Finite Group Databasebeta

← #4452726

Subgroups — #4452726

Order 512.

2566 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2659C211yesmaximal, characteristic
2C22×#398C211yesmaximal, characteristic
3#16827C218yesmaximal, complemented
4#16525C218yesmaximal, complemented
5#14914C212yesmaximal
6#14821C212yesmaximal
7#14821C212yesmaximal
8#14720C212yesmaximal
9#13965C212yesmaximal
10#13881C212yesmaximal
11#13796C211yesmaximal, characteristic
12C22×C22⋊Q8C2212yes—
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2659—28no—
18C2×#398—28no—
19C2×#398C2214yes—
20C2×#398C2218yes—
21C2×#398C2214yes—
22C2×#398C2214yes—
23C2×#395C2214yes—
24C2×#395C2214yes—
25C2×#385C2214yes—
26C2×#385C2214yes—
27C2×#382C2214yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×C4×C22⋊C4C2212yes—
31C22×#95C2211yescharacteristic
32C22×#95C2211yescharacteristic
33C22×#95C2211yescharacteristic
34#1446C2214yes—
35#1444C2214yes—
36#1444C2218yes—
37#1444C2214yes—
38#1444C2214yes—
39#1442C2218yes—
40#1441C2218yes—
41#1441C2214yes—
42#1441C2214yes—
43#1441C2218yes—
44#1441C2218yes—
45#1441C2214yes—
46C4⋊#95C2214yes—
47C4⋊#95C2218yes—
48#1434C2218yes—
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—48no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—44no—
68C22×C4⋊C4—44no—
69C22×C4⋊C4—44no—
70C22×C4⋊C4—44no—
71C22×C4⋊C4C2314yes—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2318yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2318yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C42C2314yes—
86#398—232no—
87#398—216no—
88#398—48no—
89#398—48no—
90#395—216no—
91#395—48no—
92#385—48no—
93#385—48no—
94#382—48no—
95#380—28no—
96#380—28no—
97C4×C22⋊C4Q8116yescomplemented
98C2×#95—24no—
99C2×#95—24no—
100C2×#95D418yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2318yes—
106C2×#95C2314yes—
107C2×#95C2318yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2318yes—
111C2×#95C2314yes—
112C2×#95C2318yes—
113C2×#95C2318yes—
114C2×#95C2318yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C25C2411yescharacteristic
118C22×Q8—24no—
119C22×Q8—44no—
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×C4C2×Q818yes—
150C23×C4C2414yes—
151C23×C4C2414yes—
152C23×C4C2412yes—
153C23×C4C2412yes—
154C23×C4C2412yes—
155C23×C4C2418yes—
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—28no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—216no—
176C2×C4⋊C4—216no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—416no—
185C2×C4⋊C4—416no—
186C2×C4⋊C4—416no—
187C2×C4⋊C4—416no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—48no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—48no—
192C2×C4⋊C4—48no—
193C2×C4⋊C4—48no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4C2×Q8116yes—
202C2×C42—44no—
203C2×C42—44no—
204C2×C42C2×Q8116yes—
205#95—28no—
206#95—28no—
207#95—28no—
208#95—28no—
209#95—216no—
210#95—44no—
211#95—48no—
212#95—48no—
213#95—48no—
214#95—44no—
215#95—48no—
216#95—44no—
217#95—44no—
218#95—48no—
219#95C2×D4116yes—
220#95C2×Q8116yes—
221C24—22no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24C22×D412yes—
226C24C22×Q814yes—
227C24H(5,𝔽2)14yes—
228C242−1+414yes—
229C24C2511yescharacteristic
230C2×Q8—48no—
231C2×Q8—84no—
232C2×Q8—84no—
233C2×Q8—816no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—28no—
237C22×C4—28no—
238C22×C4—28no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—28no—
243C22×C4—28no—
244C22×C4—28no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—28no—
256C22×C4—28no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—28no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—28no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—28no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—28no—
277C22×C4—24no—
278C22×C4—28no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—416no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—416no—
287C22×C4—416no—
288C22×C4—416no—
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—44no—
298C22×C4—88no—
299C22×C4—88no—
300C22×C4—88no—
301C22×C4C4⋊Q8116yes—
302C22×C4C22×D414yes—
303C22×C4C22×D414yes—
304C22×C4C22×D414yes—
305C22×C4C22×Q8116yes—
306C22×C4C22×Q814yes—
307C22×C4C22×Q814yes—
308C22×C4H(5,𝔽2)14yes—
309C22×C4H(5,𝔽2)14yes—
310C22×C42−1+414yes—
311C4⋊C4—416no—
312C4⋊C4—416no—
313C4⋊C4—416no—
314C4⋊C4—416no—
315C4⋊C4—48no—
316C4⋊C4—416no—
317C4⋊C4—84no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C4⋊C4—84no—
321C4⋊C4—816no—
322C4⋊C4—816no—
323C22⋊C4—232no—
324C22⋊C4—416no—
325C42—416no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—28no—
331C23—28no—
332C23—28no—
333C23—28no—
334C23—28no—
335C23—28no—
336C23C2×C4⋊Q814yes—
337C23#53514yes—
338C23#53614yes—
339C23#544116yes—
340C23C23×D411yescharacteristic
341C23C23×Q812yes—
342C23C2×H(5,𝔽2)12yes—
343C23C2×H(5,𝔽2)12yes—
344C23C2×H(5,𝔽2)12yes—
345C23C2×2−1+412yes—
346C23C2×2−1+412yes—
347C23#58512yes—
348Q8—816no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—28no—
352C2×C4—28no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—216no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—216no—
364C2×C4—216no—
365C2×C4—28no—
366C2×C4—28no—
367C2×C4—28no—
368C2×C4—416no—
369C2×C4—416no—
370C2×C4—416no—
371C2×C4—416no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—48no—
379C2×C4—44no—
380C2×C4—48no—
381C2×C4—48no—
382C2×C4—48no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—48no—
386C2×C4—48no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—48no—
397C2×C4—48no—
398C2×C4—48no—
399C2×C4—48no—
400C2×C4—48no—
401C2×C4—88no—
402C2×C4—88no—
403C2×C4—88no—
404C2×C4C2×C4⋊Q818yes—
405C22—24no—
406C22—24no—
407C22—24no—
408C22—24no—
409C22—24no—
410C22—232no—
411C22—24no—
412C22#266018yes—
413C22C22×C4⋊Q811yescharacteristic
414C22C2×#53511yescharacteristic
415C22C2×#53611yescharacteristic
416C22C2×#54414yes—
417C22#348411yescharacteristic
418C22#348411yescharacteristic
419C22#348511yescharacteristic
420C22#348611yescharacteristic
421C22#350114yes—
422C22#350314yes—
423C22#353414yes—
424C22#353614yes—
425C22#353614yes—
426C22#356314yes—
427C4—28no—
428C4—44no—
429C4—44no—
430C4—44no—
431C4—44no—
432C4—44no—
433C4—416no—
434C4—84no—
435C4—82no—
436C4—84no—
437C2—28no—
438C2C2×#266011yescharacteristic
439C2#3539111yescharacteristic
440C2#3557611yescharacteristic
441C2#3670711yescharacteristic
442C2#3671811yescharacteristic
443C2#3672111yescharacteristic
444C2#3695911yescharacteristic
445C2#4061918yes—
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. #16827C2semidirect
4. #16525C2semidirect
97. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26592
C22×#3982
#168272
#165252
#149142
#148212
#147202
#139652
#138812
#137962