
29C2×#354C2214yes—
30C2×#354C2211yescharacteristic
31#1964—22no—
32#1899—24no—
33#1799C2212yes—
34#1782C2214yes—
35C4×#99C2212yes—
36#1393C2212yes—
37#1393C2212yes—
38C22×#41C2312yes—
39C23×Q8C2×C412yes—
40D4×Q8—42no—
41D4×Q8—42no—
42#547—42no—
43#547—42no—
44#536—24no—
45#536—24no—
46#536—44no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8C2×C412yes—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×C4⋊Q8C2×C412yes—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#124C2312yes—
54C2×#124C2311yescharacteristic
55C2×#123—24no—
56C2×#123D414yes—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8C2×C412yes—
60C2×C22⋊Q8C2×C412yes—
61C2×#121D412yes—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4C2311yescharacteristic
66C2×#117—24no—
67C2×#117C2×C412yes—
68C22×C4⋊C4D412yes—
69#410—28no—
70#410—44no—
71C2×#99—24no—
72C2×#99—24no—
73C2×#99C2314yes—
74C2×#99C2312yes—
75C2×#99C2312yes—
76#398—24no—
77#398—22no—
78#397—24no—
79#397—22no—
80#395—24no—
81#394—24no—
82#391C2×C412yes—
83#391C2×C412yes—
84#391C2×C411yescharacteristic
85#390—21no—
86#390C2311yescharacteristic
87#389—22no—
88#389—21no—
89#389—42no—
90#388—22no—
91#388—22no—
92#387—44no—
93#385—24no—
94#385—24no—
95#384—22no—
96#384—22no—
97#382C2×C411yescharacteristic
98#381—24no—
99#380—22no—
100#380—22no—
101C4×C4⋊C4—42no—
102C4×C4⋊C4C2×C411yescharacteristic
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4C2×C412yes—
106C43C2×C411yescharacteristic
107#354—24no—
108#354—22no—
109#354—28no—
110#354—44no—
111#354D414yescomplemented
112#342—22no—
113#342—22no—
114C2×#41—22no—
115C2×#41—22no—
116C2×#41—24no—
117C2×#41—44no—
118C2×#41—82no—
119C22×Q8—22no—
120C22×Q8—24no—
121C22×Q8—42no—
122C22×Q8—44no—
123C22×Q8—82no—
124C22×Q8C22×C412yes—
125C22×Q8C22×C411yescharacteristic
126C22×Q8C22×C412yes—
127C22×D4C22⋊C411yescharacteristic
128C22×D4C2×D414yes—
129C22×D4C2411yescharacteristic
130C23×C4—22no—
131C23×C4C22×C412yes—
132C23×C4C2×D414yes—
133C23×C4C2×D412yes—
134C4⋊Q8—24no—
135C4⋊Q8—24no—
136C4⋊Q8—41no—
137C4⋊Q8—41no—
138C4⋊Q8—42no—
139C4⋊Q8—42no—
140C4⋊Q8—41no—
141C4⋊Q8—41no—
142C4⋊Q8—42no—
143C4⋊Q8—42no—
144C4⋊Q8—82no—
145#124—22no—
146#124—22no—
147#124—22no—
148#124—24no—
149#124C2×D412yes—
150#124#4112yes—
151#123—44no—
152#123—44no—
153#123—48no—
154#123—48no—
155C22⋊Q8—48no—
156C22⋊Q8—48no—
157C22⋊Q8—48no—
158C22⋊Q8—44no—
159C22⋊Q8—44no—
160#121—42no—
161#121—42no—
162C4×Q8—41no—
163C4×Q8—41no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—41no—
167C4×D4—41no—
168C4×D4—42no—
169C4×D4—42no—
170#117—24no—
171#117—42no—
172#117—48no—
173#117—84no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C414yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22⋊C414yes—
205C2×C22⋊C4C2×D412yes—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—44no—
210C2×C42—42no—
211C2×C42C22×C412yes—
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C411yescharacteristic
215#99—22no—
216#99—24no—
217#99—22no—
218#99—22no—
219#99—24no—
220#99—48no—
221#99—84no—
222#99C2×D414yes—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—24no—
228#95—24no—
229#95—42no—
230#95—44no—
231#95—42no—
232#95—44no—
233C24C2×C22⋊C412yes—
234C24C22×D412yes—
235#41—84no—
236#41—84no—
237#41—84no—
238#41—84no—
239C2×Q8—22no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—44no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—44no—
252C2×Q8—82no—
253C2×Q8—88no—
254C2×Q8—88no—
255C2×Q8C4×D412yes—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—42no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4C22×D411yescharacteristic
266C2×D4C2×#4111yescharacteristic
267C2×D4H(5,𝔽2)12yes—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—82no—
292C22×C4C2×C22⋊C414yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C23×C411yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—44no—
305C4⋊C4—48no—
306C4⋊C4—48no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—44no—
314C4⋊C4—84no—
315C4⋊C4—88no—
316C4⋊C4—84no—
317C4⋊C4#9912yes—
318C4⋊C4#9912yes—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—44no—
324C22⋊C4—42no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—42no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—42no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—42no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—22no—
340C42—22no—
341C42—41no—
342C42—42no—
343C42—42no—
344C42—41no—
345C42—41no—
346C42—88no—
347C42—84no—
348C42C4×D412yes—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23C22×C22⋊C411yescharacteristic
359C23D4212yes—
360C23#54612yes—
361Q8—42no—
362Q8—42no—
363Q8—82no—
364Q8—82no—
365Q8—88no—
366Q8—82no—
367Q8—82no—
368D4—41no—
369D4—41no—
370D4—42no—
371D4—42no—
372D4—42no—
373D4—84no—
374D4—84no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—44no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—82no—
413C2×C4—88no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—84no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—84no—
420C2×C4—82no—
421C2×C4—84no—
422C2×C4C2×#9911yescharacteristic
423C2×C4C2×#9914yes—
424C2×C4C2×#9911yescharacteristic
425C2×C4C2×C4×D411yescharacteristic
426C2×C4#51811yescharacteristic
427C22—21no—
428C22—21no—
429C22—22no—
430C22—22no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—82no—
439C22—82no—
440C22—82no—
441C22D4×C22⋊C411yescharacteristic
442C22C22×#9911yescharacteristic
443C22#288311yescharacteristic
444C4—21no—
445C4—24no—
446C4—21no—
447C4—21no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—41no—
452C4—44no—
453C4—42no—
454C4—82no—
455C4—84no—
456C4—81no—
457C4—82no—
458C4—164no—
459C4—162no—
460C4#212912yescomplemented
461C2—21no—
462C2—21no—
463C2—42no—
464C2—42no—
465C2—82no—
466C2D4×#9911yescharacteristic
467C2C2×#212911yescharacteristic
468C2#3336011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#547 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2129C2semidirect
4. #13224C2semidirect
5. #13091C2semidirect
6. #12798C2semidirect
7. #12698C2semidirect
8. C4×#354C2semidirect
9. C2×D4×Q8C4semidirect
13. #2868C4semidirect
15. #2621C4semidirect
17. #2546C4semidirect
19. #2434C4semidirect
24. #2293C4semidirect
111. #354D4semidirect
460. C4#2129semidirect

Maximal subgroups

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

subgroupindex
#393602
C2×#21292
#224062
#132242
#130912
#127982
#126982
C4×#3542