Finite Group Databasebeta

← #319146

Subgroups — #319146

Order 512.

874 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33172C211yesmaximal, characteristic
2#27950C211yesmaximal, characteristic
3#27950C211yesmaximal, characteristic
4#25856C211yesmaximal, characteristic
5#25811C211yesmaximal, characteristic
6C2×#1990C211yesmaximal, characteristic
7#10674C212yesmaximal, complemented
8#10318C212yesmaximal, complemented
9#9841C212yesmaximal, complemented
10#9833C212yesmaximal, complemented
11#8145C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#498C2212yes—
14C2×#494C2211yescharacteristic
15C2×#487C2212yes—
16C2×#467—22no—
17C22×C4×C8C2211yescharacteristic
18#2857C2211yescharacteristic
19#2857C2211yescharacteristic
20C2×#384C2211yescharacteristic
21#2083—22no—
22#2083—22no—
23#2058—22no—
24#2043—22no—
25#1990—22no—
26#1990C2212yes—
27#1990C2212yes—
28#1990C2212yes—
29#1990C2212yes—
30#1990C2212yes—
31#1969—22no—
32#1956C2212yes—
33#1956C2211yescharacteristic
34#1956C2211yescharacteristic
35#1853—22no—
36#1852—22no—
37#1852—22no—
38#1845C2212yes—
39#1845C2212yes—
40#1845C2211yescharacteristic
41#1787—22no—
42C4×C22⋊C8C2211yescharacteristic
43C4×C22⋊C8C2211yescharacteristic
44C2×#340—22no—
45C2×#322C2212yes—
46#1273C2212yes—
47C23×Q8—21no—
48C22×Q16—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⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2312yes—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C42C2311yescharacteristic
66#498—24no—
67#498—42no—
68#494—42no—
69#487—24no—
70#487—42no—
71#467—48no—
72C2×#107—22no—
73C2×#107—24no—
74C2×#107—24no—
75C2×#106—24no—
76C2×#103—24no—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—24no—
80C2×C22⋊C8D412yes—
81C2×C22⋊C8D412yes—
82C2×C4×C8—42no—
83C2×C4×C8D412yes—
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2312yes—
86C2×C4×C8C2312yes—
87C2×C4×C8C2311yescharacteristic
88#391—21no—
89#391—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384C2311yescharacteristic
98#384C2311yescharacteristic
99#384C2312yes—
100#380—21no—
101#380—21no—
102#380—21no—
103#380—21no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#336D412yes—
110#336D412yes—
111#322—42no—
112C22×Q8—22no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8—44no—
116C22×Q8—41no—
117C22×Q8—44no—
118C22×Q8—82no—
119C23×C4—41no—
120C23×C4—41no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C2×Q16—44no—
125C2×Q16—44no—
126C2×Q16—44no—
127C22×C8—24no—
128C22×C8—24no—
129C22×C8—42no—
130C22×C8C2×D412yes—
131C22×C8C2×D414yes—
132C22×C8C2×D414yes—
133C22×C8C2×D412yes—
134C4⋊Q8—24no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—44no—
143C4⋊Q8—44no—
144C4⋊Q8—42no—
145C4⋊Q8#4114yes—
146C22⋊Q8—44no—
147C22⋊Q8—44no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2411yescharacteristic
184#107—44no—
185#107—84no—
186#107—84no—
187#106—48no—
188#103—48no—
189#103—44no—
190#103—44no—
191#103—84no—
192C22⋊C8—24no—
193C22⋊C8Q1618yescomplemented
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—42no—
201#95—41no—
202#95—41no—
203#95—41no—
204#95—41no—
205C24#12711yescharacteristic
206C2×Q8—24no—
207C2×Q8—44no—
208C2×Q8—42no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—44no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—44no—
219C2×Q8—816no—
220C2×Q8—82no—
221C2×Q8—82no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4#12711yescharacteristic
242C22×C4#12711yescharacteristic
243C22×C4#12711yescharacteristic
244C22×C4C2×Q1614yes—
245C22×C4AGL(1,ℤ/8)14yes—
246C22×C4#13714yes—
247C22×C4C22×D411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4C22×D411yescharacteristic
250Q16—88no—
251C2×C8—22no—
252C2×C8—24no—
253C2×C8—22no—
254C2×C8—24no—
255C2×C8—22no—
256C2×C8—24no—
257C2×C8—22no—
258C2×C8—24no—
259C2×C8—22no—
260C2×C8—24no—
261C2×C8—48no—
262C2×C8C2×Q1618yes—
263C2×C8#13714yes—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C22⋊C4—82no—
289C22⋊C4—82no—
290C42—21no—
291C42—21no—
292C42—22no—
293C42—22no—
294C42C2×#4112yes—
295C42H(5,𝔽2)12yes—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23#49412yes—
301C23#49612yes—
302C23#49712yes—
303C23C2×#12711yescharacteristic
304Q8—48no—
305Q8—88no—
306Q8—84no—
307Q8—82no—
308Q8—82no—
309Q8—82no—
310Q8—82no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—28no—
320C2×C4—28no—
321C2×C4—44no—
322C2×C4—88no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—82no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—82no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—164no—
341C2×C4#49412yes—
342C2×C4#49612yes—
343C2×C4#49712yes—
344C2×C4#52511yescharacteristic
345C2×C4#53411yescharacteristic
346C2×C4#54612yes—
347C2×C4#54612yes—
348C2×C4C22×Q1612yes—
349C2×C4C2×AGL(1,ℤ/8)12yes—
350C2×C4C2×#13712yes—
351C2×C4C2×#13712yes—
352C2×C4#57612yes—
353C2×C4#57712yes—
354C8—24no—
355C8—28no—
356C8—42no—
357C8—44no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22#172314yes—
365C22#253911yescharacteristic
366C22C2×#49411yescharacteristic
367C22C2×#49611yescharacteristic
368C22C2×#49711yescharacteristic
369C22#315211yescharacteristic
370C22#315411yescharacteristic
371C22#315511yescharacteristic
372C4—22no—
373C4—22no—
374C4—22no—
375C4—84no—
376C4—161no—
377C4—161no—
378C4—161no—
379C4—161no—
380C4—162no—
381C4#311212yes—
382C4#319312yes—
383C4#329914yes—
384C4#331414yes—
385C2—24no—
386C2C2×#172311yescharacteristic
387C2#1963311yescharacteristic
388C2#1968011yescharacteristic
389C2#1970111yescharacteristic
390C2#2831411yescharacteristic
391C2#2833311yescharacteristic
392C2#2834211yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
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
7. #10674C2semidirect
8. #10318C2semidirect
9. #9841C2semidirect
10. #9833C2semidirect
193. C22⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#331722
#279502
#258562
#258112
C2×#19902
#106742
#103182
#98412
#98332
#81452