Finite Group Databasebeta

← #769073

Subgroups — #769073

Order 512.

2618 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#398C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic
3#15811C218yesmaximal, complemented
4#15766C218yesmaximal, complemented
5#15014C212yesmaximal
6#14821C212yesmaximal
7#14128C212yesmaximal
8#14128C212yesmaximal
9#13923C212yesmaximal
10#13880C212yesmaximal
11#13776C211yesmaximal, characteristic
12C22×#117C2212yes—
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#398—28no—
18C2×#398C2218yes—
19C2×#398C2214yes—
20#2316—28no—
21C2×#389C2214yes—
22C2×#389C2214yes—
23C2×#385C2214yes—
24C2×#385C2214yes—
25C2×#384C2214yes—
26C2×#381C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×C4×C22⋊C4C2212yes—
30C22×#95C2211yescharacteristic
31C22×#95C2211yescharacteristic
32C22×#95C2211yescharacteristic
33#1445C2214yes—
34#1444C2214yes—
35#1444C2214yes—
36#1444C2218yes—
37#1444C2214yes—
38#1442C2214yes—
39#1442C2214yes—
40#1442C2218yes—
41#1441C2214yes—
42#1441C2214yes—
43#1441C2218yes—
44#1441C2214yes—
45#1436C2218yes—
46#1436C2218yes—
47#1436C2218yes—
48#1436C2218yes—
49C4⋊#95C2214yes—
50C22⋊#95C2214yes—
51C22⋊#95C2214yes—
52C22⋊#95C2212yes—
53C24×C4C2311yescharacteristic
54C24×C4C2311yescharacteristic
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C2×#117—28no—
61C2×#117—48no—
62C22×C4⋊C4—28no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—28no—
67C22×C4⋊C4—44no—
68C22×C4⋊C4—44no—
69C22×C4⋊C4D418yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2314yes—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2318yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2318yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C42C2314yes—
85C22×C42C2314yes—
86#398—232no—
87#398—48no—
88#389—216no—
89#389—216no—
90#385—48no—
91#385—48no—
92#384—48no—
93#381Q8116yes—
94#380—28no—
95#380—28no—
96C4×C22⋊C4—28no—
97C2×#95—24no—
98C2×#95—24no—
99C2×#95—24no—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2318yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2318yes—
108C2×#95C2314yes—
109C2×#95C2318yes—
110C2×#95C2314yes—
111C2×#95C2318yes—
112C2×#95C2318yes—
113C2×#95C2318yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C25C2411yescharacteristic
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—24no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—42no—
140C23×C4C2×D414yes—
141C23×C4C2×D414yes—
142C23×C4C2×D414yes—
143C23×C4C2×Q818yes—
144C23×C4#4118yes—
145C23×C4C2414yes—
146C23×C4C2414yes—
147C23×C4C2412yes—
148C23×C4C2412yes—
149C23×C4C2412yes—
150C23×C4C2418yes—
151C23×C4C2412yes—
152C23×C4C2412yes—
153C23×C4C2412yes—
154C23×C4C2412yes—
155#117—432no—
156C2×C4⋊C4—232no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—28no—
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—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—216no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—416no—
178C2×C4⋊C4—416no—
179C2×C4⋊C4—416no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—48no—
186C2×C4⋊C4—48no—
187C2×C4⋊C4—48no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4C2×D4116yes—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4C2×Q8116yes—
196C2×C42—28no—
197C2×C42—28no—
198C2×C42—44no—
199C2×C42—44no—
200C2×C42—44no—
201C2×C42—44no—
202#95—28no—
203#95—28no—
204#95—28no—
205#95—28no—
206#95—28no—
207#95—28no—
208#95—216no—
209#95—44no—
210#95—48no—
211#95—48no—
212#95—48no—
213#95—48no—
214#95—44no—
215#95—48no—
216#95C2×Q8116yes—
217#95C2×Q8116yes—
218C24—22no—
219C24—22no—
220C24—22no—
221C24—22no—
222C24C22×D412yes—
223C24C22×Q814yes—
224C24C2×#4114yes—
225C24H(5,𝔽2)14yes—
226C24C2511yescharacteristic
227C22×C4—28no—
228C22×C4—28no—
229C22×C4—232no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—28no—
235C22×C4—28no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—28no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—28no—
242C22×C4—28no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—28no—
247C22×C4—22no—
248C22×C4—28no—
249C22×C4—24no—
250C22×C4—28no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—28no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—28no—
268C22×C4—24no—
269C22×C4—28no—
270C22×C4—28no—
271C22×C4—24no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—416no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—416no—
278C22×C4—416no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—88no—
290C22×C4—88no—
291C22×C4—88no—
292C22×C4#123116yes—
293C22×C4C4⋊Q8116yes—
294C22×C4C22×D414yes—
295C22×C4C22×D414yes—
296C22×C4C22×D414yes—
297C22×C4C22×Q8116yes—
298C22×C4C22×Q814yes—
299C22×C4C22×Q814yes—
300C22×C4C2×#4114yes—
301C22×C4C2×#4114yes—
302C22×C4H(5,𝔽2)14yes—
303C22×C42−1+414yes—
304C4⋊C4—232no—
305C4⋊C4—416no—
306C4⋊C4—416no—
307C4⋊C4—48no—
308C4⋊C4—48no—
309C4⋊C4—416no—
310C4⋊C4—84no—
311C4⋊C4—84no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C4⋊C4—816no—
315C4⋊C4—816no—
316C22⋊C4—232no—
317C22⋊C4—416no—
318C42—416no—
319C42—416no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—28no—
325C23—28no—
326C23—28no—
327C23—28no—
328C23—28no—
329C23—28no—
330C23C2×#12314yes—
331C23C2×C4⋊Q814yes—
332C23#53514yes—
333C23#554116yes—
334C23C23×D411yescharacteristic
335C23C23×Q812yes—
336C23C22×#4112yes—
337C23C2×H(5,𝔽2)12yes—
338C23C2×H(5,𝔽2)12yes—
339C23C2×2−1+412yes—
340C23#58512yes—
341C23#58512yes—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—28no—
349C2×C4—28no—
350C2×C4—232no—
351C2×C4—24no—
352C2×C4—216no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—28no—
358C2×C4—216no—
359C2×C4—216no—
360C2×C4—28no—
361C2×C4—416no—
362C2×C4—416no—
363C2×C4—416no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—48no—
369C2×C4—44no—
370C2×C4—48no—
371C2×C4—48no—
372C2×C4—48no—
373C2×C4—44no—
374C2×C4—48no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—48no—
392C2×C4—48no—
393C2×C4—88no—
394C2×C4—88no—
395C2×C4—88no—
396C2×C4C2×#12318yes—
397C2×C4C2×C4⋊Q818yes—
398C22—24no—
399C22—24no—
400C22—24no—
401C22—24no—
402C22—24no—
403C22—232no—
404C22—24no—
405C22#250218yes—
406C22C22×#12311yescharacteristic
407C22C22×C4⋊Q811yescharacteristic
408C22C2×#53511yescharacteristic
409C22C2×#55414yes—
410C22#348411yescharacteristic
411C22#348511yescharacteristic
412C22#348611yescharacteristic
413C22#348711yescharacteristic
414C22#349514yes—
415C22#350314yes—
416C22#350414yes—
417C22#350514yes—
418C22#353414yes—
419C22#355914yes—
420C4—28no—
421C4—28no—
422C4—44no—
423C4—44no—
424C4—44no—
425C4—416no—
426C4—82no—
427C4—84no—
428C4—84no—
429C4—82no—
430C2—28no—
431C2C2×#250211yescharacteristic
432C2#3535111yescharacteristic
433C2#3557611yescharacteristic
434C2#3564011yescharacteristic
435C2#3567211yescharacteristic
436C2#3671011yescharacteristic
437C2#3695811yescharacteristic
438C2#4393718yes—
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. #15811C2semidirect
4. #15766C2semidirect

Maximal subgroups

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

subgroupindex
C22×#3982
C2×#23162
#158112
#157662
#150142
#148212
#141282
#139232
#138802
#137762