Finite Group Databasebeta

← #507710

Subgroups — #507710

Order 512.

688 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#493C211yesmaximal, characteristic, complemented
2#21275C211yesmaximal, characteristic, complemented
3#13537C211yesmaximal, characteristic, complemented
4#13526C211yesmaximal, characteristic, complemented
5#13407C211yesmaximal, characteristic, complemented
6#13325C211yesmaximal, characteristic
7#13130C212yesmaximal, complemented
8#13012C212yesmaximal, complemented
9#13012C212yesmaximal, complemented
10#12935C212yesmaximal, complemented
11#12894C211yesmaximal, characteristic, complemented
12C22×D16C2212yes—
13C2×#493C2211yescharacteristic
14C2×C4×D8—21no—
15C2×C4×D8C2211yescharacteristic
16C2×C4×D8C2212yes—
17C4×#127—21no—
18#2293—21no—
19C2×#366C2212yes—
20C2×#357C2212yes—
21C2×#357C2212yes—
22C2×#357C2212yes—
23C2×#357C2212yes—
24#1970—21no—
25#1970C2211yescharacteristic
26#1970C2211yescharacteristic
27#1970C2212yes—
28#1957—21no—
29#1942C2211yescharacteristic
30#1942C2211yescharacteristic
31#1934—21no—
32#1934—21no—
33#1845C2211yescharacteristic
34C4×#107C2211yescharacteristic
35C4×#102—21no—
36C42×C8C2211yescharacteristic
37#1380C2212yes—
38#1380C2212yes—
39#1374C2211yescharacteristic
40#1374C2212yes—
41#1374C2211yescharacteristic
42#1374C2212yes—
43C22×D8—21no—
44C22×D8C2312yes—
45C22×D8C2311yescharacteristic
46C2×#127—21no—
47C2×C4×D4—21no—
48C2×C4×D4—22no—
49C2×C4×D4—41no—
50C2×C4×D4—41no—
51C2×#117—41no—
52C2×D16—24no—
53C2×D16—24no—
54C2×D16—42no—
55C22×C16C2312yes—
56C22×C16C2312yes—
57#493—24no—
58C4×D8—22no—
59C4×D8—22no—
60C4×D8—42no—
61C4×D8—42no—
62C4×D8—44no—
63C2×#107—21no—
64C2×#107C2312yes—
65C2×#107C2311yescharacteristic
66C2×#107C2311yescharacteristic
67C2×#107C2311yescharacteristic
68C2×#102—21no—
69C2×#102—22no—
70C2×#102—22no—
71C2×#102—21no—
72C2×#102—21no—
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79#390—22no—
80#390—21no—
81#390—21no—
82#390—41no—
83#390—41no—
84#389—21no—
85#389—21no—
86#388—41no—
87#388—41no—
88#384—21no—
89C4×C4⋊C4—21no—
90C4×C22⋊C4—41no—
91C43D411yescharacteristic
92#366—42no—
93#357—24no—
94#357—24no—
95#357—42no—
96#357—42no—
97#340—21no—
98#340—21no—
99#340—21no—
100C22×D4—22no—
101C22×D4—21no—
102C22×D4—41no—
103C22×D4—41no—
104C23×C4—41no—
105C23×C4—42no—
106C2×D8—24no—
107C2×D8—22no—
108C2×D8—22no—
109C2×D8—22no—
110C2×D8—21no—
111C2×D8—22no—
112C2×D8—22no—
113C2×D8—22no—
114C2×D8—22no—
115C2×D8—42no—
116C2×D8—42no—
117C2×D8#4114yes—
118C2×D8#4112yes—
119C22×C8C2×D411yescharacteristic
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C22×C8C2411yescharacteristic
123#127—24no—
124#127—42no—
125C4×D4—24no—
126C4×D4—42no—
127C4×D4—44no—
128C4×D4—44no—
129C4×D4—84no—
130C4×D4—84no—
131#117—84no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—41no—
148C2×C42—41no—
149C2×C42D811yescharacteristic
150C2×C42D811yescharacteristic
151C2×C42D811yescharacteristic
152C2×C42D811yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156D16—48no—
157C2×C16—22no—
158C2×C16—22no—
159C2×C16—22no—
160C2×C16—22no—
161C2×C16—22no—
162C2×C16—22no—
163#107—21no—
164#107—22no—
165#107—22no—
166#107—22no—
167#107—22no—
168#107—42no—
169#107#4112yes—
170#102—24no—
171#102—42no—
172#102—44no—
173#102—44no—
174#102—42no—
175C4×C8—22no—
176C4×C8—21no—
177C4×C8—21no—
178C4×C8—22no—
179C4×C8—22no—
180C4×C8—41no—
181C4×C8—41no—
182#95—41no—
183#95—41no—
184#95—41no—
185C24—41no—
186C24—42no—
187C2×D4—22no—
188C2×D4—24no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—41no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—41no—
195C2×D4—42no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—42no—
199C2×D4—41no—
200C2×D4—42no—
201C2×D4—82no—
202C2×D4—82no—
203C2×D4—82no—
204C2×D4—82no—
205C22×C4—44no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—82no—
211C22×C4—84no—
212C22×C4—84no—
213C22×C4—82no—
214C22×C4—84no—
215C22×C4C2×D811yescharacteristic
216C22×C4C2×D811yescharacteristic
217C22×C4C2×D811yescharacteristic
218C22×C4C2×D811yescharacteristic
219C22×C4C2×D811yescharacteristic
220C22×C4C2×D811yescharacteristic
221C22×C4C22×D411yescharacteristic
222D8—24no—
223D8—24no—
224D8—24no—
225D8—48no—
226D8—42no—
227D8—42no—
228D8—42no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8AGL(1,ℤ/8)12yes—
238C2×C8C2×#4111yescharacteristic
239C2×C8C2×#4112yes—
240C2×C8C2×#4111yescharacteristic
241C2×C8H(5,𝔽2)12yes—
242C4⋊C4—22no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C22⋊C4—44no—
255C22⋊C4—82no—
256C22⋊C4—84no—
257C22⋊C4—84no—
258C22⋊C4—82no—
259C22⋊C4—82no—
260C22⋊C4—84no—
261C22⋊C4—82no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—22no—
269C42—41no—
270C42—41no—
271C42—82no—
272C42D1614yescomplemented
273C16—42no—
274C16—42no—
275C23—42no—
276C23—44no—
277C23—81no—
278C23—82no—
279C23—81no—
280C23—82no—
281C23—82no—
282C23—81no—
283C23—81no—
284C23—81no—
285C23—82no—
286C23—82no—
287C23—82no—
288C23—81no—
289C23C22×D811yescharacteristic
290D4—42no—
291D4—44no—
292D4—44no—
293D4—42no—
294D4—88no—
295D4—88no—
296D4—81no—
297D4—82no—
298D4—82no—
299D4—81no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
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—88no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—82no—
326C2×C4—81no—
327C2×C4—168no—
328C2×C4C2×D1612yes—
329C2×C4C2×D1612yes—
330C2×C4C2×D1612yes—
331C2×C4#50912yes—
332C2×C4#52511yescharacteristic
333C2×C4#52511yescharacteristic
334C2×C4#53511yescharacteristic
335C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
336C2×C4#57511yescharacteristic
337C2×C4#57511yescharacteristic
338C8—22no—
339C8—41no—
340C8—41no—
341C8#55112yes—
342C8#55312yes—
343C22—82no—
344C22—84no—
345C22—84no—
346C22—82no—
347C22—84no—
348C22—82no—
349C22—162no—
350C22—161no—
351C22—161no—
352C22—162no—
353C22—162no—
354C22—161no—
355C22—161no—
356C22—162no—
357C22#310011yescharacteristic
358C22#310011yescharacteristic
359C22#320511yescharacteristic
360C22C22×D1611yescharacteristic
361C22C2×#50911yescharacteristic
362C22#341411yescharacteristic
363C22#341411yescharacteristic
364C4—22no—
365C4—22no—
366C4—22no—
367C4—22no—
368C4—41no—
369C4—41no—
370C4—82no—
371C4—161no—
372C4—161no—
373C4—162no—
374C4#259811yescharacteristic
375C4#311411yescharacteristic
376C4#311511yescharacteristic
377C4#321911yescharacteristic
378C2—162no—
379C2—164no—
380C2#2936411yescharacteristic
381C2#3366011yescharacteristic
382C2#3366011yescharacteristic
383C2#3367411yescharacteristic
384C2#3367511yescharacteristic
385C2#3377111yescharacteristic
386C2#3378511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#13325C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ 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. C4×#493C2semidirect
2. #21275C2semidirect
3. #13537C2semidirect
4. #13526C2semidirect
5. #13407C2semidirect
7. #13130C2semidirect
8. #13012C2semidirect
9. #13012C2semidirect
10. #12935C2semidirect
11. #12894C2semidirect
272. C42D16semidirect

Maximal subgroups

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

subgroupindex
C4×#4932
#212752
#135372
#135262
#134072
#133252
#131302
#130122
#129352
#128942