Finite Group Databasebeta

← #4414549

Subgroups — #4414549

Order 512.

3686 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C212yesmaximal
2#17059C211yesmaximal, characteristic, complemented
3#16815C214yesmaximal, complemented
4#16601C214yesmaximal, complemented
5#16366C212yesmaximal, complemented
6#16254C211yesmaximal, characteristic, complemented
7#14826C214yesmaximal, complemented
8#14820C214yesmaximal
9#14758C214yesmaximal, complemented
10#14627C214yesmaximal, complemented
11C2×C22⋊#95C211yesmaximal, characteristic
12C25×C4C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×C22⋊Q8C2212yes—
15C22×C22⋊Q8C2212yes—
16C23×C22⋊C4C2212yes—
17#2848—216no—
18C2×#400C2214yescomplemented
19C2×#397C2214yes—
20C2×#397C2214yes—
21C2×#395C2214yescomplemented
22C2×#385C2214yes—
23C2×#385C2214yes—
24C2×#385C2214yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2214yes—
29C2×#379C2214yes—
30C22×#95C2214yes—
31#1446C2214yescomplemented
32#1445C2214yescomplemented
33#1444C2218yescomplemented
34#1444C2214yescomplemented
35#1443C2218yescomplemented
36#1442C2218yescomplemented
37#1442C2214yescomplemented
38#1441C2218yescomplemented
39#1441C2214yescomplemented
40#1441C2214yescomplemented
41#1440C2218yes—
42#1440C2218yes—
43#1439C2214yes—
44#1439C2214yes—
45C22⋊#95C2212yes—
46C22⋊#95C2211yescharacteristic
47C22⋊#95C2218yes—
48C22⋊#95C2214yes—
49C22⋊#95C2211yescharacteristic
50C22⋊#95C2214yes—
51C26C2311yescharacteristic
52C23×Q8C2312yes—
53C24×C4—48no—
54C24×C4—44no—
55C24×C4D418yes—
56C24×C4C2314yes—
57C24×C4C2311yescharacteristic
58C24×C4C2312yes—
59C24×C4C2312yes—
60C24×C4C2311yescharacteristic
61C2×C22⋊Q8—28no—
62C2×C22⋊Q8—28no—
63C2×C22⋊Q8—28no—
64C2×C22⋊Q8—48no—
65C2×C22⋊Q8—48no—
66C2×C22⋊Q8—48no—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2314yes—
75C22×C22⋊C4—28no—
76C22×C22⋊C4—44no—
77C22×C22⋊C4—44no—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2312yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2312yes—
87#400—48no—
88#397—216no—
89#397—48no—
90#395—48no—
91#385—216no—
92#385—216no—
93#385—216no—
94#380—28no—
95#380—28no—
96#380—28no—
97#380—48no—
98#379—216no—
99C2×#95—48no—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2318yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2318yes—
108C2×#95C2314yes—
109C2×#95C2318yes—
110C2×#95C2318yes—
111C2×#95C2318yes—
112C2×#95C2318yes—
113C2×#95C2318yes—
114C25—24no—
115C25—22no—
116C25—24no—
117C25—22no—
118C25—28no—
119C25—44no—
120C25C2×D414yes—
121C25C2411yescharacteristic
122C25C2411yescharacteristic
123C25C2411yescharacteristic
124C22×Q8—24no—
125C22×Q8—28no—
126C22×Q8—44no—
127C22×Q8—44no—
128C23×C4—28no—
129C23×C4—28no—
130C23×C4—28no—
131C23×C4—28no—
132C23×C4—28no—
133C23×C4—216no—
134C23×C4—28no—
135C23×C4—432no—
136C23×C4—416no—
137C23×C4—42no—
138C23×C4—44no—
139C23×C4—44no—
140C23×C4—44no—
141C23×C4—44no—
142C23×C4—42no—
143C23×C4—44no—
144C23×C4—48no—
145C23×C4—464no—
146C23×C4C2×D414yes—
147C23×C4C2×D418yes—
148C23×C4C2×D418yes—
149C23×C4C2×D414yes—
150C23×C4C2414yes—
151C23×C4C2414yes—
152C23×C4C2414yes—
153C23×C4C2414yes—
154C23×C4C2414yes—
155C23×C4C2412yes—
156C23×C4C2414yes—
157C23×C4C2412yes—
158C22⋊Q8—432no—
159C22⋊Q8—432no—
160C22⋊Q8—432no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—48no—
174C2×C4⋊C4—48no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—48no—
178C2×C4⋊C4—44no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—216no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—216no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—48no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—432no—
203C2×C22⋊C4#4118yes—
204#95—28no—
205#95—28no—
206#95—28no—
207#95—28no—
208#95—216no—
209#95—28no—
210#95—28no—
211#95—48no—
212#95—48no—
213#95—48no—
214#95—44no—
215#95—48no—
216#95—48no—
217#95—48no—
218C24—22no—
219C24—24no—
220C24—21no—
221C24—21no—
222C24—216no—
223C24—216no—
224C24—28no—
225C24—28no—
226C24—28no—
227C24—22no—
228C24—21no—
229C24—24no—
230C24—24no—
231C24—22no—
232C24—22no—
233C24—24no—
234C24—24no—
235C24—21no—
236C24—216no—
237C24—48no—
238C24—432no—
239C24—416no—
240C24—416no—
241C24—416no—
242C24—416no—
243C24C22×D412yes—
244C24C22×D412yes—
245C24C22×D412yes—
246C24C2×#4114yes—
247C24H(5,𝔽2)12yes—
248C24H(5,𝔽2)14yes—
249C242−1+412yes—
250C242−1+418yes—
251C24C2511yescharacteristic
252C2×Q8—48no—
253C2×Q8—48no—
254C2×Q8—48no—
255C2×Q8—816no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—216no—
259C22×C4—28no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—28no—
266C22×C4—216no—
267C22×C4—216no—
268C22×C4—216no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—28no—
274C22×C4—28no—
275C22×C4—24no—
276C22×C4—28no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—28no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—28no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—28no—
296C22×C4—44no—
297C22×C4—48no—
298C22×C4—48no—
299C22×C4—48no—
300C22×C4—44no—
301C22×C4—48no—
302C22×C4—416no—
303C22×C4—432no—
304C22×C4—432no—
305C22×C4—416no—
306C22×C4—432no—
307C22×C4—4128no—
308C22×C4—416no—
309C22×C4—416no—
310C22×C4—88no—
311C22×C4—816no—
312C22×C4C22×D414yes—
313C22×C4C22×D414yes—
314C22×C4C2×#4118yes—
315C22×C4H(5,𝔽2)18yes—
316C4⋊C4—48no—
317C4⋊C4—48no—
318C4⋊C4—416no—
319C4⋊C4—416no—
320C4⋊C4—48no—
321C4⋊C4—416no—
322C4⋊C4—88no—
323C4⋊C4—88no—
324C22⋊C4—48no—
325C22⋊C4—416no—
326C22⋊C4—48no—
327C22⋊C4—416no—
328C22⋊C4—416no—
329C22⋊C4—416no—
330C22⋊C4—416no—
331C22⋊C4—416no—
332C22⋊C4—416no—
333C23—22no—
334C23—24no—
335C23—22no—
336C23—24no—
337C23—24no—
338C23—216no—
339C23—28no—
340C23—28no—
341C23—28no—
342C23—232no—
343C23—22no—
344C23—24no—
345C23—24no—
346C23—22no—
347C23—22no—
348C23—24no—
349C23—24no—
350C23—22no—
351C23—24no—
352C23—24no—
353C23—24no—
354C23—24no—
355C23—216no—
356C23—28no—
357C23—28no—
358C23—432no—
359C23—432no—
360C23—416no—
361C23—464no—
362C23—464no—
363C23—416no—
364C23—48no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23#53214yes—
372C23#53414yes—
373C23#53618yes—
374C23#53714yes—
375C23#563116yes—
376C23C23×D411yescharacteristic
377C23C22×#4112yes—
378C23C2×H(5,𝔽2)12yes—
379C23C2×H(5,𝔽2)14yes—
380C23C2×H(5,𝔽2)11yescharacteristic
381C23C2×2−1+414yes—
382C23C2×2−1+411yescharacteristic
383Q8—816no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—216no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—28no—
391C2×C4—216no—
392C2×C4—416no—
393C2×C4—432no—
394C2×C4—432no—
395C2×C4—416no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—48no—
399C2×C4—48no—
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—48no—
414C2×C4—48no—
415C2×C4—48no—
416C2×C4—48no—
417C2×C4—48no—
418C2×C4—44no—
419C2×C4—48no—
420C2×C4—44no—
421C2×C4—48no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—48no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—832no—
428C2×C4#53418yes—
429C22—28no—
430C22—24no—
431C22—28no—
432C22—216no—
433C22—216no—
434C22—24no—
435C22—28no—
436C22—28no—
437C22—24no—
438C22—24no—
439C22—464no—
440C22—44no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—44no—
446C22—48no—
447C22—48no—
448C22—48no—
449C22—48no—
450C22#281718yes—
451C22C2×#53211yescharacteristic
452C22C2×#53412yes—
453C22C2×#53411yescharacteristic
454C22C2×#53612yes—
455C22C2×#53711yescharacteristic
456C22#349012yes—
457C22#349212yes—
458C22#352018yes—
459C22C2×#56314yes—
460C22#353214yes—
461C22#353614yes—
462C22#353814yes—
463C4—28no—
464C4—44no—
465C4—48no—
466C4—84no—
467C4—84no—
468C4—84no—
469C4—84no—
470C4—84no—
471C2—28no—
472C2—44no—
473C2—44no—
474C2#3505111yescharacteristic
475C2#3508512yes—
476C2C2×#281711yescharacteristic
477C2#3743611yescharacteristic
478C2#3744212yes—
479C2#4572814yes—
480C2#4574414yes—
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. #17059C2semidirect
3. #16815C2semidirect
4. #16601C2semidirect
5. #16366C2semidirect
6. #16254C2semidirect
7. #14826C2semidirect
9. #14758C2semidirect
10. #14627C2semidirect
18. C2×#400C22semidirect
21. C2×#395C22semidirect
31. #1446C22semidirect
32. #1445C22semidirect
33. #1444C22semidirect
34. #1444C22semidirect
35. #1443C22semidirect
36. #1442C22semidirect
37. #1442C22semidirect
38. #1441C22semidirect
39. #1441C22semidirect
40. #1441C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
#170592
#168152
#166012
#163662
#162542
#148262
#148202
#147582
#146272
C2×C22⋊#952