Finite Group Databasebeta

← #508502

Subgroups — #508502

Order 512.

563 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33129C211yesmaximal, characteristic
2#20797C211yesmaximal, characteristic, complemented
3#13539C212yesmaximal, complemented
4#13528C212yesmaximal, complemented
5#13506C211yesmaximal, characteristic, complemented
6#13505C211yesmaximal, characteristic, complemented
7#13480C211yesmaximal, characteristic, complemented
8#13480C211yesmaximal, characteristic, complemented
9#12992C212yesmaximal
10#12921C212yesmaximal, complemented
11#12844C211yesmaximal, characteristic, complemented
12C2×#466C2211yescharacteristic
13C2×#427—21no—
14C2×#427C2212yes—
15C22×C4×C8C2211yescharacteristic
16#2854—21no—
17#2285—21no—
18C2×#367C2212yes—
19C2×#366C2212yes—
20C2×#357C2212yes—
21C2×#357C2212yes—
22C2×C22⋊C16C2212yes—
23#1970C2211yescharacteristic
24#1970C2211yescharacteristic
25#1942C2211yescharacteristic
26#1942C2211yescharacteristic
27#1934—21no—
28#1934—21no—
29#1893—21no—
30#1846—21no—
31#1846C2212yes—
32#1845C2211yescharacteristic
33#1845C2211yescharacteristic
34#1844—21no—
35#1808C2211yescharacteristic
36#1380C2212yes—
37#1380C2212yes—
38#1380C2212yes—
39#1380C2212yes—
40#1374C2211yescharacteristic
41#1374C2211yescharacteristic
42#1374C2212yes—
43C22×D8C2311yescharacteristic
44C23×C8D411yescharacteristic
45C23×C8C2311yescharacteristic
46C2×#123—41no—
47C2×#121—21no—
48C2×#117—22no—
49C22×C42D411yescharacteristic
50C22×C16C2312yes—
51C22×C16C2312yes—
52#466—24no—
53#427—42no—
54#427—42no—
55#427—44no—
56C2×#107—21no—
57C2×#107—21no—
58C2×#107C2311yescharacteristic
59C2×#107C2311yescharacteristic
60C2×#107C2312yes—
61C2×#107C2312yes—
62C2×#107C2311yescharacteristic
63C2×#106—22no—
64C2×#106—22no—
65C2×#102—21no—
66C2×#102—21no—
67C2×#102—21no—
68C2×C4×C8—24no—
69C2×C4×C8D411yescharacteristic
70C2×C4×C8D411yescharacteristic
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#390—21no—
74#390—21no—
75#389—21no—
76#389—21no—
77#388—41no—
78#388—41no—
79#386—41no—
80#384—21no—
81#384—21no—
82#382—22no—
83#380—21no—
84#380—41no—
85#380—41no—
86C4×C4⋊C4—41no—
87C4×C4⋊C4—41no—
88#367—42no—
89#366—42no—
90#357—42no—
91#357—42no—
92C22⋊C16—24no—
93#340—21no—
94#340—21no—
95#340—21no—
96C22×D4—21no—
97C22×D4—41no—
98C23×C4D811yescharacteristic
99C23×C4D811yescharacteristic
100C23×C4C2×D411yescharacteristic
101C2×D8—22no—
102C2×D8—21no—
103C2×D8—21no—
104C2×D8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—22no—
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C22×C8#4114yes—
115C22×C8C2411yescharacteristic
116#123—84no—
117#121—42no—
118#121—42no—
119#117—44no—
120#117—44no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—42no—
142C2×C42D811yescharacteristic
143C2×C42D811yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C16—22no—
147C2×C16—22no—
148C2×C16—22no—
149C2×C16—22no—
150C2×C16—22no—
151C2×C16—22no—
152#107—22no—
153#107—22no—
154#107—22no—
155#107—21no—
156#107—22no—
157#107—21no—
158#107—22no—
159#107—22no—
160#107—42no—
161#107—42no—
162#106—44no—
163#106—44no—
164#102—42no—
165#102—42no—
166#102—42no—
167C4×C8—22no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—22no—
171#95—42no—
172#95—42no—
173#95—41no—
174#95—41no—
175#95—41no—
176C24—41no—
177C24C2×D811yescharacteristic
178C2×D4—41no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—82no—
185C2×D4—82no—
186C2×D4—82no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4#12312yes—
203C22×C4C2×D811yescharacteristic
204C22×C4C2×D811yescharacteristic
205C22×C4C2×D811yescharacteristic
206C22×C4C2×D811yescharacteristic
207C22×C4C2×D811yescharacteristic
208C22×C4#13712yes—
209C22×C4C22×D411yescharacteristic
210D8—42no—
211D8—42no—
212C2×C8—28no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—24no—
220C2×C8—24no—
221C2×C8#12312yes—
222C2×C8AGL(1,ℤ/8)12yes—
223C2×C8#13712yes—
224C2×C8C2×#4112yes—
225C2×C8H(5,𝔽2)11yescharacteristic
226C2×C8H(5,𝔽2)11yescharacteristic
227C2×C82−1+412yes—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—82no—
239C4⋊C4—82no—
240C4⋊C4—82no—
241C4⋊C4—82no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C22⋊C4—82no—
245C22⋊C4—82no—
246C22⋊C4—84no—
247C22⋊C4—82no—
248C22⋊C4—82no—
249C22⋊C4—82no—
250C22⋊C4—82no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—22no—
256C42—22no—
257C42—84no—
258C16—42no—
259C16—42no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—81no—
267C23—81no—
268C23—81no—
269C23—81no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23#48012yes—
274C23C22×D811yescharacteristic
275D4—81no—
276D4—81no—
277D4—81no—
278D4—81no—
279D4—164no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—81no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4#48012yes—
314C2×C4#50812yes—
315C2×C4#50912yes—
316C2×C4C2×#12311yescharacteristic
317C2×C4#53511yescharacteristic
318C2×C4#53611yescharacteristic
319C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
320C2×C4C2×#13711yescharacteristic
321C2×C4C2×#13711yescharacteristic
322C8—22no—
323C8—22no—
324C8—22no—
325C8#53912yes—
326C8#53912yes—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—161no—
335C22—161no—
336C22—161no—
337C22—161no—
338C22—161no—
339C22—161no—
340C22—161no—
341C22C2×#48011yescharacteristic
342C22#320511yescharacteristic
343C22#322611yescharacteristic
344C22C2×#50811yescharacteristic
345C22C2×#50911yescharacteristic
346C22#341411yescharacteristic
347C22#341411yescharacteristic
348C4—22no—
349C4—22no—
350C4—22no—
351C4—41no—
352C4—41no—
353C4—161no—
354C4—161no—
355C4—162no—
356C4—161no—
357C4—162no—
358C4#262311yescharacteristic
359C4#309211yescharacteristic
360C4#321811yescharacteristic
361C4#323911yescharacteristic
362C2—42no—
363C2—321no—
364C2#2990611yescharacteristic
365C2#3364511yescharacteristic
366C2#3364811yescharacteristic
367C2#3377811yescharacteristic
368C2#3378011yescharacteristic
369C2#3379411yescharacteristic
370C2#3380611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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. #20797C2semidirect
3. #13539C2semidirect
4. #13528C2semidirect
5. #13506C2semidirect
6. #13505C2semidirect
7. #13480C2semidirect
8. #13480C2semidirect
10. #12921C2semidirect
11. #12844C2semidirect

Maximal subgroups

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

subgroupindex
#331292
#207972
#135392
#135282
#135062
#135052
#134802
#129922
#129212
#128442