Finite Group Databasebeta

← #4452735

Subgroups — #4452735

Order 512.

2422 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2659C211yesmaximal, characteristic
2C22×#398C211yesmaximal, characteristic
3#16835C218yesmaximal, complemented
4#16377C218yesmaximal, complemented
5#15015C212yesmaximal
6#14822C212yesmaximal
7#14822C212yesmaximal
8#14822C212yesmaximal
9#14159C212yesmaximal
10#14139C212yesmaximal
11#13796C211yesmaximal, characteristic
12C22×C22⋊Q8C2212yes—
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2659—28no—
18C2×#400C2214yes—
19C2×#400C2214yes—
20C2×#400C2214yes—
21C2×#398—28no—
22C2×#398C2214yes—
23C2×#398C2218yes—
24C2×#395C2214yes—
25C2×#391C2214yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×C4×C22⋊C4C2212yes—
29C22×#95C2211yescharacteristic
30C22×#95C2211yescharacteristic
31C22×#95C2211yescharacteristic
32#1446C2214yes—
33#1445C2214yes—
34#1445C2214yes—
35#1443C2218yes—
36#1442C2214yes—
37#1442C2218yes—
38#1442C2218yes—
39#1442C2218yes—
40#1441C2218yes—
41#1441C2214yes—
42#1441C2214yes—
43#1441C2214yes—
44#1436C2218yes—
45C4⋊#95C2214yes—
46#1434C2218yes—
47C22⋊#95C2214yes—
48C22⋊#95C2214yes—
49C22⋊#95C2214yes—
50C22⋊#95C2214yes—
51C22⋊#95C2212yes—
52C22⋊#95C2212yes—
53C23×Q8C2312yes—
54C24×C4C2311yescharacteristic
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C24×C4C2311yescharacteristic
61C2×C22⋊Q8—28no—
62C2×C22⋊Q8—28no—
63C2×C22⋊Q8—28no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—44no—
69C22×C4⋊C4—44no—
70C22×C4⋊C4—44no—
71C22×C4⋊C4—44no—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C42C2314yes—
85#400—48no—
86#400—48no—
87#400—48no—
88#398—232no—
89#398—216no—
90#395—216no—
91#391—48no—
92#380—28no—
93#380—28no—
94C4×C22⋊C4—28no—
95C2×#95—24no—
96C2×#95—24no—
97C2×#95D418yes—
98C2×#95C2314yes—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2318yes—
103C2×#95C2318yes—
104C2×#95C2318yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2318yes—
110C2×#95C2318yes—
111C2×#95C2318yes—
112C2×#95C2314yes—
113C2×#95C2318yes—
114C2×#95C2318yes—
115C2×#95C2314yes—
116C25C2411yescharacteristic
117C22×Q8—24no—
118C22×Q8—24no—
119C22×Q8—24no—
120C22×Q8—44no—
121C22×Q8—44no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4C2×D414yes—
147C23×C4C2×D414yes—
148C23×C4C2×D414yes—
149C23×C4#4118yes—
150C23×C4C2414yes—
151C23×C4C2414yes—
152C23×C4C2418yes—
153C23×C4C2412yes—
154C23×C4C2412yes—
155C23×C4C2412yes—
156C23×C4C2412yes—
157C23×C4C2412yes—
158C23×C4C2412yes—
159C23×C4C2412yes—
160C22⋊Q8—432no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—416no—
184C2×C4⋊C4—416no—
185C2×C4⋊C4—416no—
186C2×C4⋊C4—416no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
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×C42—28no—
198C2×C42—44no—
199C2×C42—44no—
200#95—28no—
201#95—28no—
202#95—28no—
203#95—28no—
204#95—28no—
205#95—216no—
206#95—216no—
207#95—216no—
208#95—48no—
209#95—44no—
210#95—48no—
211#95—48no—
212#95—48no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—48no—
217#95C2×D4116yes—
218C24—22no—
219C24—22no—
220C24—22no—
221C24—22no—
222C24C22×D412yes—
223C24C2×#4114yes—
224C242−1+414yes—
225C242−1+414yes—
226C24C2511yescharacteristic
227C2×Q8—416no—
228C2×Q8—48no—
229C2×Q8—84no—
230C2×Q8—84no—
231C2×Q8—88no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—28no—
235C22×C4—28no—
236C22×C4—28no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—28no—
256C22×C4—28no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—28no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—28no—
266C22×C4—28no—
267C22×C4—28no—
268C22×C4—28no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—28no—
272C22×C4—28no—
273C22×C4—28no—
274C22×C4—28no—
275C22×C4—24no—
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—416no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—416no—
286C22×C4—416no—
287C22×C4—416no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—88no—
298C22×C4—88no—
299C22×C4—88no—
300C22×C4#124116yes—
301C22×C4C22×D414yes—
302C22×C4C22×D414yes—
303C22×C4C22×D414yes—
304C22×C4C2×#4114yes—
305C22×C4C2×#4114yes—
306C22×C4H(5,𝔽2)14yes—
307C22×C4H(5,𝔽2)14yes—
308C22×C42−1+414yes—
309C4⋊C4—416no—
310C4⋊C4—416no—
311C4⋊C4—416no—
312C4⋊C4—416no—
313C4⋊C4—48no—
314C4⋊C4—416no—
315C4⋊C4—84no—
316C4⋊C4—84no—
317C4⋊C4—84no—
318C4⋊C4—84no—
319C22⋊C4—416no—
320C22⋊C4—416no—
321C42—88no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—28no—
327C23—28no—
328C23—28no—
329C23—28no—
330C23—28no—
331C23—28no—
332C23C2×#12414yes—
333C23#53614yes—
334C23#53614yes—
335C23#541116yes—
336C23C23×D411yescharacteristic
337C23C22×#4112yes—
338C23C2×H(5,𝔽2)12yes—
339C23C2×H(5,𝔽2)12yes—
340C23C2×2−1+412yes—
341C23C2×2−1+412yes—
342C23C2×2−1+412yes—
343C23#58512yes—
344Q8—816no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—28no—
348C2×C4—28no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—28no—
359C2×C4—28no—
360C2×C4—28no—
361C2×C4—416no—
362C2×C4—416no—
363C2×C4—416no—
364C2×C4—416no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—48no—
373C2×C4—44no—
374C2×C4—48no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—48no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—48no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—48no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4—48no—
397C2×C4—88no—
398C2×C4—88no—
399C2×C4—88no—
400C2×C4C2×#12418yes—
401C22—24no—
402C22—24no—
403C22—24no—
404C22—24no—
405C22—24no—
406C22—24no—
407C22—232no—
408C22#264218yes—
409C22C22×#12411yescharacteristic
410C22C2×#53611yescharacteristic
411C22C2×#53611yescharacteristic
412C22C2×#54114yes—
413C22#348411yescharacteristic
414C22#348411yescharacteristic
415C22#348511yescharacteristic
416C22#348611yescharacteristic
417C22#350514yes—
418C22#351114yes—
419C22#353514yes—
420C22#353614yes—
421C22#353814yes—
422C22#356314yes—
423C4—28no—
424C4—44no—
425C4—44no—
426C4—44no—
427C4—44no—
428C4—44no—
429C4—84no—
430C4—84no—
431C4—82no—
432C4—88no—
433C2—28no—
434C2C2×#264211yescharacteristic
435C2#3537711yescharacteristic
436C2#3559211yescharacteristic
437C2#3670811yescharacteristic
438C2#3671811yescharacteristic
439C2#3672011yescharacteristic
440C2#3695911yescharacteristic
441C2#4067118yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16835C2semidirect
4. #16377C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26592
C22×#3982
#168352
#163772
#150152
#148222
#141592
#141392
#137962