Finite Group Databasebeta

← #319288

Subgroups — #319288

Order 512.

937 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33170C211yesmaximal, characteristic
2#27948C211yesmaximal, characteristic, complemented
3#27906C211yesmaximal, characteristic
4#25903C211yesmaximal, characteristic
5#25856C211yesmaximal, characteristic, complemented
6C2×#1990C211yesmaximal, characteristic, complemented
7#10685C212yesmaximal, complemented
8#10310C212yesmaximal, complemented
9#9827C212yesmaximal, complemented
10#9826C212yesmaximal, complemented
11#8144C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×#492C2211yescharacteristic
14C2×#487C2212yes—
15C2×#486C2212yes—
16C2×#481—22no—
17C2×#465—22no—
18C22×C4×C8C2211yescharacteristic
19#2857C2211yescharacteristic
20#2856C2211yescharacteristic
21C2×#384C2211yescharacteristic
22#2094—22no—
23#2084—22no—
24#2043—22no—
25#2023—22no—
26#1990—22no—
27#1990C2212yes—
28#1990C2212yes—
29#1990C2212yes—
30#1990C2212yes—
31#1968—22no—
32#1957C2212yes—
33#1957C2211yescharacteristic
34#1956C2212yes—
35#1956C2211yescharacteristic
36#1874—22no—
37#1853—22no—
38#1844C2212yes—
39#1844C2212yes—
40#1844C2211yescharacteristic
41#1787—22no—
42C4×C22⋊C8C2211yescharacteristic
43C4×C22⋊C8C2211yescharacteristic
44C2×#340—22no—
45C2×#322C2212yes—
46#1273C2212yes—
47C23×Q8—21no—
48C22×SD16—22no—
49C23×C8D412yes—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8C2311yescharacteristic
56C2×C4⋊Q8C2312yes—
57C2×#127C2311yescharacteristic
58C2×C22⋊Q8—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C42C2311yescharacteristic
68#492—42no—
69#487—24no—
70#487—42no—
71#486—24no—
72#486—42no—
73#481—48no—
74#465—48no—
75C2×#106—24no—
76C2×#106—22no—
77C2×#106—24no—
78C2×#103—24no—
79C2×#103—22no—
80C2×#103—24no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—22no—
84C2×C22⋊C8D412yes—
85C2×C22⋊C8D412yes—
86C2×C4×C8—42no—
87C2×C4×C8D412yes—
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2312yes—
90C2×C4×C8C2312yes—
91C2×C4×C8C2311yescharacteristic
92#391—21no—
93#390—21no—
94#390—21no—
95#390—21no—
96#389—21no—
97#389—21no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384C2311yescharacteristic
102#384C2311yescharacteristic
103#384C2312yes—
104#380—21no—
105#380—21no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#336D412yes—
111#336D412yes—
112#322—42no—
113C22×Q8—22no—
114C22×Q8—21no—
115C22×Q8—44no—
116C22×Q8—41no—
117C22×Q8—44no—
118C22×Q8—82no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4—41no—
123C23×C4—41no—
124C23×C4—41no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C2×SD16—44no—
129C2×SD16—44no—
130C2×SD16—44no—
131C22×C8—42no—
132C22×C8—42no—
133C22×C8SD1618yes—
134C22×C8C2×D414yes—
135C22×C8C2×D414yes—
136C22×C8C2×D412yes—
137C22×C8C2×D412yes—
138C4⋊Q8—24no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—44no—
145C4⋊Q8—44no—
146C4⋊Q8—42no—
147C4⋊Q8#4114yes—
148#127—22no—
149#127—22no—
150C22⋊Q8—44no—
151#121—44no—
152#121—44no—
153#121—44no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—82no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2411yescharacteristic
189#106—48no—
190#106—44no—
191#106—84no—
192#103—48no—
193#103—44no—
194#103—84no—
195#102—48no—
196#102—48no—
197#102—44no—
198C22⋊C8—24no—
199C22⋊C8—24no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—42no—
207#95—41no—
208#95—41no—
209C24—41no—
210C24#12711yescharacteristic
211C2×Q8—24no—
212C2×Q8—44no—
213C2×Q8—42no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—42no—
217C2×Q8—44no—
218C2×Q8—42no—
219C2×Q8—44no—
220C2×Q8—816no—
221C2×Q8—82no—
222C2×Q8—82no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—82no—
234C2×D4—82no—
235C2×D4—82no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4#12711yescharacteristic
255C22×C4#12711yescharacteristic
256C22×C4#12711yescharacteristic
257C22×C4C2×SD1614yes—
258C22×C4AGL(1,ℤ/8)14yes—
259C22×C4#13714yes—
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263SD16—88no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—48no—
275C2×C8C2×SD1614yes—
276C2×C8C2×SD1614yes—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C22⋊C4—82no—
299C22⋊C4—82no—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C42—21no—
303C42—21no—
304C42—22no—
305C42—22no—
306C42C2×#4112yes—
307C42H(5,𝔽2)12yes—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23#49212yes—
320C23#49612yes—
321C23#49712yes—
322C23C2×#12711yescharacteristic
323Q8—48no—
324Q8—88no—
325Q8—84no—
326Q8—82no—
327Q8—82no—
328D4—82no—
329D4—82no—
330D4—84no—
331D4—84no—
332D4—164no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—44no—
344C2×C4—88no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—81no—
355C2×C4—82no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—164no—
360C2×C4#49212yes—
361C2×C4#49612yes—
362C2×C4#49712yes—
363C2×C4#52511yescharacteristic
364C2×C4#53411yescharacteristic
365C2×C4#54612yes—
366C2×C4#54612yes—
367C2×C4C22×SD1612yes—
368C2×C4C2×AGL(1,ℤ/8)12yes—
369C2×C4C2×#13712yes—
370C2×C4#57512yes—
371C2×C4#57712yes—
372C2×C4#57712yes—
373C8—24no—
374C8—24no—
375C8—44no—
376C8—44no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—161no—
384C22—161no—
385C22—161no—
386C22—161no—
387C22—161no—
388C22—161no—
389C22—161no—
390C22#171414yes—
391C22#253911yescharacteristic
392C22C2×#49211yescharacteristic
393C22C2×#49611yescharacteristic
394C22C2×#49711yescharacteristic
395C22#315311yescharacteristic
396C22#315411yescharacteristic
397C22#315411yescharacteristic
398C4—22no—
399C4—22no—
400C4—22no—
401C4—84no—
402C4—161no—
403C4—161no—
404C4—161no—
405C4—162no—
406C4#310312yes—
407C4#318712yes—
408C4#330614yes—
409C4#331614yes—
410C2—24no—
411C2—321no—
412C2C2×#171411yescharacteristic
413C2#1965411yescharacteristic
414C2#1967811yescharacteristic
415C2#1968611yescharacteristic
416C2#2830511yescharacteristic
417C2#2834311yescharacteristic
418C2#2834711yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #27948C2semidirect
5. #25856C2semidirect
6. C2×#1990C2semidirect
7. #10685C2semidirect
8. #10310C2semidirect
9. #9827C2semidirect
10. #9826C2semidirect
11. #8144C2semidirect

Maximal subgroups

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

subgroupindex
#331702
#279482
#279062
#259032
#258562
C2×#19902
#106852
#103102
#98272
#98262
#81442