Finite Group Databasebeta

← #5236471

Subgroups — #5236471

Order 512.

1846 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17135C211yesmaximal, characteristic
2#16997C214yesmaximal, complemented
3#16730C214yesmaximal, complemented
4#16302C214yesmaximal, complemented
5#16032C212yesmaximal
6#16014C211yesmaximal, characteristic
7#15685C214yesmaximal, complemented
8#15575C214yesmaximal, complemented
9#15436C214yesmaximal, complemented
10#14662C211yesmaximal, characteristic
11#14410C212yesmaximal
12C22×#124C2211yescharacteristic
13C2×#400C2214yes—
14C2×#398C2214yes—
15C2×#396C2214yes—
16C2×#396C2214yes—
17C2×#389C2214yes—
18C2×#388C2214yes—
19C2×#386C2212yes—
20C2×#385C2212yes—
21C2×#382C2212yes—
22C2×#381C2212yes—
23C2×#380C2214yes—
24#1446C2212yes—
25#1446C2214yes—
26#1445C2214yes—
27#1445C2214yes—
28#1444C2214yes—
29#1444C2214yes—
30#1443C2214yes—
31#1442C2214yes—
32#1442C2214yes—
33#1442C2214yes—
34#1442C2214yes—
35#1441C2214yes—
36#1441C2214yes—
37#1441C2214yes—
38#1441C2214yes—
39#1440C2214yes—
40#1440C2212yes—
41#1440C2214yes—
42#1440C2212yes—
43#1439C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1438C2211yescharacteristic
48#1438C2211yescharacteristic
49#1438C2212yes—
50#1437C2214yes—
51#1436C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1434C2214yes—
56C22⋊#95C2214yes—
57C23×Q8C2311yescharacteristic
58C23×D4C2311yescharacteristic
59C24×C4C2312yes—
60C2×#124—24no—
61C2×#124—44no—
62C22×C4⋊C4C2314yes—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C22⋊C4C2312yes—
70C22×C22⋊C4C2312yes—
71C22×C22⋊C4C2312yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2312yes—
77C22×C22⋊C4C2314yes—
78C22×C42C2311yescharacteristic
79C22×C42C2314yes—
80#400—48no—
81#398—48no—
82#396—48no—
83#396—48no—
84#389—48no—
85#388—48no—
86#386—28no—
87#385—44no—
88#382—44no—
89#381—44no—
90#380—48no—
91C2×#95C2314yes—
92C2×#95C2314yes—
93C2×#95C2314yes—
94C2×#95C2314yes—
95C2×#95C2314yes—
96C2×#95C2314yes—
97C2×#95C2314yes—
98C2×#95C2314yes—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2312yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2312yes—
114C2×#95C2314yes—
115C25C2412yes—
116C22×Q8—22no—
117C22×Q8—42no—
118C22×Q8—44no—
119C22×D4—42no—
120C22×D4—44no—
121C22×D4#4114yes—
122C23×C4—24no—
123C23×C4—28no—
124C23×C4—44no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4C2414yes—
128C23×C4C2414yes—
129C23×C4C2414yes—
130C23×C4C2411yescharacteristic
131C23×C4C2411yescharacteristic
132C23×C4C2414yes—
133C23×C4C2411yescharacteristic
134C23×C4C2414yes—
135C23×C4C2412yes—
136C23×C4C2414yes—
137#124—416no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C42—22no—
184C2×C42—24no—
185C2×C42—44no—
186C2×C42—44no—
187C2×C42—44no—
188#95—28no—
189#95—28no—
190#95—28no—
191#95—28no—
192#95—28no—
193#95—28no—
194#95—28no—
195#95—28no—
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—42no—
205#95—42no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—44no—
212C24—24no—
213C24—24no—
214C24—22no—
215C24—22no—
216C24—24no—
217C24—24no—
218C24—24no—
219C24—24no—
220C24C2×#4114yes—
221C24C2511yescharacteristic
222C2×Q8—44no—
223C2×Q8—84no—
224C2×Q8—88no—
225C2×D4—44no—
226C2×D4—48no—
227C2×D4—88no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
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—24no—
247C22×C4—21no—
248C22×C4—21no—
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—22no—
259C22×C4—22no—
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—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—48no—
281C22×C4—416no—
282C22×C4—88no—
283C22×C4—84no—
284C22×C4—84no—
285C22×C4C2×#4112yes—
286C22×C4H(5,𝔽2)14yes—
287C22×C4H(5,𝔽2)18yes—
288C22×C42−1+418yes—
289C4⋊C4—416no—
290C4⋊C4—88no—
291C4⋊C4—88no—
292C4⋊C4—88no—
293C4⋊C4—88no—
294C4⋊C4—88no—
295C4⋊C4—88no—
296C22⋊C4—48no—
297C22⋊C4—416no—
298C22⋊C4—48no—
299C22⋊C4—48no—
300C22⋊C4—88no—
301C22⋊C4—84no—
302C22⋊C4—84no—
303C22⋊C4—88no—
304C22⋊C4—88no—
305C42—28no—
306C42—88no—
307C23—24no—
308C23—24no—
309C23—24no—
310C23—28no—
311C23—28no—
312C23—44no—
313C23—44no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—44no—
322C23—44no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—44no—
327C23—44no—
328C23C22×#4111yescharacteristic
329C23C2×H(5,𝔽2)12yes—
330C23C2×H(5,𝔽2)14yes—
331C23C2×2−1+414yes—
332C23#58514yes—
333Q8—88no—
334D4—88no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—28no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
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—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—816no—
392C22—48no—
393C22—48no—
394C22—44no—
395C22—44no—
396C22—44no—
397C22—84no—
398C22—84no—
399C22—84no—
400C22—84no—
401C22#349011yescharacteristic
402C22#349112yes—
403C22#352014yes—
404C22#353614yes—
405C22#353712yes—
406C22#353812yes—
407C22#354714yes—
408C22#355114yes—
409C22#355514yes—
410C22#355814yes—
411C22#355914yes—
412C4—48no—
413C4—42no—
414C4—44no—
415C4—48no—
416C4—84no—
417C4—84no—
418C4—84no—
419C4—84no—
420C4—81no—
421C4—81no—
422C2—84no—
423C2#4719511yescharacteristic
424C2#4993314yes—
425C2#5162912yes—
426C2#5351814yes—
427C2#5487814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #16997C2semidirect
3. #16730C2semidirect
4. #16302C2semidirect
7. #15685C2semidirect
8. #15575C2semidirect
9. #15436C2semidirect

Maximal subgroups

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

subgroupindex
#171352
#169972
#167302
#163022
#160322
#160142
#156852
#155752
#154362
#146622
#144102