Finite Group Databasebeta

← #4474586

Subgroups — #4474586

Order 512.

2098 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#17290C214yesmaximal, complemented
2#17059C211yesmaximal, characteristic, complemented
3#16954C214yesmaximal, complemented
4#16903C211yesmaximal, characteristic, complemented
5#16651C214yesmaximal, complemented
6#15409C214yesmaximal, complemented
7#15393C214yesmaximal, complemented
8#15282C214yesmaximal
9#14820C212yesmaximal
10#14423C212yesmaximal, complemented
11#14338C211yesmaximal, characteristic
12C23×C22⋊C4C2211yescharacteristic
13C2×#397C2212yes—
14C2×#397C2214yes—
15C2×#397C2212yes—
16C2×#388C2214yes—
17C2×#388C2214yes—
18C2×#387C2214yescomplemented
19C2×#380C2212yes—
20C2×#380C2212yes—
21C2×#379C2212yes—
22C2×#379C2214yes—
23#1445C2214yescomplemented
24#1445C2214yescomplemented
25#1444C2214yescomplemented
26#1444C2214yescomplemented
27#1443C2214yescomplemented
28#1443C2214yescomplemented
29#1443C2214yescomplemented
30#1442C2214yescomplemented
31#1442C2214yescomplemented
32#1442C2214yescomplemented
33#1441C2214yescomplemented
34#1441C2214yescomplemented
35#1440C2214yes—
36#1440C2214yes—
37#1440C2214yes—
38#1440C2214yes—
39#1439C2214yes—
40#1439C2214yes—
41#1439C2214yes—
42#1439C2214yes—
43#1439C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2212yes—
48#1439C2212yes—
49#1437C2214yes—
50#1436C2214yescomplemented
51#1436C2214yescomplemented
52#1436C2214yescomplemented
53C22⋊#95C2211yescharacteristic
54C22⋊#95C2211yescharacteristic
55C22⋊#95C2212yes—
56C22⋊#95C2214yes—
57C26C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2314yes—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2314yes—
66C22×C22⋊C4—24no—
67C22×C22⋊C4—44no—
68C22×C22⋊C4C2314yes—
69C22×C22⋊C4C2312yes—
70C22×C22⋊C4C2312yes—
71C22×C22⋊C4C2312yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2312yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C42C2314yes—
81#397—44no—
82#397—44no—
83#397—48no—
84#388—48no—
85#388—48no—
86#387—48no—
87#380—28no—
88#380—44no—
89#379—44no—
90#379—48no—
91C2×#95C2314yes—
92C2×#95C2314yes—
93C2×#95C2314yes—
94C2×#95C2314yes—
95C2×#95C2312yes—
96C2×#95C2312yes—
97C2×#95C2314yes—
98C2×#95C2314yes—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C25—22no—
115C25—22no—
116C25—22no—
117C25—24no—
118C25—42no—
119C25—44no—
120C25—44no—
121C25C2411yescharacteristic
122C25C2411yescharacteristic
123C25C2411yescharacteristic
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—42no—
127C23×C4—44no—
128C23×C4—44no—
129C23×C4#4114yes—
130C23×C4C2414yes—
131C23×C4C2414yes—
132C23×C4C2414yes—
133C23×C4C2414yes—
134C23×C4C2414yes—
135C23×C4C2414yes—
136C23×C4C2412yes—
137C23×C4C2412yes—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—44no—
155C2×C22⋊C4—28no—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—28no—
158C2×C22⋊C4—28no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—416no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C42—44no—
193C2×C42—44no—
194C2×C42—44no—
195#95—28no—
196#95—28no—
197#95—28no—
198#95—28no—
199#95—44no—
200#95—44no—
201#95—44no—
202#95—44no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—42no—
214#95—42no—
215#95—44no—
216#95—44no—
217#95—44no—
218C24—21no—
219C24—22no—
220C24—21no—
221C24—24no—
222C24—24no—
223C24—28no—
224C24—21no—
225C24—21no—
226C24—22no—
227C24—22no—
228C24—21no—
229C24—22no—
230C24—22no—
231C24—21no—
232C24—24no—
233C24—24no—
234C24—24no—
235C24—24no—
236C24—48no—
237C24—48no—
238C24—44no—
239C24—416no—
240C24—44no—
241C24—84no—
242C24—84no—
243C24—88no—
244C24—88no—
245C24—88no—
246C24—88no—
247C24—88no—
248C24C2×#4112yes—
249C24H(5,𝔽2)12yes—
250C24H(5,𝔽2)14yes—
251C242−1+412yes—
252C24C2511yescharacteristic
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—28no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—44no—
305C22×C4—48no—
306C22×C4—88no—
307C22×C4—84no—
308C22×C4—88no—
309C22×C4C2×#4114yes—
310C22×C4H(5,𝔽2)14yes—
311C4⋊C4—416no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C4⋊C4—88no—
315C4⋊C4—88no—
316C4⋊C4—88no—
317C22⋊C4—48no—
318C22⋊C4—48no—
319C22⋊C4—416no—
320C22⋊C4—416no—
321C22⋊C4—88no—
322C22⋊C4—88no—
323C22⋊C4—84no—
324C22⋊C4—88no—
325C22⋊C4—84no—
326C22⋊C4—88no—
327C22⋊C4—88no—
328C22⋊C4—88no—
329C42—88no—
330C23—22no—
331C23—24no—
332C23—22no—
333C23—24no—
334C23—28no—
335C23—24no—
336C23—24no—
337C23—28no—
338C23—22no—
339C23—22no—
340C23—28no—
341C23—24no—
342C23—24no—
343C23—28no—
344C23—416no—
345C23—44no—
346C23—44no—
347C23—44no—
348C23—44no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—44no—
358C23—44no—
359C23—44no—
360C23—42no—
361C23—42no—
362C23—44no—
363C23—44no—
364C23—44no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23—44no—
369C23—816no—
370C23—816no—
371C23—832no—
372C23—832no—
373C23—88no—
374C23—88no—
375C23#54014yes—
376C23C22×#4111yescharacteristic
377C23C2×H(5,𝔽2)12yes—
378C23C2×H(5,𝔽2)12yes—
379C23C2×H(5,𝔽2)11yescharacteristic
380C23C2×2−1+411yescharacteristic
381C23#58514yes—
382C23#58514yes—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—28no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—88no—
436C2×C4—88no—
437C22—24no—
438C22—28no—
439C22—42no—
440C22—48no—
441C22—42no—
442C22—44no—
443C22—48no—
444C22—42no—
445C22—42no—
446C22—48no—
447C22—44no—
448C22—44no—
449C22—48no—
450C22—832no—
451C22—81no—
452C22—81no—
453C22—84no—
454C22—81no—
455C22—81no—
456C22—82no—
457C22—82no—
458C22—84no—
459C22C2×#54011yescharacteristic
460C22#349011yescharacteristic
461C22#349112yes—
462C22#349211yescharacteristic
463C22#351012yes—
464C22#353314yes—
465C22#353314yes—
466C22#353612yes—
467C22#353712yes—
468C22#354014yes—
469C22#356014yes—
470C22#356014yes—
471C22#356414yes—
472C4—44no—
473C4—44no—
474C4—84no—
475C4—84no—
476C4—84no—
477C4—84no—
478C4—84no—
479C4—84no—
480C2—44no—
481C2—82no—
482C2—82no—
483C2#3672711yescharacteristic
484C2#4138912yes—
485C2#4645711yescharacteristic
486C2#4699811yescharacteristic
487C2#4879412yes—
488C2#5415114yes—
489C2#5651614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17290C2semidirect
2. #17059C2semidirect
3. #16954C2semidirect
4. #16903C2semidirect
5. #16651C2semidirect
6. #15409C2semidirect
7. #15393C2semidirect
10. #14423C2semidirect
18. C2×#387C22semidirect
23. #1445C22semidirect
24. #1445C22semidirect
25. #1444C22semidirect
26. #1444C22semidirect
27. #1443C22semidirect
28. #1443C22semidirect
29. #1443C22semidirect
30. #1442C22semidirect
31. #1442C22semidirect
32. #1442C22semidirect
33. #1441C22semidirect
34. #1441C22semidirect
50. #1436C22semidirect
51. #1436C22semidirect
52. #1436C22semidirect

Maximal subgroups

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

subgroupindex
#172902
#170592
#169542
#169032
#166512
#154092
#153932
#152822
#148202
#144232
#143382