Finite Group Databasebeta

← #505567

Subgroups — #505567

Order 512.

686 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21224C211yesmaximal, characteristic
2#20817C211yesmaximal, characteristic, complemented
3#13531C211yesmaximal, characteristic, complemented
4#13526C211yesmaximal, characteristic, complemented
5#13481C211yesmaximal, characteristic, complemented
6#13480C211yesmaximal, characteristic, complemented
7#13133C212yesmaximal, complemented
8#13131C212yesmaximal, complemented
9#12921C212yesmaximal, complemented
10#12844C211yesmaximal, characteristic, complemented
11#12733C212yesmaximal
12C22×#135C2211yescharacteristic
13C2×#427C2211yescharacteristic
14C2×#416—21no—
15C22×C4×C8C2211yescharacteristic
16#2297—21no—
17#2285—21no—
18C2×#367C2212yes—
19C2×#366C2212yes—
20C2×#358C2212yes—
21C2×#358C2212yes—
22C2×#357C2212yes—
23C2×#357C2212yes—
24C2×C22⋊C16C2212yes—
25#1970C2211yescharacteristic
26#1970C2211yescharacteristic
27#1969C2211yescharacteristic
28#1969C2211yescharacteristic
29#1968—21no—
30#1968—21no—
31#1845C2211yescharacteristic
32#1845C2211yescharacteristic
33#1834—21no—
34#1808—21no—
35#1808C2212yes—
36C4×#107—21no—
37C4×#107C2212yes—
38#1380C2212yes—
39#1380C2212yes—
40#1374C2212yes—
41#1374C2211yescharacteristic
42#1374C2211yescharacteristic
43C22×#41—21no—
44C2×#135—22no—
45C2×#135—22no—
46C2×#135—24no—
47C2×#135—42no—
48C22×Q16C2311yescharacteristic
49C22×SD16—21no—
50C22×D8C2311yescharacteristic
51C23×C8D411yescharacteristic
52C23×C8C2311yescharacteristic
53C2×#117—21no—
54C2×#117—41no—
55C22×C42D411yescharacteristic
56C22×C16C2312yes—
57C22×C16C2312yes—
58#427—42no—
59C2×#107—21no—
60C2×#107—21no—
61C2×#107C2312yes—
62C2×#107C2312yes—
63C2×#107C2311yescharacteristic
64C2×#107C2311yescharacteristic
65C2×#106—21no—
66#416—44no—
67C2×#103—21no—
68C2×#103—21no—
69C2×#102—21no—
70C2×#102—21no—
71C2×C4×C8—24no—
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#391—21no—
77#391—21no—
78#390—21no—
79#390—21no—
80#386—41no—
81#386—41no—
82#384—21no—
83#384—21no—
84#382—41no—
85#380—22no—
86#380—41no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—41no—
89C4×C22⋊C4—41no—
90C4×C22⋊C4—41no—
91#367—42no—
92#366—42no—
93#358—24no—
94#358—42no—
95#357—24no—
96#357—42no—
97C22⋊C16—24no—
98#340—22no—
99#340—22no—
100C2×#41—24no—
101C2×#41—42no—
102C2×#41—42no—
103C2×#41—42no—
104C2×#41—42no—
105C2×#41—42no—
106C22×Q8—21no—
107C22×D4—21no—
108C22×D4—41no—
109C23×C4—41no—
110C23×C4D811yescharacteristic
111C23×C4D811yescharacteristic
112C23×C4C2×D411yescharacteristic
113#135—48no—
114#135—48no—
115C2×Q16—22no—
116C2×Q16—21no—
117C2×Q16—22no—
118C2×Q16#4112yes—
119C2×SD16—24no—
120C2×SD16—42no—
121C2×SD16—42no—
122C2×D8—22no—
123C2×D8—21no—
124C2×D8—22no—
125C2×D8#4112yes—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—41no—
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134C22×C8C2×D411yescharacteristic
135C22×C8#4114yes—
136C22×C8C2411yescharacteristic
137#117—42no—
138#117—42no—
139#117—84no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—41no—
157C2×C42—42no—
158C2×C42D811yescharacteristic
159C2×C42D811yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C16—22no—
163C2×C16—22no—
164C2×C16—22no—
165C2×C16—22no—
166C2×C16—22no—
167C2×C16—22no—
168#107—22no—
169#107—22no—
170#107—22no—
171#107—22no—
172#107—22no—
173#107—22no—
174#107—42no—
175#107—42no—
176#106—42no—
177#103—42no—
178#103—42no—
179#102—42no—
180#102—42no—
181C4×C8—22no—
182C4×C8—22no—
183C4×C8—22no—
184C4×C8—22no—
185#95—41no—
186#95—41no—
187#95—41no—
188#95—41no—
189#95—42no—
190#95—42no—
191C24—41no—
192C24C2×D811yescharacteristic
193#41—48no—
194#41—48no—
195#41—84no—
196#41—84no—
197C2×Q8—22no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×D4—22no—
204C2×D4—44no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—82no—
211C2×D4—82no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—41no—
222C22×C4—44no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4#12312yes—
231C22×C4C2×D811yescharacteristic
232C22×C4C2×D811yescharacteristic
233C22×C4C2×D811yescharacteristic
234C22×C4C2×D811yescharacteristic
235C22×C4C2×D811yescharacteristic
236C22×C4#13712yes—
237C22×C4C22×D411yescharacteristic
238Q16—24no—
239Q16—42no—
240SD16—48no—
241D8—24no—
242D8—42no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—44no—
253C2×C8#12312yes—
254C2×C8AGL(1,ℤ/8)12yes—
255C2×C8C2×#4112yes—
256C2×C8C2×#4111yescharacteristic
257C2×C8C2×#4111yescharacteristic
258C2×C8H(5,𝔽2)12yes—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C22⋊C4—82no—
272C22⋊C4—82no—
273C22⋊C4—84no—
274C22⋊C4—82no—
275C22⋊C4—82no—
276C22⋊C4—82no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—22no—
282C42—22no—
283C42—82no—
284C42—84no—
285C16—42no—
286C16—42no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—42no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23#48012yes—
301C23C22×D811yescharacteristic
302Q8—42no—
303Q8—42no—
304Q8—81no—
305Q8—81no—
306D4—42no—
307D4—42no—
308D4—88no—
309D4—81no—
310D4—81no—
311C2×C4—28no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—88no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—82no—
333C2×C4—81no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—82no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—82no—
344C2×C4#48012yes—
345C2×C4#50812yes—
346C2×C4#50912yes—
347C2×C4#51012yes—
348C2×C4C2×#12311yescharacteristic
349C2×C4#52511yescharacteristic
350C2×C4#53511yescharacteristic
351C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
352C2×C4C2×#13711yescharacteristic
353C2×C4#57511yescharacteristic
354C8—22no—
355C8—22no—
356C8—41no—
357C8#55112yes—
358C8#55112yes—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—161no—
370C22—161no—
371C22—161no—
372C22—161no—
373C22C2×#48011yescharacteristic
374C22#310011yescharacteristic
375C22#320511yescharacteristic
376C22C2×#50811yescharacteristic
377C22C2×#50911yescharacteristic
378C22C2×#51011yescharacteristic
379C22#341411yescharacteristic
380C4—22no—
381C4—22no—
382C4—22no—
383C4—22no—
384C4—41no—
385C4—82no—
386C4—162no—
387C4—161no—
388C4—162no—
389C4—161no—
390C4#248211yescharacteristic
391C4#309411yescharacteristic
392C4#311011yescharacteristic
393C4#321511yescharacteristic
394C2—42no—
395C2—162no—
396C2#2758211yescharacteristic
397C2#3364511yescharacteristic
398C2#3364611yescharacteristic
399C2#3366711yescharacteristic
400C2#3367511yescharacteristic
401C2#3377311yescharacteristic
402C2#3378211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×#135C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20817C2semidirect
3. #13531C2semidirect
4. #13526C2semidirect
5. #13481C2semidirect
6. #13480C2semidirect
7. #13133C2semidirect
8. #13131C2semidirect
9. #12921C2semidirect
10. #12844C2semidirect

Maximal subgroups

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

subgroupindex
#212242
#208172
#135312
#135262
#134812
#134802
#131332
#131312
#129212
#128442
#127332