Finite Group Databasebeta

← #4474581

Subgroups — #4474581

Order 512.

1742 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17256C214yesmaximal, complemented
2#17059C211yesmaximal, characteristic
3#16996C214yesmaximal, complemented
4#16904C211yesmaximal, characteristic
5#16615C214yesmaximal, complemented
6#15460C214yesmaximal, complemented
7#15398C214yesmaximal
8#15386C214yesmaximal
9#14821C212yesmaximal
10#14423C212yesmaximal
11#14339C211yesmaximal, characteristic
12C23×C4⋊C4C2211yescharacteristic
13C2×#398C2212yes—
14C2×#398C2212yes—
15C2×#397C2214yes—
16C2×#389C2214yes—
17C2×#388C2214yes—
18C2×#387C2214yes—
19C2×#385C2214yes—
20C2×#381C2212yes—
21C2×#380C2212yes—
22C2×#380C2212yes—
23#1445C2214yes—
24#1445C2214yes—
25#1445C2214yes—
26#1445C2214yes—
27#1444C2214yes—
28#1444C2214yes—
29#1444C2214yes—
30#1443C2214yes—
31#1443C2214yes—
32#1443C2214yes—
33#1443C2214yes—
34#1442C2214yes—
35#1442C2214yes—
36#1442C2214yes—
37#1441C2214yes—
38#1441C2214yes—
39#1441C2214yes—
40#1441C2214yes—
41#1441C2214yes—
42#1441C2214yes—
43#1441C2214yes—
44#1441C2212yes—
45#1441C2214yes—
46#1441C2212yes—
47#1440C2214yes—
48#1439C2214yes—
49#1436C2214yes—
50#1436C2214yes—
51#1436C2214yes—
52#1436C2214yes—
53C22⋊#95C2211yescharacteristic
54C22⋊#95C2211yescharacteristic
55C22⋊#95C2212yes—
56C22⋊#95C2214yes—
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C22×C4⋊C4—24no—
61C22×C4⋊C4—44no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2314yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2312yes—
76C22×C42C2314yes—
77#398—44no—
78#398—44no—
79#397—48no—
80#389—48no—
81#388—48no—
82#387—48no—
83#385—48no—
84#381—44no—
85#380—28no—
86#380—44no—
87C2×#95C2314yes—
88C2×#95C2314yes—
89C2×#95C2314yes—
90C2×#95C2314yes—
91C2×#95C2314yes—
92C2×#95C2314yes—
93C2×#95C2314yes—
94C2×#95C2314yes—
95C2×#95C2314yes—
96C2×#95C2312yes—
97C2×#95C2312yes—
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—
114C25C2411yescharacteristic
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—44no—
124C23×C4—44no—
125C23×C4#4114yes—
126C23×C4C2414yes—
127C23×C4C2414yes—
128C23×C4C2414yes—
129C23×C4C2414yes—
130C23×C4C2414yes—
131C23×C4C2412yes—
132C23×C4C2411yescharacteristic
133C23×C4C2412yes—
134C23×C4C2414yes—
135C23×C4C2411yescharacteristic
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—416no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—44no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178C2×C42—44no—
179C2×C42—44no—
180C2×C42—44no—
181#95—28no—
182#95—28no—
183#95—28no—
184#95—28no—
185#95—28no—
186#95—28no—
187#95—44no—
188#95—44no—
189#95—44no—
190#95—44no—
191#95—44no—
192#95—44no—
193#95—44no—
194#95—44no—
195#95—44no—
196#95—44no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—44no—
201#95—44no—
202#95—44no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—42no—
207#95—42no—
208C24—21no—
209C24—22no—
210C24—24no—
211C24—21no—
212C24—24no—
213C24C2×#4112yes—
214C24H(5,𝔽2)14yes—
215C24C2511yescharacteristic
216C22×C4—28no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—24no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—21no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—28no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
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—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
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—24no—
275C22×C4—44no—
276C22×C4—48no—
277C22×C4—48no—
278C22×C4—48no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—88no—
282C22×C4—84no—
283C22×C4—88no—
284C22×C4C2×#4114yes—
285C22×C4H(5,𝔽2)14yes—
286C22×C4H(5,𝔽2)12yes—
287C22×C42−1+418yes—
288C22×C42−1+412yes—
289C4⋊C4—48no—
290C4⋊C4—48no—
291C4⋊C4—416no—
292C4⋊C4—416no—
293C4⋊C4—84no—
294C4⋊C4—84no—
295C4⋊C4—88no—
296C4⋊C4—88no—
297C4⋊C4—88no—
298C4⋊C4—88no—
299C22⋊C4—416no—
300C22⋊C4—416no—
301C22⋊C4—84no—
302C22⋊C4—84no—
303C42—88no—
304C23—22no—
305C23—24no—
306C23—22no—
307C23—24no—
308C23—28no—
309C23—24no—
310C23—24no—
311C23—28no—
312C23—44no—
313C23—44no—
314C23—44no—
315C23—44no—
316C23#54014yes—
317C23C22×#4111yescharacteristic
318C23C2×H(5,𝔽2)12yes—
319C23C2×H(5,𝔽2)12yes—
320C23C2×H(5,𝔽2)11yescharacteristic
321C23C2×2−1+411yescharacteristic
322C23C2×2−1+414yes—
323C23#58514yes—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—28no—
331C2×C4—28no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—28no—
335C2×C4—28no—
336C2×C4—416no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
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—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—88no—
388C2×C4—88no—
389C22—24no—
390C22—28no—
391C22—44no—
392C22—48no—
393C22—42no—
394C22—42no—
395C22—48no—
396C22C2×#54011yescharacteristic
397C22#349011yescharacteristic
398C22#349112yes—
399C22#349211yescharacteristic
400C22#350912yes—
401C22#353314yes—
402C22#353514yes—
403C22#353612yes—
404C22#353712yes—
405C22#354114yes—
406C22#355214yes—
407C22#355514yes—
408C22#355914yes—
409C4—48no—
410C4—42no—
411C4—42no—
412C4—44no—
413C4—44no—
414C4—84no—
415C4—84no—
416C4—84no—
417C4—84no—
418C4—84no—
419C2—44no—
420C2#3672711yescharacteristic
421C2#4139512yes—
422C2#4645311yescharacteristic
423C2#4699911yescharacteristic
424C2#5018412yes—
425C2#5585814yes—
426C2#5677014yes—
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
1. #17256C2semidirect
3. #16996C2semidirect
5. #16615C2semidirect
6. #15460C2semidirect

Maximal subgroups

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

subgroupindex
#172562
#170592
#169962
#169042
#166152
#154602
#153982
#153862
#148212
#144232
#143392