Finite Group Databasebeta

← #499448

Subgroups — #499448

Order 512.

744 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D16C212yesmaximal, complemented
2C4×#493C211yesmaximal, characteristic, complemented
3#21275C211yesmaximal, characteristic, complemented
4#13505C211yesmaximal, characteristic, complemented
5#13480C211yesmaximal, characteristic, complemented
6#13474C212yesmaximal, complemented
7#13448C212yesmaximal, complemented
8#13407C211yesmaximal, characteristic, complemented
9#13325C211yesmaximal, characteristic
10C4×C4⋊C16C211yesmaximal, characteristic, complemented
11C4×#357C212yesmaximal, complemented
12C22×D16C2212yes—
13C2×#493C2211yescharacteristic
14C2×C4×D8—21no—
15C2×C4×D8C2212yes—
16C2×C4×D8C2211yescharacteristic
17C4×#127—21no—
18#2293—21no—
19C4×D16—28no—
20C2×#366C2212yes—
21C2×C4⋊C16C2212yes—
22C2×#357C2212yes—
23C2×#357C2212yes—
24C2×#357C2212yes—
25C2×#357C2212yes—
26C2×C4×C16C2212yes—
27#1970—21no—
28#1970C2211yescharacteristic
29#1970C2211yescharacteristic
30#1970C2212yes—
31#1957—21no—
32#1942C2211yescharacteristic
33#1942C2211yescharacteristic
34#1934—21no—
35#1934—21no—
36#1845C2211yescharacteristic
37C4×#107C2211yescharacteristic
38C4×#102—21no—
39C42×C8C2211yescharacteristic
40#1380C2212yes—
41#1380C2212yes—
42#1374C2211yescharacteristic
43#1374C2211yescharacteristic
44C22×D8—21no—
45C22×D8C2311yescharacteristic
46C22×D8C2312yes—
47C2×#127—21no—
48C2×C4×D4—21no—
49C2×C4×D4—22no—
50C2×C4×D4—41no—
51C2×C4×D4—41no—
52C2×#117—41no—
53C2×D16—24no—
54C2×D16—24no—
55C2×D16—24no—
56C22×C16C2312yes—
57C22×C16C2312yes—
58#493—24no—
59C4×D8—24no—
60C4×D8—22no—
61C4×D8—24no—
62C4×D8—44no—
63C4×D8D414yescomplemented
64C2×#107—21no—
65C2×#107C2312yes—
66C2×#107C2311yescharacteristic
67C2×#107C2311yescharacteristic
68C2×#107C2311yescharacteristic
69C2×#102—21no—
70C2×#102—22no—
71C2×#102—22no—
72C2×#102—21no—
73C2×#102—21no—
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8D411yescharacteristic
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2311yescharacteristic
80#390—22no—
81#390—21no—
82#390—21no—
83#390—41no—
84#390—41no—
85#389—21no—
86#389—21no—
87#388—41no—
88#388—41no—
89#384—21no—
90C4×C4⋊C4—21no—
91C4×C22⋊C4—41no—
92C43D411yescharacteristic
93#366—24no—
94C4⋊C16—24no—
95#357—24no—
96#357—24no—
97#357—24no—
98#357—24no—
99C4×C16—24no—
100#340—21no—
101#340—21no—
102#340—21no—
103C22×D4—22no—
104C22×D4—21no—
105C22×D4—41no—
106C22×D4—41no—
107C23×C4—41no—
108C23×C4—42no—
109C2×D8—24no—
110C2×D8—22no—
111C2×D8—22no—
112C2×D8—22no—
113C2×D8—22no—
114C2×D8—22no—
115C2×D8—22no—
116C2×D8—21no—
117C2×D8—22no—
118C2×D8—42no—
119C2×D8—42no—
120C2×D8C2×D412yes—
121C2×D8#4114yes—
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D411yescharacteristic
124C22×C8C2×D411yescharacteristic
125C22×C8C2411yescharacteristic
126#127—24no—
127#127—42no—
128C4×D4—24no—
129C4×D4—44no—
130C4×D4—42no—
131C4×D4—44no—
132C4×D4—84no—
133C4×D4—84no—
134#117—84no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C42—41no—
152C2×C42D811yescharacteristic
153C2×C42D811yescharacteristic
154C2×C42D811yescharacteristic
155C2×C42D811yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159D16—216no—
160C2×C16—22no—
161C2×C16—22no—
162C2×C16—22no—
163C2×C16—22no—
164C2×C16—22no—
165C2×C16#4114yes—
166#107—21no—
167#107—22no—
168#107—22no—
169#107—22no—
170#107—22no—
171#107—42no—
172#107C2×D412yes—
173#102—24no—
174#102—42no—
175#102—44no—
176#102—44no—
177#102—42no—
178C4×C8—22no—
179C4×C8—22no—
180C4×C8—22no—
181C4×C8—22no—
182C4×C8—41no—
183C4×C8C2×D412yes—
184C4×C8#4112yes—
185#95—41no—
186#95—41no—
187#95—41no—
188C24—41no—
189C24—42no—
190C2×D4—22no—
191C2×D4—24no—
192C2×D4—44no—
193C2×D4—44no—
194C2×D4—42no—
195C2×D4—41no—
196C2×D4—42no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—42no—
204C2×D4—82no—
205C2×D4—82no—
206C2×D4—82no—
207C2×D4—82no—
208C22×C4—41no—
209C22×C4—44no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4—84no—
216C22×C4—84no—
217C22×C4—84no—
218C22×C4C2×D811yescharacteristic
219C22×C4C2×D811yescharacteristic
220C22×C4C2×D811yescharacteristic
221C22×C4C2×D811yescharacteristic
222C22×C4C2×D811yescharacteristic
223C22×C4C2×D811yescharacteristic
224C22×C4C22×D411yescharacteristic
225D8—24no—
226D8—24no—
227D8—24no—
228D8—48no—
229D8—42no—
230D8—42no—
231D8—42no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8#12112yes—
240C2×C8#13512yes—
241C2×C8C22×D411yescharacteristic
242C2×C8C2×#4112yes—
243C2×C8C2×#4112yes—
244C2×C8C2×#4111yescharacteristic
245C4⋊C4—22no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—41no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C22⋊C4—44no—
258C22⋊C4—82no—
259C22⋊C4—84no—
260C22⋊C4—84no—
261C22⋊C4—82no—
262C22⋊C4—82no—
263C22⋊C4—84no—
264C22⋊C4—82no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—22no—
270C42—22no—
271C42—41no—
272C42—82no—
273C42D1614yescomplemented
274C42#12112yes—
275C42AGL(1,ℤ/8)12yes—
276C16—24no—
277C16—42no—
278C23—44no—
279C23—42no—
280C23—81no—
281C23—81no—
282C23—82no—
283C23—81no—
284C23—82no—
285C23—82no—
286C23—81no—
287C23—81no—
288C23—82no—
289C23—81no—
290C23—82no—
291C23—82no—
292C23C22×D811yescharacteristic
293D4—44no—
294D4—42no—
295D4—44no—
296D4—42no—
297D4—88no—
298D4—88no—
299D4—81no—
300D4—82no—
301D4—81no—
302D4—82no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—88no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—82no—
322C2×C4—81no—
323C2×C4—82no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—82no—
327C2×C4—168no—
328C2×C4#45912yes—
329C2×C4#45912yes—
330C2×C4C2×D1612yes—
331C2×C4C2×D1612yes—
332C2×C4C2×D1612yes—
333C2×C4#50812yes—
334C2×C4#50912yes—
335C2×C4C2×#12111yescharacteristic
336C2×C4#52511yescharacteristic
337C2×C4#53211yescharacteristic
338C2×C4C2×#13511yescharacteristic
339C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
340C2×C4#57511yescharacteristic
341C8—22no—
342C8—22no—
343C8—41no—
344C8#52512yes—
345C8#52912yes—
346C22—82no—
347C22—82no—
348C22—84no—
349C22—82no—
350C22—84no—
351C22—84no—
352C22—162no—
353C22—162no—
354C22—161no—
355C22—161no—
356C22—161no—
357C22—162no—
358C22—162no—
359C22—161no—
360C22C2×#45911yescharacteristic
361C22#310011yescharacteristic
362C22#316911yescharacteristic
363C22C22×D1611yescharacteristic
364C22C2×#50811yescharacteristic
365C22C2×#50911yescharacteristic
366C22#341411yescharacteristic
367C4—22no—
368C4—22no—
369C4—22no—
370C4—22no—
371C4—41no—
372C4—82no—
373C4—162no—
374C4—161no—
375C4—161no—
376C4#220614yes—
377C4#239211yescharacteristic
378C4#303511yescharacteristic
379C4#310511yescharacteristic
380C4#317111yescharacteristic
381C2—162no—
382C2—164no—
383C2#2367611yescharacteristic
384C2C2×#220611yescharacteristic
385C2#3357911yescharacteristic
386C2#3366011yescharacteristic
387C2#3366511yescharacteristic
388C2#3373311yescharacteristic
389C2#3373711yescharacteristic
subgroupG/H
centerC22×C4C2×D8
commutator, Frattini, cosocleC22×C8C24
socleC23C22×D8
prime radical#13325C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C4×D16C2semidirect
2. C4×#493C2semidirect
3. #21275C2semidirect
4. #13505C2semidirect
5. #13480C2semidirect
6. #13474C2semidirect
7. #13448C2semidirect
8. #13407C2semidirect
10. C4×C4⋊C16C2semidirect
11. C4×#357C2semidirect
63. C4×D8D4semidirect
273. C42D16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D162
C4×#4932
#212752
#135052
#134802
#134742
#134482
#134072
#133252
C4×C4⋊C162
C4×#3572