Finite Group Databasebeta

← #4386834

Subgroups — #4386834

Order 512.

1946 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16343C214yesmaximal
2#16239C214yesmaximal, complemented
3#15872C214yesmaximal, complemented
4#15766C214yesmaximal, complemented
5#15629C214yesmaximal, complemented
6#14753C214yesmaximal
7#14674C211yesmaximal, characteristic
8#14297C212yesmaximal
9#14165C211yesmaximal, characteristic
10#13880C212yesmaximal
11#13776C211yesmaximal, characteristic
12C23×C4⋊C4C2211yescharacteristic
13C2×#398C2214yes—
14C2×#398C2214yes—
15C2×#398C2214yes—
16C2×#389C2214yes—
17C2×#387C2214yes—
18C2×#385C2214yes—
19C2×#385C2214yes—
20C2×#384C2212yes—
21C2×#384C2212yes—
22C2×#381C2211yescharacteristic
23C2×#381C2211yescharacteristic
24C2×#380C2212yes—
25C2×C4×C22⋊C4C2212yes—
26C2×C4×C22⋊C4C2212yes—
27#1445C2214yes—
28#1445C2214yes—
29#1444C2214yes—
30#1444C2214yes—
31#1444C2214yes—
32#1444C2214yes—
33#1443C2214yes—
34#1442C2214yes—
35#1442C2214yes—
36#1442C2214yes—
37#1442C2214yes—
38#1441C2214yes—
39#1441C2214yes—
40#1441C2214yes—
41#1441C2214yes—
42#1439C2214yes—
43#1439C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1436C2214yes—
47#1436C2214yes—
48#1436C2214yes—
49#1436C2214yes—
50#1436C2214yes—
51#1436C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54C4⋊#95C2212yes—
55C4⋊#95C2212yes—
56C22⋊#95C2212yes—
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C22×C4⋊C4—24no—
61C22×C4⋊C4—44no—
62C22×C4⋊C4C2314yes—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2314yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2312yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2312yes—
78C22×C42C2312yes—
79C22×C42C2312yes—
80C22×C42C2314yes—
81#398—48no—
82#398—48no—
83#398—48no—
84#389—48no—
85#387—48no—
86#385—48no—
87#385—48no—
88#384—44no—
89#384—44no—
90#381—24no—
91#381Q818yes—
92#380—28no—
93C4×C22⋊C4—28no—
94C4×C22⋊C4—44no—
95C2×#95C2314yes—
96C2×#95C2314yes—
97C2×#95C2314yes—
98C2×#95C2314yes—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C25C2411yescharacteristic
119C23×C4—22no—
120C23×C4—24no—
121C23×C4—22no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—44no—
126C23×C4—44no—
127C23×C4C2×Q814yes—
128C23×C4#4114yes—
129C23×C4#4114yes—
130C23×C4C2414yes—
131C23×C4C2414yes—
132C23×C4C2414yes—
133C23×C4C2411yescharacteristic
134C23×C4C2411yescharacteristic
135C23×C4C2414yes—
136C23×C4C2414yes—
137C23×C4C2412yes—
138C23×C4C2414yes—
139C23×C4C2412yes—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—416no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4C2×Q818yes—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—42no—
192C2×C42—42no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42—44no—
196C2×C42—44no—
197C2×C42—44no—
198#95—28no—
199#95—28no—
200#95—28no—
201#95—28no—
202#95—28no—
203#95—28no—
204#95—28no—
205#95—28no—
206#95—28no—
207#95—28no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95C2×Q8116yes—
221C24—24no—
222C24—22no—
223C24—24no—
224C24—22no—
225C24C22×Q812yes—
226C24C2×#4112yes—
227C24C2×#4112yes—
228C24C2511yescharacteristic
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—28no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—44no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—88no—
289C22×C4—84no—
290C22×C4—84no—
291C22×C4—88no—
292C22×C4#125116yes—
293C22×C4C22×Q818yes—
294C22×C4C22×Q814yes—
295C22×C4C2×#4114yes—
296C22×C4C2×#4114yes—
297C22×C4H(5,𝔽2)18yes—
298C22×C4H(5,𝔽2)14yes—
299C22×C42−1+418yes—
300C4⋊C4—416no—
301C4⋊C4—416no—
302C4⋊C4—48no—
303C4⋊C4—48no—
304C4⋊C4—84no—
305C4⋊C4—84no—
306C4⋊C4—88no—
307C4⋊C4—88no—
308C4⋊C4—88no—
309C4⋊C4—88no—
310C4⋊C4—88no—
311C4⋊C4—88no—
312C22⋊C4—232no—
313C22⋊C4—48no—
314C22⋊C4—48no—
315C22⋊C4—88no—
316C42—48no—
317C42—48no—
318C42—88no—
319C23—24no—
320C23—22no—
321C23—22no—
322C23—28no—
323C23—24no—
324C23—28no—
325C23—24no—
326C23—24no—
327C23—44no—
328C23—44no—
329C23—44no—
330C23—44no—
331C23C2×#12514yes—
332C23C23×Q811yescharacteristic
333C23C22×#4111yescharacteristic
334C23C22×#4111yescharacteristic
335C23C2×H(5,𝔽2)12yes—
336C23C2×H(5,𝔽2)14yes—
337C23C2×2−1+414yes—
338C23#58512yes—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—28no—
347C2×C4—28no—
348C2×C4—28no—
349C2×C4—28no—
350C2×C4—28no—
351C2×C4—28no—
352C2×C4—28no—
353C2×C4—28no—
354C2×C4—28no—
355C2×C4—28no—
356C2×C4—416no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—88no—
400C2×C4—88no—
401C2×C4C2×#12518yes—
402C22—24no—
403C22—28no—
404C22—44no—
405C22—48no—
406C22—48no—
407C22—42no—
408C22—42no—
409C22C22×#12511yescharacteristic
410C22#345912yes—
411C22#349011yescharacteristic
412C22#349111yescharacteristic
413C22#349214yes—
414C22#350112yes—
415C22#350314yes—
416C22#350812yes—
417C22#350912yes—
418C22#353514yes—
419C22#353714yes—
420C22#355214yes—
421C22#355414yes—
422C4—28no—
423C4—48no—
424C4—48no—
425C4—48no—
426C4—44no—
427C4—84no—
428C4—84no—
429C4—84no—
430C4—81no—
431C4—81no—
432C2—44no—
433C2#3499411yescharacteristic
434C2#3555512yes—
435C2#4097814yes—
436C2#4645311yescharacteristic
437C2#4645411yescharacteristic
438C2#4685412yes—
439C2#5069714yes—
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
2. #16239C2semidirect
3. #15872C2semidirect
4. #15766C2semidirect
5. #15629C2semidirect

Maximal subgroups

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

subgroupindex
#163432
#162392
#158722
#157662
#156292
#147532
#146742
#142972
#141652
#138802
#137762