Finite Group Databasebeta

← #414740

Subgroups — #414740

Order 512.

1750 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35011C211yesmaximal, characteristic
2C2×#2627C212yesmaximal
3C2×#1996C212yesmaximal
4#13776C211yesmaximal, characteristic
5#11344C214yesmaximal
6#11070C214yesmaximal
7#10846C211yesmaximal, characteristic
8C22×C22⋊Q8C2212yes—
9C23×C4⋊C4C2211yescharacteristic
10#2857—22no—
11#2659—22no—
12#2627—28no—
13#2627—28no—
14#2583—24no—
15#2502—22no—
16#2434—22no—
17C2×#400—22no—
18C2×#398—24no—
19C2×#395—22no—
20#2315C2214yes—
21C2×#385C2214yes—
22C2×#381C2214yes—
23C2×#381C2212yes—
24C2×#380C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27#1996—28no—
28C2×#342C2218yes—
29C2×#342C2214yes—
30#1436—22no—
31#1436—22no—
32C22⋊#95C2212yes—
33C23×Q8D412yes—
34C24×C4C2311yescharacteristic
35C24×C4C2311yescharacteristic
36C24×C4C2311yescharacteristic
37C2×C4⋊Q8—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—24no—
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—24no—
46C22×C4⋊C4—24no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—24no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4D414yes—
55C22×C4⋊C4D414yes—
56C22×C4⋊C4D412yes—
57C22×C22⋊C4C2314yes—
58C22×C22⋊C4C2314yes—
59C22×C22⋊C4C2312yes—
60C22×C22⋊C4C2312yes—
61C22×C42—21no—
62C22×C42D412yes—
63#400—216no—
64#398—28no—
65#398—232no—
66#395—28no—
67#395—216no—
68#391—24no—
69#389—28no—
70#385—28no—
71#385—28no—
72#385—28no—
73#385—28no—
74#384—22no—
75#381—28no—
76#381—28no—
77#381—24no—
78#381Q818yescomplemented
79#380—28no—
80#380—24no—
81#380D418yes—
82C4×C22⋊C4—22no—
83C4×C22⋊C4—22no—
84C4×C22⋊C4—24no—
85C2×#95—24no—
86C2×#95—22no—
87C2×#95—24no—
88C2×#95—24no—
89C2×#95—24no—
90C2×#95—21no—
91C2×#95—24no—
92C2×#95D412yes—
93#342—216no—
94#342—28no—
95C25C2411yescharacteristic
96C22×Q8—22no—
97C22×Q8—24no—
98C22×Q8—22no—
99C22×Q8—24no—
100C22×Q8—84no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—24no—
107C23×C4—24no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—82no—
114C23×C4—82no—
115C23×C4C2×D412yes—
116C23×C4C2×D414yes—
117C23×C4C2×D414yes—
118C23×C4C2×D412yes—
119C23×C4C2×D414yes—
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D414yes—
126C23×C4C2×Q814yes—
127C23×C4#4112yes—
128C4⋊Q8—84no—
129C22⋊Q8—44no—
130C22⋊Q8—44no—
131C22⋊Q8—44no—
132C22⋊Q8—44no—
133C22⋊Q8—44no—
134C22⋊Q8—44no—
135C22⋊Q8—44no—
136C22⋊Q8—44no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—216no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—84no—
161C2×C4⋊C4—88no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4C2×D418yes—
177C2×C22⋊C4C2×Q814yes—
178C2×C22⋊C4C2×Q814yes—
179C2×C22⋊C4#4114yes—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—42no—
183C2×C42—82no—
184C2×C42—82no—
185#95—216no—
186#95—216no—
187#95—216no—
188#95—216no—
189#95—28no—
190#95—24no—
191#95—216no—
192#95—42no—
193#95—84no—
194C24—42no—
195C24—42no—
196C24C22×D412yes—
197C24C22×D411yescharacteristic
198C24C22×D411yescharacteristic
199C24C22×D412yes—
200C24C22×Q812yes—
201C24C2×#4111yescharacteristic
202C24C2×#4112yes—
203C242−1+414yes—
204C2×Q8—48no—
205C2×Q8—44no—
206C2×Q8—44no—
207C2×Q8—84no—
208C2×Q8—88no—
209C2×Q8—88no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—28no—
218C22×C4—28no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—28no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—48no—
243C22×C4—44no—
244C22×C4—48no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—48no—
248C22×C4—48no—
249C22×C4—48no—
250C22×C4—44no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—48no—
257C22×C4—84no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—84no—
262C22×C4—82no—
263C22×C4#12012yes—
264C22×C4#12011yescharacteristic
265C22×C4#12011yescharacteristic
266C22×C4#12014yes—
267C22×C4#12014yes—
268C22×C4#12114yes—
269C22×C4#12112yes—
270C22×C4#12112yes—
271C22×C4#12114yes—
272C22×C4C22⋊Q818yes—
273C22×C4#12712yes—
274C22×C4#12712yes—
275C22×C4C4⋊Q814yescomplemented
276C4⋊C4—44no—
277C4⋊C4—48no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—48no—
282C4⋊C4—48no—
283C4⋊C4—88no—
284C4⋊C4—88no—
285C4⋊C4—816no—
286C4⋊C4—88no—
287C22⋊C4—28no—
288C22⋊C4—28no—
289C22⋊C4—216no—
290C22⋊C4—28no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—44no—
298C42—44no—
299C42—82no—
300C42—84no—
301C23—22no—
302C23—24no—
303C23—22no—
304C23—24no—
305C23—41no—
306C23—42no—
307C23—41no—
308C23—41no—
309C23—42no—
310C23—44no—
311C23—44no—
312C23—41no—
313C23—48no—
314C23C2×#12012yes—
315C23C2×#12011yescharacteristic
316C23C2×#12111yescharacteristic
317C23C2×#12112yes—
318C23C2×C22⋊Q812yes—
319C23C2×#12711yescharacteristic
320C23C2×C4⋊Q811yescharacteristic
321C23#53211yescharacteristic
322C23#53312yes—
323C23#53612yes—
324C23#53612yes—
325C23#54112yes—
326C23#54714yes—
327C23#54814yes—
328C23D4×Q814yes—
329C23D4×Q814yes—
330Q8—88no—
331Q8—88no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—28no—
337C2×C4—28no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—48no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—48no—
355C2×C4—48no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—48no—
359C2×C4—48no—
360C2×C4—48no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—48no—
366C2×C4—48no—
367C2×C4—48no—
368C2×C4—44no—
369C2×C4—84no—
370C2×C4—82no—
371C2×C4—84no—
372C2×C4—816no—
373C2×C4—82no—
374C2×C4—84no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4#39212yes—
380C2×C4#39214yes—
381C2×C4#39212yes—
382C22—24no—
383C22—24no—
384C22—28no—
385C22—28no—
386C22—22no—
387C22—21no—
388C22—22no—
389C22—21no—
390C22—22no—
391C22—22no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—44no—
400C22—44no—
401C22C2×#39211yescharacteristic
402C22#240912yes—
403C22#242914yes—
404C22#243212yes—
405C22#243412yes—
406C22#254212yes—
407C22#262111yescharacteristic
408C22#285711yescharacteristic
409C22#302612yes—
410C22#302812yes—
411C4—22no—
412C4—24no—
413C4—22no—
414C4—48no—
415C4—44no—
416C4—88no—
417C4—84no—
418C4—82no—
419C4—84no—
420C4—84no—
421C2—22no—
422C2—22no—
423C2—21no—
424C2—21no—
425C2—24no—
426C2—22no—
427C2#1517811yescharacteristic
428C2#2344411yescharacteristic
429C2#2344611yescharacteristic
430C2#2435314yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
78. #381Q8semidirect
275. C22×C4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#350112
C2×#26272
C2×#19962
#137762
#113442
#110702
#108462