Finite Group Databasebeta

← #4416995

Subgroups — #4416995

Order 512.

2098 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C212yesmaximal
2#17044C218yesmaximal, complemented
3#16675C214yesmaximal
4#15239C212yesmaximal, complemented
5#15239C212yesmaximal, complemented
6#15233C212yesmaximal
7#15228C212yesmaximal, complemented
8#15228C212yesmaximal, complemented
9#15195C211yesmaximal, characteristic
10#15150C211yesmaximal, characteristic
11#14627C214yesmaximal
12C4×#381C211yesmaximal, characteristic
13C22×#125C2212yes—
14C23×C42C2211yescharacteristic
15#2851—216no—
16C2×#397C2214yes—
17C2×#389C2212yes—
18C2×#389C2212yes—
19C2×#387C2214yes—
20C2×#387C2214yes—
21C2×#385C2214yes—
22C2×#385C2214yes—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#382C2212yes—
26C2×#382C2212yes—
27C2×#381C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32#1445C2214yes—
33#1444C2218yes—
34#1444C2218yes—
35#1443C2218yes—
36#1443C2218yes—
37#1443C2218yes—
38#1442C2218yes—
39#1442C2218yes—
40#1442C2214yes—
41#1441C2218yes—
42#1441C2218yes—
43#1440C2218yes—
44#1439C2214yes—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48C4⋊#95C2211yescharacteristic
49C4⋊#95C2212yes—
50C4⋊#95C2212yes—
51C4⋊#95C2212yes—
52#1434C2214yes—
53C22⋊#95C2212yes—
54C4×#95C2212yes—
55C4×#95C2212yes—
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C2×#125—28no—
60C2×#125—44no—
61C2×#125—44no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2314yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2312yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2312yes—
76C22×C42—42no—
77C22×C42—42no—
78C22×C42D418yes—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2312yes—
83C22×C42C2312yes—
84#397—216no—
85#389—28no—
86#389—28no—
87#387—216no—
88#387—216no—
89#385—216no—
90#385—216no—
91#384—42no—
92#384—42no—
93#382—28no—
94#382—44no—
95#381—28no—
96#380—28no—
97#380—28no—
98C4×C22⋊C4—42no—
99C4×C22⋊C4—42no—
100C2×#95C2318yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2312yes—
113C2×#95C2312yes—
114C2×#95C2318yes—
115C2×#95C2314yes—
116C2×#95C2318yes—
117C2×#95C2318yes—
118C2×#95C2318yes—
119C2×#95C2318yes—
120C25C2411yescharacteristic
121C23×C4—24no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—44no—
127C23×C4—42no—
128C23×C4—41no—
129C23×C4—41no—
130C23×C4—41no—
131C23×C4—41no—
132C23×C4C2×D414yes—
133C23×C4C2×D414yes—
134C23×C4C2×D414yes—
135C23×C4C2414yes—
136C23×C4C2414yes—
137C23×C4C2414yes—
138C23×C4C2414yes—
139C23×C4C2414yes—
140C23×C4C2412yes—
141C23×C4C2412yes—
142C23×C4C2411yescharacteristic
143C23×C4C2412yes—
144C23×C4C2412yes—
145C23×C4C2411yescharacteristic
146#125—432no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—48no—
165C2×C4⋊C4—48no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—48no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—216no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—48no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4#4118yes—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—416no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42C2×D418yes—
197C2×C42C2×D418yes—
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—216no—
208#95—44no—
209#95—48no—
210#95—44no—
211#95—44no—
212#95—48no—
213#95—48no—
214#95—48no—
215#95—42no—
216#95—42no—
217#95—48no—
218C24—21no—
219C24—21no—
220C24—22no—
221C24—22no—
222C24—24no—
223C24—22no—
224C24C22×D412yes—
225C24C2×#4114yes—
226C24C2511yescharacteristic
227C22×C4—28no—
228C22×C4—24no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—28no—
249C22×C4—216no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—28no—
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—28no—
271C22×C4—24no—
272C22×C4—28no—
273C22×C4—28no—
274C22×C4—28no—
275C22×C4—28no—
276C22×C4—28no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—48no—
281C22×C4—48no—
282C22×C4—48no—
283C22×C4—48no—
284C22×C4—48no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—48no—
288C22×C4C22×D412yes—
289C22×C4C22×D414yes—
290C22×C4C22×D414yes—
291C22×C4C22×D412yes—
292C22×C4C2×#4114yes—
293C22×C4C2×#4114yes—
294C22×C4H(5,𝔽2)12yes—
295C22×C4H(5,𝔽2)12yes—
296C22×C42−1+414yes—
297C22×C42−1+414yes—
298C4⋊C4—416no—
299C4⋊C4—416no—
300C4⋊C4—416no—
301C4⋊C4—416no—
302C4⋊C4—48no—
303C4⋊C4—48no—
304C4⋊C4—48no—
305C4⋊C4—48no—
306C4⋊C4—88no—
307C4⋊C4—88no—
308C22⋊C4—48no—
309C22⋊C4—416no—
310C22⋊C4—48no—
311C22⋊C4—416no—
312C42—216no—
313C42—48no—
314C42—44no—
315C42—44no—
316C42—48no—
317C23—22no—
318C23—24no—
319C23—24no—
320C23—22no—
321C23—24no—
322C23—24no—
323C23—28no—
324C23—28no—
325C23—41no—
326C23—41no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—44no—
331C23—44no—
332C23#53214yes—
333C23C23×D411yescharacteristic
334C23C22×#4112yes—
335C23C2×H(5,𝔽2)11yescharacteristic
336C23C2×H(5,𝔽2)11yescharacteristic
337C23C2×2−1+412yes—
338C23C2×2−1+412yes—
339C23#58512yes—
340C23#58514yes—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—28no—
352C2×C4—28no—
353C2×C4—416no—
354C2×C4—416no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—416no—
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—48no—
382C2×C4—44no—
383C2×C4—48no—
384C2×C4—48no—
385C2×C4—48no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—44no—
391C2×C4—48no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4#53414yes—
397C2×C4#53618yes—
398C2×C4#56314yes—
399C2×C4#56314yes—
400C2×C4#56318yes—
401C22—28no—
402C22—24no—
403C22—42no—
404C22—42no—
405C22—44no—
406C22—44no—
407C22—48no—
408C22—44no—
409C22C2×#53211yescharacteristic
410C22C2×#53411yescharacteristic
411C22C2×#53612yes—
412C22#344912yes—
413C22#344911yescharacteristic
414C22#350512yes—
415C22#350612yes—
416C22#351514yes—
417C22#351514yes—
418C22C2×#56311yescharacteristic
419C22C2×#56311yescharacteristic
420C22C2×#56312yes—
421C22#353912yes—
422C22#354012yes—
423C22#356514yes—
424C22#356514yes—
425C4—42no—
426C4—44no—
427C4—44no—
428C4—44no—
429C4—44no—
430C4—84no—
431C4—84no—
432C4—84no—
433C4—84no—
434C4—84no—
435C4#281718yes—
436C2—44no—
437C2#3503211yescharacteristic
438C2#3513912yes—
439C2C2×#281711yescharacteristic
440C2#3745511yescharacteristic
441C2#3754512yes—
442C2#4568412yes—
443C2#4569612yes—
444C2#4570914yes—
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. #17044C2semidirect
4. #15239C2semidirect
5. #15239C2semidirect
7. #15228C2semidirect
8. #15228C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
#170442
#166752
#152392
#152332
#152282
#151952
#151502
#146272
C4×#3812