Finite Group Databasebeta

← #2044149

Subgroups — #2044149

Order 512.

1898 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17073C211yesmaximal, characteristic, complemented
2#17070C211yesmaximal, characteristic, complemented
3#16835C214yesmaximal, complemented
4#16096C214yesmaximal, complemented
5#15512C214yesmaximal
6#15490C214yesmaximal
7#15431C214yesmaximal
8#14697C212yesmaximal
9#14682C211yesmaximal, characteristic
10#14446C212yesmaximal, complemented
11#14394C214yesmaximal, complemented
12C23×C42C2211yescharacteristic
13C2×#400C2214yes—
14C2×#398C2214yes—
15C2×#398C2214yes—
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×#387C2214yes—
19C2×#385C2214yes—
20C2×#382C2214yes—
21C2×#382C2212yes—
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×#380C2212yes—
25#1445C2214yes—
26#1444C2214yes—
27#1443C2214yes—
28#1442C2214yes—
29#1442C2214yes—
30#1442C2214yes—
31#1442C2214yes—
32#1442C2214yes—
33#1442C2214yes—
34#1442C2214yes—
35#1441C2214yes—
36#1441C2214yes—
37#1441C2214yes—
38#1441C2214yes—
39#1441C2214yes—
40#1439C2214yes—
41#1439C2214yes—
42#1439C2214yes—
43#1437C2214yes—
44#1436C2214yes—
45#1436C2214yes—
46#1436C2214yes—
47#1436C2214yes—
48#1436C2214yes—
49C4⋊#95C2211yescharacteristic
50C4⋊#95C2211yescharacteristic
51C4⋊#95C2212yes—
52C4⋊#95C2212yes—
53#1434C2214yes—
54#1434C2214yes—
55C22⋊#95C2214yes—
56C22⋊#95C2214yes—
57C24×C4C2311yescharacteristic
58C24×C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2314yes—
61C22×C4⋊C4C2312yes—
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×C22⋊C4C2314yes—
71C22×C22⋊C4C2314yes—
72C22×C22⋊C4C2314yes—
73C22×C42—24no—
74C22×C42—44no—
75C22×C42C2311yescharacteristic
76C22×C42C2312yes—
77C22×C42C2314yes—
78C22×C42C2311yescharacteristic
79#400—48no—
80#398—48no—
81#398—48no—
82#389—28no—
83#389—44no—
84#387—216no—
85#385—48no—
86#382—44no—
87#382—48no—
88#380—28no—
89#380—44no—
90#380—44no—
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×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C25C2411yescharacteristic
117C23×C4—22no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—42no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4C2414yes—
130C23×C4C2414yes—
131C23×C4C2414yes—
132C23×C4C2414yes—
133C23×C4C2414yes—
134C23×C4C2414yes—
135C23×C4C2412yes—
136C23×C4C2411yescharacteristic
137C23×C4C2412yes—
138C23×C4C2411yescharacteristic
139C2×C4⋊C4—24no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
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#4118yes—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C42—22no—
182C2×C42—24no—
183C2×C42—28no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—416no—
187C2×C42—44no—
188C2×C42—42no—
189C2×C42—42no—
190C2×C42—44no—
191C2×C42#4114yes—
192#95—28no—
193#95—28no—
194#95—28no—
195#95—28no—
196#95—28no—
197#95—28no—
198#95—28no—
199#95—28no—
200#95—28no—
201#95—28no—
202#95—28no—
203#95—28no—
204#95—28no—
205#95—44no—
206#95—44no—
207#95—44no—
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—
217C24—22no—
218C24—21no—
219C24—21no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24—22no—
224C242−1+414yes—
225C242−1+412yes—
226C24C2511yescharacteristic
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—216no—
235C22×C4—28no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
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—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—48no—
288C22×C4—48no—
289C22×C4—48no—
290C22×C4—48no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—44no—
294C22×C4—48no—
295C22×C4—44no—
296C22×C4—48no—
297C22×C4C2×#4118yes—
298C22×C4C2×#4112yes—
299C22×C4C2×#4114yes—
300C22×C4C2×#4114yes—
301C22×C4H(5,𝔽2)12yes—
302C22×C4H(5,𝔽2)14yes—
303C22×C4H(5,𝔽2)14yes—
304C22×C42−1+414yes—
305C4⋊C4—48no—
306C4⋊C4—48no—
307C4⋊C4—416no—
308C4⋊C4—84no—
309C4⋊C4—84no—
310C4⋊C4—84no—
311C4⋊C4—84no—
312C4⋊C4—88no—
313C4⋊C4—88no—
314C4⋊C4—88no—
315C4⋊C4—88no—
316C22⋊C4—416no—
317C22⋊C4—88no—
318C22⋊C4—88no—
319C42—28no—
320C42—44no—
321C42—48no—
322C42—416no—
323C23—22no—
324C23—24no—
325C23—24no—
326C23—24no—
327C23—24no—
328C23—24no—
329C23—24no—
330C23—24no—
331C23—22no—
332C23—24no—
333C23—41no—
334C23—42no—
335C23—42no—
336C23—41no—
337C23—41no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—41no—
342C23—42no—
343C23#56414yes—
344C23C22×#4111yescharacteristic
345C23C22×#4112yes—
346C23C2×H(5,𝔽2)12yes—
347C23C2×H(5,𝔽2)12yes—
348C23C2×H(5,𝔽2)11yescharacteristic
349C23C2×2−1+412yes—
350C23C2×2−1+412yes—
351C23C2×2−1+411yescharacteristic
352C23#58512yes—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—28no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—28no—
368C2×C4—28no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—432no—
372C2×C4—416no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
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—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4#54114yes—
420C2×C4#55518yes—
421C22—24no—
422C22—28no—
423C22—44no—
424C22—44no—
425C22—44no—
426C22—42no—
427C22—42no—
428C22—44no—
429C22—44no—
430C22C2×#54111yescharacteristic
431C22#345912yes—
432C22C2×#55512yes—
433C22#348311yescharacteristic
434C22#349011yescharacteristic
435C22#349212yes—
436C22#349211yescharacteristic
437C22#349512yes—
438C22#351512yes—
439C22#352012yes—
440C22#352312yes—
441C22C2×#56411yescharacteristic
442C22#353412yes—
443C22#353612yes—
444C22#353612yes—
445C22#353812yes—
446C22#354911yescharacteristic
447C22#355212yes—
448C22#355311yescharacteristic
449C22#355412yes—
450C22#356112yes—
451C4—28no—
452C4—24no—
453C4—48no—
454C4—44no—
455C4—42no—
456C4—84no—
457C4—84no—
458C4—84no—
459C4—84no—
460C4—84no—
461C2—44no—
462C2#3672811yescharacteristic
463C2#3716612yes—
464C2#3794812yes—
465C2#3882112yes—
466C2#3894312yes—
467C2#4614411yescharacteristic
468C2#4620012yes—
469C2#5320412yes—
470C2#5923311yescharacteristic
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. #17073C2semidirect
2. #17070C2semidirect
3. #16835C2semidirect
4. #16096C2semidirect
10. #14446C2semidirect
11. #14394C2semidirect

Maximal subgroups

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

subgroupindex
#170732
#170702
#168352
#160962
#155122
#154902
#154312
#146972
#146822
#144462
#143942