Finite Group Databasebeta

← #215456

Subgroups — #215456

Order 512.

1302 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35166C211yesmaximal, characteristic
2#21916C211yesmaximal, characteristic, complemented
3#13950C211yesmaximal, characteristic, complemented
4#9715C212yesmaximal, complemented
5#9712C212yesmaximal, complemented
6#9684C212yesmaximal, complemented
7#9684C212yesmaximal, complemented
8#8332C212yesmaximal, complemented
9#8332C212yesmaximal, complemented
10C2×#537—21no—
11C2×#537—21no—
12C2×#532C2212yes—
13#2856C2212yes—
14#2856C2212yes—
15#2656—21no—
16#2656—21no—
17#2542C414yescomplemented
18#2541C414yescomplemented
19#2540—24no—
20C2×C4×C4⋊C4C2211yescharacteristic
21#1917—24no—
22#1817C2212yes—
23#1797C2212yes—
24#1749C2212yes—
25C2×#327C2214yes—
26C2×#327C2214yes—
27C2×#327C2214yes—
28C2×#327C2214yes—
29#1445—21no—
30#1443—21no—
31#1436—21no—
32#1434—21no—
33#1434C2211yescharacteristic
34#1434C2211yescharacteristic
35#1434C2211yescharacteristic
36#1434C2211yescharacteristic
37#1434C2211yescharacteristic
38C4×#95C2211yescharacteristic
39C22×#41—22no—
40#537—44no—
41#537—44no—
42#537—44no—
43#537—44no—
44#532—44no—
45#532—44no—
46C2×C4⋊Q8C2×C412yes—
47C2×#127C2×C412yes—
48C2×#127C2×C412yes—
49C2×#127C2×C412yes—
50C2×#124—22no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8—24no—
53C2×#121—22no—
54C2×#121—22no—
55C2×#121—22no—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121—24no—
59C2×#121—24no—
60C2×#121C2314yes—
61C2×C4×D4—22no—
62C2×C4×D4C2×C414yes—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×C4⋊C8—22no—
70C2×C22⋊C8C2314yes—
71C2×#97—22no—
72#399—24no—
73#399—24no—
74#392—24no—
75#392—24no—
76#390—22no—
77#390—22no—
78#390—22no—
79#390—22no—
80#390C2×C414yes—
81#390C2×C414yes—
82#384C2×C412yes—
83#384C2×C412yes—
84C4×C4⋊C4C2×C414yes—
85C4×C4⋊C4D414yes—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95D411yescharacteristic
98C2×#95D411yescharacteristic
99C2×#95C2311yescharacteristic
100C2×#95C2311yescharacteristic
101C2×#95C2311yescharacteristic
102C2×#95C2311yescharacteristic
103#336—22no—
104#336—24no—
105#336—22no—
106#327—44no—
107#327—44no—
108#327—44no—
109#327—44no—
110C2×#41—44no—
111C2×#41—44no—
112C2×#41—44no—
113C2×#41—44no—
114C2×#41—44no—
115C2×#41—44no—
116C22×Q8—22no—
117C22×D4—24no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4C22×C414yes—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—24no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—24no—
138C4⋊Q8—44no—
139#127—44no—
140#127—82no—
141#127—82no—
142#124—48no—
143C22⋊Q8—416no—
144C22⋊Q8—416no—
145#121—24no—
146#121—24no—
147#121—24no—
148#121—24no—
149#121—416no—
150#121—416no—
151#121—44no—
152#121—44no—
153#121—44no—
154#121—44no—
155#121—44no—
156#121—44no—
157#121—44no—
158#121—44no—
159#121—44no—
160#121—44no—
161C4×D4—44no—
162C4×D4—44no—
163C4×D4—48no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4C22⋊C414yes—
174C2×C4⋊C4C22⋊C414yes—
175C2×C4⋊C4C22×C414yes—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—42no—
185C2×C42—42no—
186C2×C42—81no—
187C2×C42—81no—
188C2×C42—81no—
189C2×C42C22×C412yes—
190C2×C42C22×C412yes—
191C2×C42C22×C412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C4⋊C8—82no—
196C22⋊C8—24no—
197C22⋊C8—24no—
198#97—82no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—42no—
204#95—42no—
205#95—42no—
206#95—42no—
207#95—42no—
208#95—42no—
209#95—42no—
210#95—42no—
211#95—81no—
212#95—81no—
213#95—81no—
214#95C22⋊C414yes—
215#95C22⋊C414yes—
216C24—24no—
217C24C22×D411yescharacteristic
218#41—88no—
219#41—88no—
220#41—88no—
221#41—88no—
222C2×Q8—44no—
223C2×Q8—44no—
224C2×Q8—44no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—44no—
228C2×D4—44no—
229C2×D4—44no—
230C2×D4—44no—
231C2×D4—44no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4—44no—
235C2×D4—44no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4—44no—
240C2×D4—44no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—82no—
244C2×D4—82no—
245C2×D4—82no—
246C2×D4—82no—
247C2×D4—82no—
248C2×D4—82no—
249C2×D4—82no—
250C2×D4—82no—
251C2×D4—82no—
252C2×D4—82no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—48no—
281C22×C4—48no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—48no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4AGL(1,ℤ/8)12yes—
316C22×C4AGL(1,ℤ/8)12yes—
317C22×C4AGL(1,ℤ/8)12yes—
318C22×C4AGL(1,ℤ/8)12yes—
319C22×C4C23×C411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4H(5,𝔽2)12yes—
322C22×C4H(5,𝔽2)12yes—
323C2×C8—44no—
324C2×C8—44no—
325C2×C8—44no—
326C4⋊C4—28no—
327C4⋊C4—28no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C22⋊C4—48no—
336C22⋊C4—48no—
337C22⋊C4—48no—
338C22⋊C4—48no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—48no—
342C42—24no—
343C42—24no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—84no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—44no—
354C23—44no—
355C23—44no—
356C23—44no—
357C23—44no—
358C23—44no—
359C23—44no—
360C23C22×C22⋊C411yescharacteristic
361C23#53511yescharacteristic
362C23#53511yescharacteristic
363C23C2×AGL(1,ℤ/8)11yescharacteristic
364C23C2×AGL(1,ℤ/8)11yescharacteristic
365C23C2×AGL(1,ℤ/8)11yescharacteristic
366C23C2×AGL(1,ℤ/8)11yescharacteristic
367Q8—88no—
368D4—84no—
369D4—84no—
370D4—88no—
371D4—88no—
372D4—84no—
373D4—84no—
374D4—84no—
375D4—84no—
376D4—164no—
377D4—164no—
378D4—164no—
379D4—164no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
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—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—816no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4#41814yes—
449C2×C4#41814yes—
450C2×C4#51812yes—
451C2×C4#53512yes—
452C8—84no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—84no—
466C22—84no—
467C22—84no—
468C22—84no—
469C22—84no—
470C22—84no—
471C22—84no—
472C22#230911yescharacteristic
473C22#288311yescharacteristic
474C22#288411yescharacteristic
475C22C2×#41811yescharacteristic
476C22C2×#41811yescharacteristic
477C22#290111yescharacteristic
478C22#290111yescharacteristic
479C22#321711yescharacteristic
480C22#321711yescharacteristic
481C22#321711yescharacteristic
482C22#321711yescharacteristic
483C4—22no—
484C4—22no—
485C4—24no—
486C4—24no—
487C4—42no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—82no—
492C4—82no—
493C4—84no—
494C4—161no—
495C4—161no—
496C2—21no—
497C2—21no—
498C2—21no—
499C2—21no—
500C2—164no—
501C2#1762011yescharacteristic
502C2#1762611yescharacteristic
503C2#1763111yescharacteristic
504C2#1763111yescharacteristic
505C2#2179611yescharacteristic
506C2#2181011yescharacteristic
507C2#2181711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35166C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21916C2semidirect
3. #13950C2semidirect
4. #9715C2semidirect
5. #9712C2semidirect
6. #9684C2semidirect
7. #9684C2semidirect
8. #8332C2semidirect
9. #8332C2semidirect
17. #2542C4semidirect
18. #2541C4semidirect

Maximal subgroups

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

subgroupindex
#351662
#219162
#139502
#97152
#97122
#96842
#83322