Finite Group Databasebeta

← #415249

Subgroups — #415249

Order 512.

1134 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35173C211yesmaximal, characteristic
2#21323C212yesmaximal
3C2×#1986C212yesmaximal
4#13909C211yesmaximal, characteristic
5#11296C212yesmaximal
6#11296C212yesmaximal
7#11211C212yesmaximal
8#11210C212yesmaximal
9C4×#343C211yesmaximal, characteristic
10C2×#533C2212yes—
11C2×C4×M16C2212yes—
12#2857—22no—
13#2659—22no—
14#2584—24no—
15#2567—22no—
16#2546—22no—
17#2316C2212yes—
18#2316C2212yes—
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1993—22no—
25#1993—22no—
26#1991—24no—
27#1986—28no—
28C2×#343C2212yes—
29C2×#343C2212yes—
30#1736C2214yes—
31#1736C2214yes—
32C2×#339C2214yes—
33C2×#339C2214yes—
34#1441—22no—
35#1436—22no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C22×M16C2314yes—
39#533—24no—
40#533—24no—
41#533—24no—
42#533—24no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8D412yes—
45C2×C4⋊Q8D412yes—
46C2×#125D414yes—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×#117C2314yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C42D412yes—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C4×M16—24no—
63C4×M16—24no—
64C2×#97—22no—
65C2×C4×C8—22no—
66#398—28no—
67#395—28no—
68#391—24no—
69#389—24no—
70#389—24no—
71#389—24no—
72#389—24no—
73#389—24no—
74#389—24no—
75#384—22no—
76#384—24no—
77#382—24no—
78#382—22no—
79#382—22no—
80#380—24no—
81#380—24no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4D414yes—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95D412yes—
93#343—24no—
94#343Q818yescomplemented
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#339—28no—
100#339—28no—
101C22×Q8—22no—
102C22×Q8—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2411yescharacteristic
112C2×M16—24no—
113C2×M16—24no—
114C2×M16—24no—
115C2×M16—24no—
116C2×M16—24no—
117C2×M16C2×Q818yes—
118C22×C8—24no—
119C4⋊Q8—24no—
120C4⋊Q8—24no—
121C4⋊Q8—24no—
122C4⋊Q8—24no—
123C4⋊Q8—84no—
124#125—28no—
125#125—28no—
126C22⋊Q8—44no—
127C22⋊Q8—44no—
128C22⋊Q8—44no—
129C22⋊Q8—44no—
130C4×Q8—44no—
131C4×Q8—44no—
132C4×Q8—44no—
133C4×Q8—44no—
134#117—24no—
135#117—24no—
136#117—24no—
137#117—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4C2×D414yes—
163C2×C4⋊C4C2×D414yes—
164C2×C4⋊C4C2×Q814yes—
165C2×C4⋊C4#4114yes—
166C2×C22⋊C4—24no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—24no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—82no—
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42#4114yes—
183C2×C42#4112yes—
184#97—44no—
185C4×C8—44no—
186#95—28no—
187#95—28no—
188#95—24no—
189#95—24no—
190#95—44no—
191#95—42no—
192#95—41no—
193#95—41no—
194#95—44no—
195#95—82no—
196C24C22×D411yescharacteristic
197C2×Q8—44no—
198C2×Q8—44no—
199C2×Q8—44no—
200C2×Q8—44no—
201C2×Q8—82no—
202C2×Q8—82no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4#12012yes—
242C22×C4#12011yescharacteristic
243C22×C4#12011yescharacteristic
244C22×C4#12112yes—
245C22×C4#12112yes—
246C22×C4#12414yes—
247C22×C4#12412yes—
248C22×C4#12412yes—
249C22×C4C4⋊Q814yes—
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D412yes—
252C22×C4C22×Q812yes—
253C22×C4C2×#4112yes—
254C22×C4C2×#4111yescharacteristic
255C22×C4C2×#4112yes—
256C22×C42−1+412yes—
257C22×C42−1+412yes—
258M16—28no—
259M16—28no—
260M16—44no—
261M16—44no—
262C2×C8—28no—
263C2×C8—44no—
264C2×C8—44no—
265C4⋊C4—28no—
266C4⋊C4—28no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—48no—
272C4⋊C4—48no—
273C4⋊C4—48no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—84no—
279C4⋊C4—84no—
280C4⋊C4—84no—
281C4⋊C4—84no—
282C4⋊C4—88no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C42—24no—
286C42—24no—
287C42—44no—
288C42—44no—
289C42—44no—
290C42—42no—
291C42—42no—
292C42—82no—
293C42#12114yes—
294C42#12114yes—
295C42#12314yes—
296C42#12314yes—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23C2×#12011yescharacteristic
302C23C2×#12111yescharacteristic
303C23C2×#12411yescharacteristic
304C23C2×#12411yescharacteristic
305C23C2×C4⋊Q811yescharacteristic
306C23#53611yescharacteristic
307C23#53611yescharacteristic
308Q8—88no—
309Q8—88no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4#39412yes—
361C2×C4#39414yes—
362C2×C4#39412yes—
363C2×C4C2×#12112yes—
364C2×C4C2×#12312yes—
365C2×C4#53314yes—
366C2×C4#54114yes—
367C2×C4#54712yes—
368C2×C4#54814yes—
369C2×C4D4×Q812yes—
370C2×C4D4×Q814yes—
371C2×C4#55612yes—
372C2×C4#55712yes—
373C8—48no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22C2×#39411yescharacteristic
387C22#243411yescharacteristic
388C22#243611yescharacteristic
389C22#254611yescharacteristic
390C22#254711yescharacteristic
391C22#264211yescharacteristic
392C22#286211yescharacteristic
393C22#301811yescharacteristic
394C22#301811yescharacteristic
395C22#301911yescharacteristic
396C22#302011yescharacteristic
397C4—24no—
398C4—22no—
399C4—22no—
400C4—44no—
401C4—44no—
402C4—44no—
403C4—82no—
404C4—84no—
405C4—84no—
406C4—161no—
407C4—161no—
408C4#242914yes—
409C4#244614yes—
410C2—21no—
411C2—21no—
412C2—21no—
413C2—21no—
414C2#1518811yescharacteristic
415C2#2349811yescharacteristic
416C2#2349911yescharacteristic
417C2#2582711yescharacteristic
418C2#2582811yescharacteristic
419C2#2592111yescharacteristic
420C2#2592211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
94. #343Q8semidirect

Maximal subgroups

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

subgroupindex
#351732
#213232
C2×#19862
#139092
#112962
#112112
#112102
C4×#3432