Finite Group Databasebeta

← #505666

Subgroups — #505666

Order 512.

640 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2234C212yesmaximal
2C4×#466C211yesmaximal, characteristic
3#20797C211yesmaximal, characteristic, complemented
4#13506C211yesmaximal, characteristic, complemented
5#13505C211yesmaximal, characteristic, complemented
6#13480C211yesmaximal, characteristic, complemented
7#13480C211yesmaximal, characteristic, complemented
8#13178C212yesmaximal, complemented
9#12868C212yesmaximal, complemented
10#12813C212yesmaximal, complemented
11#12783C211yesmaximal, characteristic, complemented
12C2×#466C2211yescharacteristic
13C2×C4×D8C2211yescharacteristic
14C2×#427—21no—
15C2×#427C2212yes—
16C22×C4×C8C2211yescharacteristic
17C4×#121—21no—
18#2285—21no—
19#2234—28no—
20C2×#367C2212yes—
21C2×#366C2212yes—
22C2×C4⋊C16C2212yes—
23C2×#357C2212yes—
24C2×#357C2212yes—
25C2×C22⋊C16C2212yes—
26#1970C2211yescharacteristic
27#1942C2211yescharacteristic
28#1934—21no—
29#1893—21no—
30#1846—21no—
31#1846C2212yes—
32#1845C2211yescharacteristic
33#1845C2211yescharacteristic
34#1844—21no—
35#1808C2211yescharacteristic
36C4×#107C2211yescharacteristic
37C4×#102—21no—
38#1380C2212yes—
39#1380C2212yes—
40#1380C2212yes—
41#1380C2212yes—
42#1374C2211yescharacteristic
43#1374C2211yescharacteristic
44C22×D8C2311yescharacteristic
45C23×C8D411yescharacteristic
46C23×C8C2311yescharacteristic
47C2×#121—21no—
48C2×C4×D4—21no—
49C2×C4×D4—41no—
50C2×#117—22no—
51C22×C42D411yescharacteristic
52C22×C16C2312yes—
53C22×C16C2312yes—
54#466—22no—
55#466D414yes—
56C4×D8—24no—
57#427—24no—
58#427—24no—
59#427—44no—
60C2×#107—21no—
61C2×#107—21no—
62C2×#107C2311yescharacteristic
63C2×#107C2311yescharacteristic
64C2×#107C2312yes—
65C2×#107C2312yes—
66C2×#107C2311yescharacteristic
67C2×#106—22no—
68C2×#106—22no—
69C2×#102—21no—
70C2×#102—21no—
71C2×#102—21no—
72C2×C4×C8—24no—
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77#390—21no—
78#389—21no—
79#388—41no—
80#386—41no—
81#384—21no—
82#384—21no—
83#382—22no—
84#380—21no—
85#380—41no—
86#380—41no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—41no—
89C4×C4⋊C4—41no—
90C4×C22⋊C4—41no—
91#367—24no—
92#366—24no—
93C4⋊C16—24no—
94#357—24no—
95#357—24no—
96C22⋊C16—24no—
97#340—21no—
98#340—21no—
99#340—21no—
100C22×D4—21no—
101C22×D4—41no—
102C23×C4—41no—
103C23×C4D811yescharacteristic
104C23×C4D811yescharacteristic
105C23×C4C2×D411yescharacteristic
106C2×D8—22no—
107C2×D8—21no—
108C2×D8—22no—
109C2×D8C2×D412yes—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D412yes—
117C22×C8C2×D411yescharacteristic
118C22×C8C2×D411yescharacteristic
119C22×C8#4112yes—
120C22×C8C2411yescharacteristic
121#121—24no—
122#121—42no—
123C4×D4—42no—
124C4×D4—42no—
125C4×D4—84no—
126#117—44no—
127#117—44no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—41no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—42no—
147C2×C42—41no—
148C2×C42D811yescharacteristic
149C2×C42D811yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C16—22no—
153C2×C16—22no—
154C2×C16—22no—
155C2×C16—22no—
156C2×C16—22no—
157C2×C16—22no—
158#107—21no—
159#107—22no—
160#107—22no—
161#107—22no—
162#107—22no—
163#107—22no—
164#107—22no—
165#107—42no—
166#107—42no—
167#107C2×D412yes—
168#106—44no—
169#106—44no—
170#102—24no—
171#102—42no—
172#102—42no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8#4114yes—
177#95—41no—
178#95—41no—
179#95—41no—
180#95—42no—
181#95—42no—
182C24—41no—
183C24C2×D811yescharacteristic
184C2×D4—22no—
185C2×D4—44no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—82no—
192C2×D4—82no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—82no—
209C22×C4—82no—
210C22×C4—82no—
211C22×C4#12112yes—
212C22×C4C2×D811yescharacteristic
213C22×C4C2×D811yescharacteristic
214C22×C4C2×D811yescharacteristic
215C22×C4C2×D811yescharacteristic
216C22×C4C2×D811yescharacteristic
217C22×C4AGL(1,ℤ/8)12yes—
218C22×C4C22×D411yescharacteristic
219D8—24no—
220D8—42no—
221C2×C8—28no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—24no—
229C2×C8—24no—
230C2×C8#12112yes—
231C2×C8#12312yes—
232C2×C8#13712yes—
233C2×C8C22×D411yescharacteristic
234C2×C8C2×#4112yes—
235C2×C8C2×#4111yescharacteristic
236C2×C82−1+412yes—
237C4⋊C4—22no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C22⋊C4—44no—
253C22⋊C4—82no—
254C22⋊C4—84no—
255C22⋊C4—82no—
256C22⋊C4—82no—
257C22⋊C4—82no—
258C42—21no—
259C42—21no—
260C42—22no—
261C42—22no—
262C42—82no—
263C42—84no—
264C42#12312yes—
265C42#13712yes—
266C16—42no—
267C16—42no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—42no—
275C23—81no—
276C23—81no—
277C23—81no—
278C23—81no—
279C23—81no—
280C23—81no—
281C23#45912yes—
282C23C22×D811yescharacteristic
283D4—42no—
284D4—42no—
285D4—88no—
286D4—81no—
287D4—81no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—164no—
320C2×C4#45912yes—
321C2×C4#48012yes—
322C2×C4#48012yes—
323C2×C4#50812yes—
324C2×C4#50912yes—
325C2×C4C2×#12111yescharacteristic
326C2×C4C2×#12311yescharacteristic
327C2×C4#53611yescharacteristic
328C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
329C2×C4C2×#13711yescharacteristic
330C2×C4C2×#13711yescharacteristic
331C8—22no—
332C8—22no—
333C8—22no—
334C8#54712yes—
335C8#55212yes—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—82no—
344C22—82no—
345C22—82no—
346C22—161no—
347C22—161no—
348C22—161no—
349C22—161no—
350C22C2×#45911yescharacteristic
351C22C2×#48011yescharacteristic
352C22#322611yescharacteristic
353C22C2×#50811yescharacteristic
354C22C2×#50911yescharacteristic
355C22#341411yescharacteristic
356C22#341411yescharacteristic
357C4—22no—
358C4—22no—
359C4—22no—
360C4—22no—
361C4—22no—
362C4—82no—
363C4—161no—
364C4—162no—
365C4—162no—
366C4—161no—
367C4#248611yescharacteristic
368C4#303911yescharacteristic
369C4#309211yescharacteristic
370C4#323811yescharacteristic
371C2—42no—
372C2—162no—
373C2#2764411yescharacteristic
374C2#3357911yescharacteristic
375C2#3358011yescharacteristic
376C2#3364511yescharacteristic
377C2#3364811yescharacteristic
378C2#3379411yescharacteristic
379C2#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
3. #20797C2semidirect
4. #13506C2semidirect
5. #13505C2semidirect
6. #13480C2semidirect
7. #13480C2semidirect
8. #13178C2semidirect
9. #12868C2semidirect
10. #12813C2semidirect
11. #12783C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22342
C4×#4662
#207972
#135062
#135052
#134802
#131782
#128682
#128132
#127832