Finite Group Databasebeta

← #5955356

Subgroups — #5955356

Order 512.

5226 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2315C211yesmaximal, characteristic, complemented
2C2×#2308C212yesmaximal
3C22×#386C211yesmaximal, characteristic
4#15019C214yesmaximal, complemented
5#14820C214yesmaximal, complemented
6#14737C2116yesmaximal, complemented
7#13794C211yesmaximal, characteristic, complemented
8C2×C22⋊#95C212yesmaximal, complemented
9C24×D4C2211yescharacteristic
10C25×C4C2211yescharacteristic
11C23×C22⋊C4C2212yes—
12C23×C22⋊C4C2212yes—
13C23×C22⋊C4C2211yescharacteristic
14C23×C22⋊C4C2212yes—
15C2×#397C2218yescomplemented
16#2315—28no—
17#2308—216no—
18C2×#386—28no—
19C2×#386C2218yes—
20C2×#386C2218yes—
21C2×#386C2218yes—
22C2×#385C2218yescomplemented
23C2×#381C2214yes—
24C2×#380C2214yes—
25C2×#379C2214yes—
26C22×#95C2212yes—
27C22×#95C2218yescomplemented
28#1445C2218yescomplemented
29#1443C22116yescomplemented
30#1442C2218yescomplemented
31#1440C22116yescomplemented
32#1439C2218yescomplemented
33C22⋊#95C2218yes—
34C22⋊#95C2214yes—
35C22⋊#95C2218yescomplemented
36C22⋊#95C2218yescomplemented
37C26C2311yescharacteristic
38C26C2311yescharacteristic
39C23×D4—24no—
40C23×D4—24no—
41C23×D4—28no—
42C23×D4—28no—
43C23×D4D418yes—
44C23×D4C2314yes—
45C24×C4—24no—
46C24×C4—28no—
47C24×C4—48no—
48C24×C4C2311yescharacteristic
49C24×C4C2312yes—
50C24×C4C2314yes—
51C24×C4C2314yes—
52C24×C4C2314yes—
53C24×C4C2312yes—
54C22×C4⋊C4C2314yes—
55C22×C4⋊C4C2314yes—
56C22×C22⋊C4—28no—
57C22×C22⋊C4—28no—
58C22×C22⋊C4—28no—
59C22×C22⋊C4—28no—
60C22×C22⋊C4—28no—
61C22×C22⋊C4—28no—
62C22×C22⋊C4—28no—
63C22×C22⋊C4—28no—
64C22×C22⋊C4—48no—
65C22×C22⋊C4D418yes—
66C22×C22⋊C4C2314yes—
67C22×C22⋊C4C2314yes—
68C22×C22⋊C4C2314yes—
69C22×C22⋊C4C2314yes—
70C22×C22⋊C4C2318yes—
71C22×C22⋊C4C2314yes—
72C22×C22⋊C4C23116yes—
73#397—416no—
74#386—232no—
75#386—232no—
76#386—232no—
77#385—232no—
78#381—216no—
79#380—216no—
80#379—216no—
81C2×#95—28no—
82C2×#95—416no—
83C2×#95C23116yes—
84C2×#95C2318yes—
85C2×#95C2318yes—
86C2×#95C23116yes—
87C2×#95C2318yes—
88C2×#95C2318yes—
89C2×#95C23116yes—
90C25—22no—
91C25—22no—
92C25—24no—
93C25—24no—
94C25—44no—
95C25—44no—
96C25—44no—
97C25—44no—
98C25C2×D414yes—
99C25C2×D414yes—
100C25C2×D414yes—
101C25#4114yes—
102C25#4118yes—
103C25#4118yes—
104C25C2412yes—
105C25C2411yescharacteristic
106C25C2412yes—
107C22×D4—232no—
108C22×D4—264no—
109C22×D4—28no—
110C22×D4—28no—
111C22×D4—28no—
112C22×D4—28no—
113C22×D4—28no—
114C22×D4—28no—
115C22×D4—416no—
116C22×D4—416no—
117C22×D4—432no—
118C22×D4C2×D4116yes—
119C23×C4—24no—
120C23×C4—22no—
121C23×C4—28no—
122C23×C4—22no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—28no—
126C23×C4—28no—
127C23×C4—216no—
128C23×C4—28no—
129C23×C4—28no—
130C23×C4—28no—
131C23×C4—28no—
132C23×C4—28no—
133C23×C4—28no—
134C23×C4—28no—
135C23×C4—28no—
136C23×C4—28no—
137C23×C4—44no—
138C23×C4—48no—
139C23×C4—44no—
140C23×C4—44no—
141C23×C4—48no—
142C23×C4—48no—
143C23×C4—48no—
144C23×C4—48no—
145C23×C4—432no—
146C23×C4—464no—
147C23×C4—416no—
148C23×C4C2×D414yes—
149C23×C4C2×D418yes—
150C23×C4C2418yes—
151C23×C4C2418yes—
152C23×C4C2414yes—
153C23×C4C2412yes—
154C23×C4C2414yes—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4—48no—
159C2×C22⋊C4—232no—
160C2×C22⋊C4—28no—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—216no—
175C2×C22⋊C4—216no—
176C2×C22⋊C4—232no—
177C2×C22⋊C4—48no—
178C2×C22⋊C4—432no—
179C2×C22⋊C4—432no—
180C2×C22⋊C4—432no—
181C2×C22⋊C4—416no—
182C2×C22⋊C4—416no—
183C2×C22⋊C4C2×D4116yes—
184#95—216no—
185#95—216no—
186#95—232no—
187#95—232no—
188#95—416no—
189#95—48no—
190#95—48no—
191C24—28no—
192C24—24no—
193C24—28no—
194C24—24no—
195C24—28no—
196C24—22no—
197C24—22no—
198C24—24no—
199C24—24no—
200C24—28no—
201C24—28no—
202C24—216no—
203C24—48no—
204C24—416no—
205C24—48no—
206C24—416no—
207C24—416no—
208C24—416no—
209C24—432no—
210C24—416no—
211C24—48no—
212C24—416no—
213C24—48no—
214C24—44no—
215C24—44no—
216C24—48no—
217C24—88no—
218C24—88no—
219C24—816no—
220C24#121116yes—
221C24#123116yes—
222C24#123132yes—
223C24#124116yes—
224C24#126132yes—
225C24C22×D414yes—
226C24C22×D412yes—
227C24C22×D414yes—
228C24C22×D412yes—
229C24C22×D414yes—
230C24C2×#4112yes—
231C24C2×#4114yes—
232C24C2×#4114yes—
233C24C2×#4114yes—
234C24C2×#4114yes—
235C24C2×#4114yes—
236C24C2×#4114yes—
237C24C2×#4114yes—
238C24C2511yescharacteristic
239C2×D4—232no—
240C2×D4—232no—
241C2×D4—232no—
242C2×D4—416no—
243C2×D4—416no—
244C2×D4—416no—
245C2×D4—416no—
246C2×D4—4128no—
247C22×C4—216no—
248C22×C4—232no—
249C22×C4—28no—
250C22×C4—216no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—28no—
254C22×C4—28no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—24no—
261C22×C4—28no—
262C22×C4—216no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—28no—
266C22×C4—28no—
267C22×C4—28no—
268C22×C4—28no—
269C22×C4—216no—
270C22×C4—216no—
271C22×C4—216no—
272C22×C4—216no—
273C22×C4—48no—
274C22×C4—48no—
275C22×C4—432no—
276C22×C4—432no—
277C22×C4—432no—
278C22×C4—48no—
279C22×C4—416no—
280C22×C4—4128no—
281C22×C4—44no—
282C22×C4—48no—
283C22×C4—416no—
284C22×C4—44no—
285C22×C4—48no—
286C22×C4—432no—
287C22×C4—432no—
288C22×C4—416no—
289C22×C4—416no—
290C22×C4—816no—
291C22×C4—88no—
292C22×C4—88no—
293C22×C4—88no—
294C22×C4—88no—
295C22×C4—816no—
296C22×C4—816no—
297C22×C4C22×D414yes—
298C22×C4C22×D418yes—
299C22×C4H(5,𝔽2)18yes—
300C4⋊C4—88no—
301C4⋊C4—88no—
302C22⋊C4—232no—
303C22⋊C4—432no—
304C22⋊C4—416no—
305C22⋊C4—416no—
306C22⋊C4—416no—
307C22⋊C4—416no—
308C22⋊C4—832no—
309C23—216no—
310C23—28no—
311C23—24no—
312C23—28no—
313C23—24no—
314C23—28no—
315C23—28no—
316C23—28no—
317C23—28no—
318C23—28no—
319C23—28no—
320C23—28no—
321C23—216no—
322C23—28no—
323C23—216no—
324C23—216no—
325C23—216no—
326C23—24no—
327C23—28no—
328C23—216no—
329C23—416no—
330C23—432no—
331C23—432no—
332C23—464no—
333C23—416no—
334C23—464no—
335C23—432no—
336C23—416no—
337C23—42no—
338C23—42no—
339C23—44no—
340C23—48no—
341C23—44no—
342C23—44no—
343C23—48no—
344C23—48no—
345C23—416no—
346C23—832no—
347C23—816no—
348C23—88no—
349C23—88no—
350C23—832no—
351C23#397164yescomplemented
352C23C2×#12114yes—
353C23C2×#12118yes—
354C23C2×#12318yes—
355C23C2×#12318yes—
356C23C2×#12318yes—
357C23C2×#12314yes—
358C23C2×#12318yes—
359C23C2×#12414yes—
360C23C2×#12418yes—
361C23C2×#12618yes—
362C23C2×#12618yes—
363C23C2×#12618yes—
364C23C23×D411yescharacteristic
365C23C23×D411yescharacteristic
366C23C22×#4111yescharacteristic
367C23C22×#4112yes—
368C23C22×#4112yes—
369C23C2×H(5,𝔽2)14yes—
370C23#58514yes—
371D4—464no—
372C2×C4—24no—
373C2×C4—28no—
374C2×C4—28no—
375C2×C4—28no—
376C2×C4—28no—
377C2×C4—28no—
378C2×C4—216no—
379C2×C4—432no—
380C2×C4—464no—
381C2×C4—416no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—48no—
388C2×C4—416no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—416no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4—416no—
397C2×C4—416no—
398C2×C4—416no—
399C2×C4—816no—
400C2×C4—832no—
401C2×C4—832no—
402C22—28no—
403C22—216no—
404C22—216no—
405C22—216no—
406C22—216no—
407C22—24no—
408C22—28no—
409C22—24no—
410C22—28no—
411C22—216no—
412C22—28no—
413C22—216no—
414C22—216no—
415C22—216no—
416C22—464no—
417C22—416no—
418C22—48no—
419C22—44no—
420C22—832no—
421C22—84no—
422C22—82no—
423C22—82no—
424C22—88no—
425C22C2×#39718yes—
426C22C2×#397116yes—
427C22C22×#12111yescharacteristic
428C22C22×#12311yescharacteristic
429C22C22×#12312yes—
430C22C22×#12411yescharacteristic
431C22C22×#12612yes—
432C22#345914yes—
433C22#348412yes—
434C22#348512yes—
435C22#349014yes—
436C22#349114yes—
437C22#349614yes—
438C22#350518yes—
439C4—48no—
440C4—48no—
441C4—44no—
442C4—88no—
443C4—88no—
444C2—28no—
445C2—216no—
446C2—84no—
447C2C22×#39711yescharacteristic
448C2#3498412yes—
449C2#3499514yes—
450C2#3535112yes—
451C2#3536014yes—
452C2#3536711yescharacteristic
453C2#3545511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2315C2semidirect
4. #15019C2semidirect
5. #14820C2semidirect
6. #14737C2semidirect
7. #13794C2semidirect
8. C2×C22⋊#95C2semidirect
15. C2×#397C22semidirect
22. C2×#385C22semidirect
27. C22×#95C22semidirect
28. #1445C22semidirect
29. #1443C22semidirect
30. #1442C22semidirect
31. #1440C22semidirect
32. #1439C22semidirect
35. C22⋊#95C22semidirect
36. C22⋊#95C22semidirect
351. C23#397semidirect

Maximal subgroups

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

subgroupindex
C2×#23152
C2×#23082
C22×#3862
#150192
#148202
#147372
#137942
C2×C22⋊#952