Finite Group Databasebeta

← #241149

Subgroups — #241149

Order 512.

996 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34642C211yesmaximal, characteristic
2#21981C212yesmaximal
3#14277C211yesmaximal, characteristic
4#10570C214yesmaximal
5#10568C214yesmaximal
6#8026C212yesmaximal
7#7906C211yesmaximal, characteristic
8C2×#533—21no—
9C2×#431C2212yes—
10#2660—22no—
11#2660—22no—
12#2659—21no—
13#2659—21no—
14#2659C2212yes—
15#2659C2212yes—
16#2399—24no—
17C2×#389—21no—
18#2285—21no—
19#2285C2212yes—
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#376C2211yescharacteristic
23#1948—24no—
24#1853C2218yes—
25#1844—22no—
26#1834—22no—
27#1444—21no—
28#1444C2212yes—
29#1442—21no—
30#1442C2212yes—
31C4⋊#95—21no—
32#1294C2218yes—
33#1280C2212yes—
34#1280C2212yes—
35#533—44no—
36#533—44no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×#125—22no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—22no—
42C2×C4×Q8—22no—
43C2×C4×Q8—22no—
44C2×#117—22no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4C2312yes—
50C22×C42D411yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42C2311yescharacteristic
53#431—24no—
54#431Q818yescomplemented
55C2×#106—28no—
56C2×C4⋊C8—22no—
57C2×C22⋊C8C2314yes—
58C2×C4×C8—22no—
59#398—24no—
60#398—24no—
61#398—24no—
62#398—24no—
63#395—24no—
64#395—24no—
65#395—24no—
66#395—24no—
67#389—44no—
68#384—21no—
69#384—21no—
70#384—22no—
71#384—22no—
72#384D412yes—
73#384D412yes—
74#382—22no—
75#382—22no—
76#382—22no—
77#382Q814yescomplemented
78#380—22no—
79#380—22no—
80#380—22no—
81#380C2314yes—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4D414yes—
86#376—22no—
87#376—22no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95D411yescharacteristic
93C2×#95D411yescharacteristic
94C2×#95C2311yescharacteristic
95C2×#95C2311yescharacteristic
96C2×#95C2312yes—
97#340—28no—
98C22×Q8—22no—
99C22×Q8—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2411yescharacteristic
109C22×C8—24no—
110C4⋊Q8—44no—
111C4⋊Q8—44no—
112#125—48no—
113C22⋊Q8—44no—
114C22⋊Q8—44no—
115C22⋊Q8—44no—
116C22⋊Q8—44no—
117C4×Q8—48no—
118C4×Q8—48no—
119#117—44no—
120#117—44no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—82no—
144C2×C4⋊C4—82no—
145C2×C4⋊C4—82no—
146C2×C4⋊C4—82no—
147C2×C4⋊C4C2×D414yes—
148C2×C4⋊C4C2×D414yes—
149C2×C22⋊C4—24no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—24no—
157C2×C42SD1614yes—
158C2×C42SD1614yes—
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42C2×Q814yes—
162C2×C42C2×Q812yes—
163C2×C42#4112yes—
164#106—88no—
165C4⋊C8—28no—
166C22⋊C8—24no—
167C22⋊C8C2×Q818yes—
168C4×C8—28no—
169#95—24no—
170#95—24no—
171#95—22no—
172#95—22no—
173#95—24no—
174#95—24no—
175#95—22no—
176#95—22no—
177#95—24no—
178#95—22no—
179#95—44no—
180#95—82no—
181#95—82no—
182#95C2×Q814yes—
183C24C22×D411yescharacteristic
184C2×Q8—44no—
185C2×Q8—44no—
186C2×Q8—82no—
187C2×Q8—82no—
188C2×Q8—82no—
189C2×Q8—82no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4#12012yes—
227C22×C4#12011yescharacteristic
228C22×C4#12011yescharacteristic
229C22×C4C22⋊Q812yes—
230C22×C4C22⋊Q812yes—
231C22×C4C4⋊Q812yes—
232C22×C4C4⋊Q812yes—
233C22×C4C2×SD1612yes—
234C22×C4C2×SD1612yes—
235C22×C4C2×SD1612yes—
236C22×C4C2×SD1612yes—
237C22×C4C2×SD1612yes—
238C22×C4C2×SD1612yes—
239C22×C4AGL(1,ℤ/8)12yes—
240C22×C4#13712yes—
241C22×C4C22×D412yes—
242C22×C4C22×Q812yes—
243C22×C4C22×Q811yescharacteristic
244C22×C4C2×#4111yescharacteristic
245C2×C8—28no—
246C2×C8—44no—
247C2×C8—44no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—48no—
251C4⋊C4—48no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—48no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—48no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—84no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—88no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C42—24no—
271C42—24no—
272C42—24no—
273C42—24no—
274C42—48no—
275C42C22⋊Q814yes—
276C42C22⋊Q814yes—
277C42C4⋊Q814yes—
278C42C4⋊Q814yes—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23C2×#12011yescharacteristic
284C23C2×C22⋊Q811yescharacteristic
285C23C2×C4⋊Q811yescharacteristic
286C23C22×SD1611yescharacteristic
287C23C22×SD1611yescharacteristic
288C23C2×AGL(1,ℤ/8)11yescharacteristic
289C23C2×#13711yescharacteristic
290Q8—88no—
291Q8—88no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4#39514yes—
342C2×C4#39512yes—
343C2×C4#39512yes—
344C2×C4#44812yes—
345C2×C4#44812yes—
346C2×C4#45012yes—
347C2×C4#45012yes—
348C2×C4#47514yes—
349C2×C4#47614yes—
350C2×C4#49814yes—
351C2×C4#50114yes—
352C2×C4C2×C22⋊Q812yes—
353C2×C4C2×C4⋊Q812yes—
354C8—48no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22C2×#39511yescharacteristic
368C22C2×#45011yescharacteristic
369C22C2×#44811yescharacteristic
370C22#302111yescharacteristic
371C22#302211yescharacteristic
372C22#302711yescharacteristic
373C22#302811yescharacteristic
374C22C2×#47611yescharacteristic
375C22C2×#47511yescharacteristic
376C22C2×#49811yescharacteristic
377C22C2×#50111yescharacteristic
378C4—24no—
379C4—22no—
380C4—22no—
381C4—42no—
382C4—42no—
383C4—42no—
384C4—42no—
385C4—84no—
386C4—84no—
387C4—84no—
388C4—162no—
389C4#205814yes—
390C4#205814yes—
391C2—21no—
392C2—21no—
393C2—21no—
394C2—21no—
395C2#1830111yescharacteristic
396C2#1830211yescharacteristic
397C2#1830911yescharacteristic
398C2#1831011yescharacteristic
399C2C2×#205811yescharacteristic
400C2#2356211yescharacteristic
401C2#2356611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
54. #431Q8semidirect
77. #382Q8semidirect

Maximal subgroups

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

subgroupindex
#346422
#219812
#142772
#105702
#105682
#80262
#79062