Finite Group Databasebeta

← #493971

Subgroups — #493971

Order 512.

1079 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34276C211yesmaximal, characteristic
2#20725C212yesmaximal
3#12857C212yesmaximal
4#12800C212yesmaximal, complemented
5#12798C212yesmaximal, complemented
6#12698C212yesmaximal
7#12627C212yesmaximal, complemented
8C4×#354C212yesmaximal, complemented
9C2×#536C2211yescharacteristic
10C2×#520C2211yescharacteristic
11C2×#519C411yescharacteristic, complemented
12#2362—22no—
13#2360—22no—
14#2359—22no—
15#2359—22no—
16#2327C411yescharacteristic, complemented
17#2327C411yescharacteristic, complemented
18#2326C2211yescharacteristic
19#2326C2211yescharacteristic
20#2317C411yescharacteristic, complemented
21#2313—22no—
22C4×C4⋊Q8C411yescharacteristic, complemented
23C4×C4⋊Q8C411yescharacteristic, complemented
24C4×#124C2211yescharacteristic
25C4×#124C2211yescharacteristic
26#2297C2211yescharacteristic
27C2×#391C411yescharacteristic, complemented
28C4×#117C411yescharacteristic, complemented
29C2×#354C2214yes—
30C2×#354C2214yes—
31#1820—22no—
32#1799—22no—
33#1799C2214yes—
34#1782C2214yes—
35C4×#99—22no—
36C4×#99C2214yes—
37#1393C2214yes—
38#1393C2214yes—
39C22×#41C2311yescharacteristic
40C23×Q8C2×C411yescharacteristic
41#536—24no—
42#536—24no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#124C2311yescharacteristic
46C2×#124C2311yescharacteristic
47C2×#123—22no—
48C2×C22⋊Q8—22no—
49#520—44no—
50#519—44no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×D4D412yes—
60C2×C4×D4D412yes—
61C2×C4×D4C2311yescharacteristic
62C2×C4×D4C2311yescharacteristic
63C2×#117—21no—
64C2×#117—21no—
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C2×#117D412yes—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4D412yes—
71C22×C42C2×C411yescharacteristic
72C2×#99—24no—
73C2×#99C2318yes—
74#391—22no—
75#391C2×C411yescharacteristic
76#391C2×C411yescharacteristic
77#391C2×C411yescharacteristic
78#391C2×C411yescharacteristic
79#391C2×C411yescharacteristic
80#391C2×C411yescharacteristic
81#390C2311yescharacteristic
82#390C2311yescharacteristic
83#389—22no—
84#389—22no—
85#389—22no—
86#388—22no—
87#388—22no—
88#388—22no—
89#388—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—48no—
95#386—22no—
96#385—22no—
97#385—22no—
98#385—22no—
99#384C2×C411yescharacteristic
100#384C2×C411yescharacteristic
101#382C2×C411yescharacteristic
102#382C2×C411yescharacteristic
103#381—24no—
104#380—24no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—24no—
115#376C2×C411yescharacteristic
116#376C2×C411yescharacteristic
117C2×#95—22no—
118C43C2×C411yescharacteristic
119C43C2×C411yescharacteristic
120#354—28no—
121#354—28no—
122#342—24no—
123#342—24no—
124C2×#41—22no—
125C2×#41—22no—
126C2×#41—42no—
127C2×#41—42no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8C22×C411yescharacteristic
134C22×Q8C22×C411yescharacteristic
135C22×Q8C22×C411yescharacteristic
136C22×D4C2×D412yes—
137C22×D4C2411yescharacteristic
138C23×C4—21no—
139C23×C4—21no—
140C23×C4C22⋊C412yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149#124—22no—
150#124—22no—
151#124#4114yes—
152#124#4114yes—
153#123—216no—
154C22⋊Q8—216no—
155C4×Q8—42no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×Q8—42no—
160C4×Q8—42no—
161C4×Q8—44no—
162C4×Q8—44no—
163C4×D4—42no—
164C4×D4—42no—
165C4×D4—44no—
166C4×D4—44no—
167#117—24no—
168#117—22no—
169#117—22no—
170#117—44no—
171#117—44no—
172#117—44no—
173#117—42no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4C22⋊C411yescharacteristic
203C2×C22⋊C4C22⋊C411yescharacteristic
204C2×C22⋊C4C2×D412yes—
205C2×C22⋊C4C2×D412yes—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—42no—
217C2×C42—44no—
218C2×C42C22⋊C412yes—
219C2×C42C22⋊C412yes—
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226#99—28no—
227#99—28no—
228#99—48no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—44no—
242#95—44no—
243#95—48no—
244C24C2×C22⋊C411yescharacteristic
245C24C22×D412yes—
246#41—48no—
247#41—48no—
248C2×Q8—24no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—48no—
256C2×Q8—48no—
257C2×Q8—42no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8#11712yes—
264C2×Q8#11712yes—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—42no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4C2×#4112yes—
271C2×D4C2×#4112yes—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—48no—
296C22×C4—81no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C23×C411yescharacteristic
309C4⋊C4—24no—
310C4⋊C4—28no—
311C4⋊C4—28no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—164no—
327C4⋊C4#9914yes—
328C22⋊C4—24no—
329C22⋊C4—28no—
330C22⋊C4—28no—
331C22⋊C4—22no—
332C22⋊C4—22no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—44no—
336C22⋊C4—42no—
337C22⋊C4—48no—
338C22⋊C4—48no—
339C22⋊C4—82no—
340C42—21no—
341C42—21no—
342C42—22no—
343C42—22no—
344C42—42no—
345C42—42no—
346C42—41no—
347C42—41no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—42no—
355C42—42no—
356C42—81no—
357C42—81no—
358C42—164no—
359C42#11712yes—
360C42#11712yes—
361C23—21no—
362C23—22no—
363C23—21no—
364C23—22no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—42no—
369C23#41012yes—
370C23C22×C22⋊C411yescharacteristic
371C23#52514yes—
372Q8—42no—
373Q8—42no—
374Q8—48no—
375Q8—42no—
376Q8—42no—
377Q8—42no—
378Q8—42no—
379D4—42no—
380D4—42no—
381D4—48no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—48no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—48no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—84no—
426C2×C4—84no—
427C2×C4—84no—
428C2×C4—88no—
429C2×C4—88no—
430C2×C4—88no—
431C2×C4—162no—
432C2×C4C2×#9912yes—
433C2×C4C2×#9912yes—
434C2×C4C2×#9912yes—
435C2×C4C2×#11711yescharacteristic
436C2×C4C2×#11711yescharacteristic
437C22—21no—
438C22—21no—
439C22—22no—
440C22—24no—
441C22—42no—
442C22—44no—
443C22—42no—
444C22—42no—
445C22—41no—
446C22—42no—
447C22—41no—
448C22—82no—
449C22#228011yescharacteristic
450C22C22×#9911yescharacteristic
451C22C2×#41011yescharacteristic
452C4—22no—
453C4—22no—
454C4—22no—
455C4—21no—
456C4—21no—
457C4—42no—
458C4—42no—
459C4—42no—
460C4—41no—
461C4—41no—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—81no—
468C4—81no—
469C4—81no—
470C4—168no—
471C2—21no—
472C2—21no—
473C2—44no—
474C2—42no—
475C2#2072211yescharacteristic
476C2#3336011yescharacteristic
477C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #12800C2semidirect
5. #12798C2semidirect
7. #12627C2semidirect
8. C4×#354C2semidirect
11. C2×#519C4semidirect
16. #2327C4semidirect
17. #2327C4semidirect
20. #2317C4semidirect
22. C4×C4⋊Q8C4semidirect
23. C4×C4⋊Q8C4semidirect
27. C2×#391C4semidirect
28. C4×#117C4semidirect

Maximal subgroups

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

subgroupindex
#342762
#207252
#128572
#128002
#127982
#126982
#126272
C4×#3542