Finite Group Databasebeta

← #493978

Subgroups — #493978

Order 512.

1084 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34211C211yesmaximal, characteristic
2#20730C212yesmaximal, complemented
3#12862C212yesmaximal, complemented
4#12803C212yesmaximal, complemented
5#12801C212yesmaximal, complemented
6#12699C212yesmaximal
7#12632C212yesmaximal, complemented
8C4×#355C212yesmaximal, complemented
9C22×#124C2211yescharacteristic
10C2×#519C2211yescharacteristic
11C2×#518C411yescharacteristic
12#2921—22no—
13#2367—22no—
14#2367—22no—
15#2342—22no—
16#2342—22no—
17#2326C411yescharacteristic
18#2326C411yescharacteristic
19#2326C2211yescharacteristic
20#2326C2211yescharacteristic
21#2310C2211yescharacteristic
22#2309C411yescharacteristic
23C4×#124C411yescharacteristic
24C4×#124C411yescharacteristic
25C4×#124C2211yescharacteristic
26C4×#124C2211yescharacteristic
27#2285C411yescharacteristic
28C4×#117C411yescharacteristic
29C2×#355C2214yes—
30C2×#355C2214yes—
31#1822—22no—
32#1801—22no—
33#1801C2214yescomplemented
34#1784C2214yes—
35C4×#101—22no—
36C4×#101C2214yescomplemented
37#1394C2214yescomplemented
38#1394C2214yescomplemented
39C23×Q8C2311yescharacteristic
40C23×D4C2×C411yescharacteristic
41C2×#124—22no—
42C2×#124—24no—
43C2×#124—24no—
44C2×#124C2×C411yescharacteristic
45C2×#124C2×C411yescharacteristic
46C2×#124C2311yescharacteristic
47C2×#124C2311yescharacteristic
48#519—44no—
49#518—44no—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8D412yes—
53C2×C4×Q8D412yes—
54C2×C4×Q8C2311yescharacteristic
55C2×C4×Q8C2311yescharacteristic
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×#117C2×C411yescharacteristic
63C2×#117C2×C411yescharacteristic
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117D412yes—
67C2×#117D412yes—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C42C2×C411yescharacteristic
72#433—48no—
73#423—24no—
74C2×#101—24no—
75C2×#101C2318yes—
76#407—24no—
77C4×M16—24no—
78#391—22no—
79#391C2×C411yescharacteristic
80#391C2×C411yescharacteristic
81#391C2311yescharacteristic
82#391C2311yescharacteristic
83#390—22no—
84#390C2×C411yescharacteristic
85#390C2×C411yescharacteristic
86#390C2×C411yescharacteristic
87#390C2×C411yescharacteristic
88#388—22no—
89#388—22no—
90#388—22no—
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#385—22no—
97#385—22no—
98#385—22no—
99#385—22no—
100#384C2×C411yescharacteristic
101#384C2×C411yescharacteristic
102#382C2×C411yescharacteristic
103#382C2×C411yescharacteristic
104#381—22no—
105#380—22no—
106#379—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116#376C2×C411yescharacteristic
117#376C2×C411yescharacteristic
118C43C2×C411yescharacteristic
119C43C2×C411yescharacteristic
120#355—28no—
121#355—28no—
122#343—24no—
123#343—24no—
124C25—21no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8C22×C411yescharacteristic
130C22×Q8C2×D412yes—
131C22×Q8C2411yescharacteristic
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—21no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4C22×C411yescharacteristic
138C22×D4C22×C411yescharacteristic
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D412yes—
146C2×M16—28no—
147#124—216no—
148#124—22no—
149#124—22no—
150#124—22no—
151#124—22no—
152#124—22no—
153#124—22no—
154#124#4114yes—
155#124#4114yes—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×Q8—42no—
160C4×Q8—44no—
161C4×Q8—44no—
162C4×D4—42no—
163C4×D4—42no—
164C4×D4—42no—
165C4×D4—42no—
166C4×D4—44no—
167C4×D4—44no—
168#117—24no—
169#117—28no—
170#117—42no—
171#117—44no—
172#117—42no—
173#117—42no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C22⋊C4—216no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4C22⋊C411yescharacteristic
205C2×C22⋊C4C22⋊C412yes—
206C2×C22⋊C4C22⋊C411yescharacteristic
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42C22⋊C412yes—
216C2×C42C22⋊C412yes—
217C2×C42C22⋊C412yes—
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C4⋊C8—48no—
227#101—28no—
228#101—28no—
229#101—48no—
230C22⋊C8—48no—
231#97—44no—
232C4×C8—44no—
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—22no—
242#95—22no—
243C24—24no—
244C24—21no—
245C24—21no—
246C24—42no—
247C24—42no—
248C24—82no—
249C24—82no—
250C24C2×C22⋊C411yescharacteristic
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—41no—
256C2×Q8—42no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—48no—
261C2×Q8—48no—
262C2×Q8C2×#4112yes—
263C2×Q8C2×#4112yes—
264C2×D4—24no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—48no—
273C2×D4—48no—
274C2×D4—42no—
275C2×D4#11712yes—
276C2×D4#11712yes—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D412yes—
311M16—48no—
312M16—48no—
313C2×C8—48no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C22⋊C4—28no—
328C22⋊C4—28no—
329C22⋊C4—22no—
330C22⋊C4—24no—
331C22⋊C4—28no—
332C22⋊C4—28no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4#9914yes—
344C42—24no—
345C42—21no—
346C42—21no—
347C42—24no—
348C42—24no—
349C42—22no—
350C42—22no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—41no—
355C42—41no—
356C42—42no—
357C42—81no—
358C42—81no—
359C42—81no—
360C42—81no—
361C42#11712yes—
362C42#11712yes—
363C23—21no—
364C23—21no—
365C23—22no—
366C23—22no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—48no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—88no—
379C23—162no—
380C23—162no—
381C23—164no—
382C23C2×#9912yes—
383C23C22×C22⋊C411yescharacteristic
384Q8—42no—
385Q8—42no—
386Q8—48no—
387Q8—42no—
388Q8—42no—
389D4—42no—
390D4—42no—
391D4—42no—
392D4—42no—
393D4—48no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—48no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—84no—
433C2×C4—162no—
434C2×C4—162no—
435C2×C4—162no—
436C2×C4—162no—
437C2×C4C2×#9912yes—
438C2×C4C2×#9912yes—
439C2×C4#41012yes—
440C2×C4C2×#11711yescharacteristic
441C2×C4C2×#11711yescharacteristic
442C2×C4#52514yes—
443C8—88no—
444C22—21no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—42no—
450C22—42no—
451C22—44no—
452C22—44no—
453C22—42no—
454C22—42no—
455C22—82no—
456C22—82no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—168no—
462C22#228011yescharacteristic
463C22C22×#9911yescharacteristic
464C22C2×#41011yescharacteristic
465C4—22no—
466C4—22no—
467C4—24no—
468C4—21no—
469C4—21no—
470C4—22no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—41no—
476C4—41no—
477C4—42no—
478C4—81no—
479C4—81no—
480C4—81no—
481C4—81no—
482C2—21no—
483C2—21no—
484C2—42no—
485C2—82no—
486C2—82no—
487C2#2072211yescharacteristic
488C2#3337011yescharacteristic
489C2#3337011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20730C2semidirect
3. #12862C2semidirect
4. #12803C2semidirect
5. #12801C2semidirect
7. #12632C2semidirect
8. C4×#355C2semidirect
33. #1801C22semidirect
36. C4×#101C22semidirect
37. #1394C22semidirect
38. #1394C22semidirect

Maximal subgroups

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

subgroupindex
#342112
#207302
#128622
#128032
#128012
#126992
#126322
C4×#3552