Finite Group Databasebeta

← #761455

Subgroups — #761455

Order 512.

4170 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
1C22×#385C212yesmaximal
2#16987C214yesmaximal, complemented
3#15204C214yesmaximal, complemented
4#15092C214yesmaximal, complemented
5#14955C214yesmaximal, complemented
6#14127C211yesmaximal, characteristic
7#14127C212yesmaximal
8#14127C211yesmaximal, characteristic
9#14044C214yesmaximal, complemented
10#14043C214yesmaximal, complemented
11C4×#379C211yesmaximal, characteristic
12C25×C4C2211yescharacteristic
13C23×C4⋊C4C2212yes—
14C23×C4⋊C4C2212yes—
15C23×C22⋊C4C2212yes—
16C23×C22⋊C4C2212yes—
17C2×#389C2214yescomplemented
18C2×#385—216no—
19C2×#385C2214yes—
20C2×#385C2214yes—
21C2×#385C2214yes—
22C2×#385C2214yes—
23C2×#385C2214yes—
24C2×#385C2214yes—
25C2×#385C2214yes—
26C2×#385C2214yes—
27C2×#379C2211yescharacteristic
28C2×#379C2211yescharacteristic
29C2×C4×C22⋊C4C2212yes—
30C2×C4×C22⋊C4C2214yes—
31C22×#95C2212yes—
32C22×#95C2212yes—
33#1444C2218yescomplemented
34#1444C2214yescomplemented
35#1442C2218yescomplemented
36#1442C2214yescomplemented
37#1441C2218yescomplemented
38#1441C2218yescomplemented
39#1441C2218yescomplemented
40#1439C2214yes—
41#1439C2218yes—
42#1439C2214yes—
43#1436C2218yescomplemented
44C4⋊#95C2214yescomplemented
45C22⋊#95C2214yes—
46C22⋊#95C2214yes—
47C22⋊#95C2214yes—
48C22⋊#95C2214yes—
49C22⋊#95C2212yes—
50C22⋊#95C2214yes—
51C26C2311yescharacteristic
52C24×C4—44no—
53C24×C4—48no—
54C24×C4D418yes—
55C24×C4C2312yes—
56C24×C4C2312yes—
57C24×C4C2312yes—
58C24×C4C2312yes—
59C24×C4C2312yes—
60C24×C4C2312yes—
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C22×C4⋊C4—28no—
64C22×C4⋊C4—28no—
65C22×C4⋊C4—48no—
66C22×C4⋊C4—48no—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2314yes—
72C22×C4⋊C4C2318yes—
73C22×C4⋊C4C2314yes—
74C22×C22⋊C4—28no—
75C22×C22⋊C4—216no—
76C22×C22⋊C4—28no—
77C22×C22⋊C4—48no—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2312yes—
85C22×C22⋊C4C2312yes—
86C22×C42C2314yes—
87#389—48no—
88#385—216no—
89#385—216no—
90#385—216no—
91#385—216no—
92#385—216no—
93#385—216no—
94#385—48no—
95#385—48no—
96#379Q818yes—
97#379Q818yes—
98C4×C22⋊C4—44no—
99C4×C22⋊C4—48no—
100C2×#95—28no—
101C2×#95—28no—
102C2×#95C2318yes—
103C2×#95C2318yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2318yes—
111C2×#95C2318yes—
112C2×#95C2314yes—
113C2×#95C2318yes—
114C2×#95C2318yes—
115C25—28no—
116C25—28no—
117C25—42no—
118C25—42no—
119C25C2×D414yes—
120C25C2×Q814yes—
121C25C2×Q814yes—
122C25C2411yescharacteristic
123C25C2412yes—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—28no—
127C23×C4—28no—
128C23×C4—28no—
129C23×C4—28no—
130C23×C4—28no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—28no—
135C23×C4—28no—
136C23×C4—28no—
137C23×C4—28no—
138C23×C4—416no—
139C23×C4—44no—
140C23×C4—432no—
141C23×C4—44no—
142C23×C4—44no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—42no—
147C23×C4—44no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—464no—
153C23×C4C2×D418yes—
154C23×C4C2×D418yes—
155C23×C4C2×D414yes—
156C23×C4C2×D414yes—
157C23×C4C2414yes—
158C23×C4C2414yes—
159C23×C4C2412yes—
160C23×C4C2411yescharacteristic
161C23×C4C2414yes—
162C23×C4C2414yes—
163C23×C4C2414yes—
164C23×C4C2414yes—
165C23×C4C2411yescharacteristic
166C2×C4⋊C4—28no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—216no—
171C2×C4⋊C4—216no—
172C2×C4⋊C4—216no—
173C2×C4⋊C4—48no—
174C2×C4⋊C4—48no—
175C2×C4⋊C4—48no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—432no—
178C2×C4⋊C4—432no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—44no—
183C2×C22⋊C4—264no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—432no—
204C2×C22⋊C4C2×Q8116yes—
205C2×C22⋊C4C2×Q8116yes—
206C2×C22⋊C4C2×Q818yes—
207C2×C22⋊C4C2×Q818yes—
208C2×C22⋊C4#41116yes—
209C2×C42—28no—
210C2×C42—48no—
211#95—28no—
212#95—28no—
213#95—28no—
214#95—28no—
215#95—216no—
216#95—216no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—48no—
221#95—48no—
222#95—48no—
223#95—48no—
224C24—24no—
225C24—24no—
226C24—24no—
227C24—24no—
228C24—216no—
229C24—216no—
230C24—232no—
231C24—232no—
232C24—24no—
233C24—22no—
234C24—24no—
235C24—22no—
236C24—44no—
237C24—44no—
238C24—48no—
239C24—48no—
240C24—416no—
241C24—416no—
242C24—432no—
243C24C4⋊Q8116yes—
244C24C22×D412yes—
245C24C22×D414yes—
246C24C22×Q812yes—
247C24C22×Q812yes—
248C24C22×Q814yes—
249C24C22×Q814yes—
250C24C2×#4118yes—
251C24H(5,𝔽2)18yes—
252C24C2511yescharacteristic
253C22×C4—24no—
254C22×C4—216no—
255C22×C4—216no—
256C22×C4—216no—
257C22×C4—216no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—28no—
261C22×C4—28no—
262C22×C4—28no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—28no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—28no—
272C22×C4—28no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—28no—
278C22×C4—24no—
279C22×C4—216no—
280C22×C4—28no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—28no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—28no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—44no—
294C22×C4—48no—
295C22×C4—48no—
296C22×C4—48no—
297C22×C4—432no—
298C22×C4—432no—
299C22×C4—432no—
300C22×C4—416no—
301C22×C4—48no—
302C22×C4—48no—
303C22×C4—432no—
304C22×C4—4128no—
305C22×C4—416no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—48no—
309C22×C4—48no—
310C22×C4—48no—
311C22×C4—48no—
312C22×C4—48no—
313C22×C4—48no—
314C22×C4—816no—
315C22×C4—816no—
316C22×C4C22×D414yes—
317C22×C4C22×D412yes—
318C22×C4C22×D412yes—
319C22×C4C22×Q818yes—
320C22×C4C22×Q818yes—
321C22×C4C2×#4118yes—
322C22×C4C2×#4118yes—
323C22×C4H(5,𝔽2)14yes—
324C4⋊C4—416no—
325C4⋊C4—416no—
326C4⋊C4—416no—
327C4⋊C4—416no—
328C4⋊C4—416no—
329C4⋊C4—816no—
330C4⋊C4—88no—
331C22⋊C4—232no—
332C22⋊C4—232no—
333C22⋊C4—232no—
334C22⋊C4—232no—
335C22⋊C4—48no—
336C22⋊C4—48no—
337C22⋊C4—416no—
338C22⋊C4—416no—
339C42—88no—
340C23—24no—
341C23—24no—
342C23—24no—
343C23—24no—
344C23—28no—
345C23—28no—
346C23—216no—
347C23—28no—
348C23—28no—
349C23—216no—
350C23—216no—
351C23—216no—
352C23—264no—
353C23—24no—
354C23—24no—
355C23—22no—
356C23—22no—
357C23—28no—
358C23—28no—
359C23—24no—
360C23—24no—
361C23—416no—
362C23—416no—
363C23—464no—
364C23—464no—
365C23—416no—
366C23—432no—
367C23—44no—
368C23—48no—
369C23—44no—
370C23#52518yes—
371C23C2×C4⋊Q814yes—
372C23C2×C4⋊Q818yes—
373C23#53418yes—
374C23#554116yes—
375C23#554116yes—
376C23C23×D411yescharacteristic
377C23C23×Q811yescharacteristic
378C23C23×Q811yescharacteristic
379C23C22×#4114yes—
380C23C2×H(5,𝔽2)14yes—
381C23C2×H(5,𝔽2)12yes—
382C23#58512yes—
383C2×C4—24no—
384C2×C4—28no—
385C2×C4—28no—
386C2×C4—28no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—216no—
393C2×C4—216no—
394C2×C4—216no—
395C2×C4—28no—
396C2×C4—28no—
397C2×C4—216no—
398C2×C4—432no—
399C2×C4—416no—
400C2×C4—432no—
401C2×C4—432no—
402C2×C4—432no—
403C2×C4—416no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—48no—
411C2×C4—48no—
412C2×C4—48no—
413C2×C4—48no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—48no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—48no—
425C2×C4—48no—
426C2×C4—48no—
427C2×C4—48no—
428C2×C4—816no—
429C2×C4—816no—
430C2×C4#53414yes—
431C22—232no—
432C22—232no—
433C22—28no—
434C22—28no—
435C22—28no—
436C22—28no—
437C22—28no—
438C22—28no—
439C22—28no—
440C22—24no—
441C22—464no—
442C22—48no—
443C22—48no—
444C22—42no—
445C22—42no—
446C22—44no—
447C22#2583116yes—
448C22C2×#52512yes—
449C22C22×C4⋊Q811yescharacteristic
450C22C2×#53412yes—
451C22C2×#53411yescharacteristic
452C22#344911yescharacteristic
453C22C2×#55414yes—
454C22C2×#55414yes—
455C22#348114yes—
456C22#350112yes—
457C22#350112yes—
458C22#351318yes—
459C22#353214yes—
460C4—24no—
461C4—48no—
462C4—48no—
463C4—44no—
464C4—42no—
465C4—48no—
466C4—48no—
467C4—84no—
468C4—84no—
469C2—216no—
470C2—44no—
471C2C2×#258312yes—
472C2#3495611yescharacteristic
473C2#3501111yescharacteristic
474C2#3506212yes—
475C2#3743611yescharacteristic
476C2#5991414yes—
477C2#5991514yes—
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
2. #16987C2semidirect
3. #15204C2semidirect
4. #15092C2semidirect
5. #14955C2semidirect
9. #14044C2semidirect
10. #14043C2semidirect
17. C2×#389C22semidirect
33. #1444C22semidirect
34. #1444C22semidirect
35. #1442C22semidirect
36. #1442C22semidirect
37. #1441C22semidirect
38. #1441C22semidirect
39. #1441C22semidirect
43. #1436C22semidirect
44. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3852
#169872
#152042
#150922
#149552
#141272
#140442
#140432
C4×#3792