Finite Group Databasebeta

← #556613

Subgroups — #556613

Order 512.

2422 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic
2C2×#2295C212yesmaximal
3#14634C214yesmaximal
4#14314C214yesmaximal
5#14266C214yesmaximal
6#14261C214yesmaximal
7#14151C211yesmaximal, characteristic
8#14046C214yesmaximal
9#14021C214yesmaximal
10#13926C212yesmaximal
11C4×#384C211yesmaximal, characteristic
12C22×C4×Q8C2212yes—
13#2582C4116yescomplemented
14C2×#398C2214yes—
15#2295—216no—
16C2×#391C2212yes—
17C2×#391C2212yes—
18C2×#391C2214yes—
19C2×#391C2214yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#389C2214yes—
23C2×#389C2214yes—
24C2×#389C2214yes—
25C2×#387C2214yes—
26C2×#387C2214yes—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2214yes—
31C2×C4×C4⋊C4C2214yes—
32C2×C4×C4⋊C4C2212yes—
33C2×#376C2214yes—
34C2×C43C2211yescharacteristic
35#1445C2214yes—
36#1443C2218yes—
37#1443C2218yes—
38#1442C2218yes—
39#1442C2218yes—
40#1442C2214yes—
41#1441C2214yes—
42#1441C2218yes—
43#1441C2218yes—
44#1436C2218yes—
45C4⋊#95C2212yes—
46C4⋊#95C2214yes—
47C4⋊#95C2214yes—
48C4⋊#95C2214yes—
49C4⋊#95C2214yes—
50C4⋊#95C2214yes—
51#1434C2214yes—
52C23×Q8C2312yes—
53C2×C4×Q8—28no—
54C2×C4×Q8—28no—
55C2×C4×Q8C2×C4116yes—
56C22×C4⋊C4C2314yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2314yes—
60C22×C4⋊C4C2314yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2314yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2314yes—
68C22×C42C2314yes—
69C22×C42C2314yes—
70C22×C42C2312yes—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73C22×C42C2314yes—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76#398C2×C4132yes—
77#391—28no—
78#391—28no—
79#391—48no—
80#391C2×C4132yes—
81#389—28no—
82#389—28no—
83#389—48no—
84#389—48no—
85#389C2×C4132yes—
86#387—216no—
87#387—216no—
88#384—24no—
89#384C2×C418yes—
90C4×C4⋊C4—28no—
91C4×C4⋊C4—216no—
92C4×C4⋊C4—44no—
93C4×C4⋊C4—48no—
94#376—216no—
95C2×#95C2314yes—
96C2×#95C2314yes—
97C2×#95C2314yes—
98C2×#95C2314yes—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2318yes—
104C2×#95C2318yes—
105C2×#95C2318yes—
106C2×#95C2318yes—
107C2×#95C2314yes—
108C2×#95C2318yes—
109C2×#95C2318yes—
110C2×#95C2314yes—
111C2×#95C2318yes—
112C43D418yes—
113C22×Q8—24no—
114C22×Q8—24no—
115C22×Q8—24no—
116C22×Q8—24no—
117C22×Q8—24no—
118C22×Q8—24no—
119C22×Q8C22×C418yes—
120C23×C4C2414yes—
121C23×C4C2414yes—
122C23×C4C2414yes—
123C23×C4C2414yes—
124C23×C4C2414yes—
125C23×C4C2414yes—
126C23×C4C2412yes—
127C23×C4C2411yescharacteristic
128C23×C4C2411yescharacteristic
129C23×C4C2411yescharacteristic
130C23×C4C2412yes—
131C4×Q8—264no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—48no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4C22×C418yes—
163C2×C4⋊C4C22×C4116yes—
164C2×C4⋊C4C22×C4116yes—
165C2×C4⋊C4C22×C418yes—
166C2×C4⋊C4C22×C4116yes—
167C2×C42—28no—
168C2×C42—28no—
169C2×C42—24no—
170C2×C42—22no—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—22no—
174C2×C42—24no—
175C2×C42—28no—
176C2×C42—28no—
177C2×C42—28no—
178C2×C42—28no—
179C2×C42—41no—
180C2×C42—41no—
181C2×C42—48no—
182C2×C42C22×C4116yes—
183C2×C42C22×C414yes—
184C2×C42C2×D418yes—
185C2×C42C2×D418yes—
186C2×C42C2×D414yes—
187C2×C42C2×D414yes—
188C2×C42C2×D414yes—
189C2×C42#4114yes—
190#95—28no—
191#95—28no—
192#95—28no—
193#95—28no—
194#95—28no—
195#95—28no—
196#95—28no—
197#95—48no—
198#95—44no—
199#95—48no—
200#95—44no—
201#95—48no—
202#95—48no—
203#95—48no—
204#95—48no—
205#95C22×C4132yes—
206#95C22×C4116yes—
207C24C2511yescharacteristic
208C2×Q8—216no—
209C2×Q8—216no—
210C2×Q8—216no—
211C2×Q8—48no—
212C2×Q8—48no—
213C2×Q8—48no—
214C2×Q8—48no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—28no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—28no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—28no—
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—28no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4C23×C418yes—
267C22×C4C23×C418yes—
268C22×C4C23×C418yes—
269C22×C4C23×C412yes—
270C22×C4C23×C414yes—
271C22×C4C22×D412yes—
272C22×C4C22×D414yes—
273C22×C4C22×D412yes—
274C22×C4C22×D412yes—
275C22×C4C22×D414yes—
276C22×C4C2×#4112yes—
277C22×C4C2×#4114yes—
278C22×C4H(5,𝔽2)12yes—
279C22×C4H(5,𝔽2)14yes—
280C22×C42−1+412yes—
281C22×C42−1+414yes—
282C22×C42−1+414yes—
283C22×C42−1+414yes—
284C4⋊C4—216no—
285C4⋊C4—232no—
286C4⋊C4—232no—
287C4⋊C4—48no—
288C4⋊C4—416no—
289C4⋊C4—48no—
290C4⋊C4—416no—
291C4⋊C4—48no—
292C4⋊C4—416no—
293C4⋊C4—48no—
294C4⋊C4—416no—
295C4⋊C4—88no—
296C42—232no—
297C42—216no—
298C42—28no—
299C42—216no—
300C42—44no—
301C42—416no—
302C42—88no—
303C42C4×D4116yes—
304C23C24×C411yescharacteristic
305C23C23×D411yescharacteristic
306C23C22×#4111yescharacteristic
307C23C2×H(5,𝔽2)12yes—
308C23C2×H(5,𝔽2)11yescharacteristic
309C23C2×2−1+412yes—
310C23C2×2−1+412yes—
311C23C2×2−1+412yes—
312C23C2×2−1+411yescharacteristic
313C23#58511yescharacteristic
314C23#58511yescharacteristic
315Q8—432no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—28no—
327C2×C4—28no—
328C2×C4—28no—
329C2×C4—216no—
330C2×C4—28no—
331C2×C4—28no—
332C2×C4—28no—
333C2×C4—28no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—44no—
351C2×C4—48no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—48no—
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—48no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4C2×C4×D414yes—
377C2×C4C2×C4×D418yes—
378C2×C4#51918yes—
379C2×C4#52018yes—
380C2×C4#53618yes—
381C2×C4#53714yes—
382C2×C4#53718yes—
383C2×C4#54118yes—
384C2×C4#54218yes—
385C22C22×C4×D411yescharacteristic
386C22C2×#51912yes—
387C22C2×#52012yes—
388C22C4×H(5,𝔽2)11yescharacteristic
389C22C4×2−1+411yescharacteristic
390C22C2×#53612yes—
391C22C2×#53711yescharacteristic
392C22C2×#53712yes—
393C22#344911yescharacteristic
394C22C2×#54112yes—
395C22C2×#54212yes—
396C22#346011yescharacteristic
397C22#346111yescharacteristic
398C22#354311yescharacteristic
399C22#354411yescharacteristic
400C22#355011yescharacteristic
401C22#355011yescharacteristic
402C22#355212yes—
403C22#355412yes—
404C22#355412yes—
405C22#355512yes—
406C22#355512yes—
407C22#355612yes—
408C4—24no—
409C4—24no—
410C4—42no—
411C4—48no—
412C4—48no—
413C4—48no—
414C4—44no—
415C4—84no—
416C4—84no—
417C4—84no—
418C4#2323116yes—
419C2C2×#232312yes—
420C2C4×#53711yescharacteristic
421C2#3448812yes—
422C2#3449111yescharacteristic
423C2#3449811yescharacteristic
424C2#3762011yescharacteristic
425C2#3762212yes—
426C2#3762911yescharacteristic
427C2#3763012yes—
428C2#3763811yescharacteristic
429C2#3764111yescharacteristic
subgroupG/H
center, Frattini, socle, prime radical, cosocleC24C25
commutatorC23C24×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2582C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
C2×#22952
#146342
#143142
#142662
#142612
#141512
#140462
#140212
#139262
C4×#3842