Finite Group Databasebeta

← #239640

Subgroups — #239640

Order 512.

1306 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2#21833C211yesmaximal, characteristic, complemented
3#21359C211yesmaximal, characteristic, complemented
4#20646C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic, complemented
6C2×#1478C211yesmaximal, characteristic
7#10417C218yesmaximal, complemented
8#7901C211yesmaximal, characteristic, complemented
9C22×C4⋊Q8C2211yescharacteristic
10C22×#124—21no—
11C22×C4×D4C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#418—21no—
16#2547—22no—
17#2547—22no—
18#2547—22no—
19#2547C2214yes—
20C2×#397—22no—
21C2×#391—21no—
22#2285C2211yescharacteristic
23C2×#380—22no—
24C2×C4×C4⋊C4C2211yescharacteristic
25#1990C2214yescomplemented
26#1988—22no—
27#1948C2214yescomplemented
28#1853—22no—
29#1823—22no—
30#1804—22no—
31#1802C2214yes—
32C4×#106—22no—
33#1736—21no—
34#1546—24no—
35C2×#332C2214yescomplemented
36#1478—24no—
37C2×D4⋊C8C2214yes—
38#1279C2214yescomplemented
39C23×Q8—21no—
40C23×D4D411yescharacteristic
41C24×C4—21no—
42C22×M16—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8D412yes—
50C2×C4⋊Q8C2312yes—
51C2×#124—24no—
52C2×#124—24no—
53C2×#124—44no—
54C2×#124—44no—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4D412yes—
62C2×C4×D4C2312yes—
63C2×#117—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D412yes—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D412yes—
68C22×C22⋊C4—24no—
69C22×C22⋊C4—21no—
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72#431—22no—
73#431—22no—
74#431—22no—
75#431D414yes—
76C2×#106—24no—
77#423—22no—
78#423—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2312yes—
83#418—44no—
84C2×#102—24no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8C2312yes—
89#397—28no—
90#397—28no—
91#391—24no—
92#391—24no—
93#384C2312yes—
94#382—21no—
95#380—28no—
96#380—28no—
97#380—21no—
98#380—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4D412yes—
104C4×C4⋊C4C2312yes—
105C2×#95—24no—
106C2×#95—21no—
107#340—24no—
108#340—28no—
109#340—24no—
110#332—28no—
111D4⋊C8—28no—
112C25—21no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—42no—
116C22×Q8—42no—
117C22×Q8—42no—
118C22×Q8—42no—
119C22×Q8—42no—
120C22×Q8—42no—
121C22×D4—21no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4SD1614yes—
128C22×D4C2×D412yes—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—48no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C2×M16—42no—
141C2×M16—42no—
142C2×M16—42no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156#124—48no—
157#124—416no—
158#124—48no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—42no—
166C4×D4—42no—
167C4×D4—42no—
168C4×D4—42no—
169#117—42no—
170#117—42no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D414yes—
188C2×C4⋊C4C2×D414yes—
189C2×C22⋊C4—216no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—416no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—41no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—41no—
201C2×C42C2×D412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42#4112yes—
206C2×C42#4112yes—
207C2×C42#4112yes—
208C2×C42C2411yescharacteristic
209#106—48no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—42no—
214C4⋊C8—42no—
215C4⋊C8C2×D414yes—
216#102—48no—
217C22⋊C8—24no—
218C22⋊C8—42no—
219C4×C8—24no—
220C4×C8—22no—
221C4×C8—22no—
222#95—216no—
223#95—22no—
224#95—22no—
225#95—41no—
226#95—41no—
227C24—22no—
228C24—24no—
229C24—44no—
230C24—44no—
231C24—42no—
232C24—42no—
233C24#12011yescharacteristic
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—44no—
241C2×Q8—44no—
242C2×Q8—44no—
243C2×Q8—84no—
244C2×Q8—84no—
245C2×Q8—84no—
246C2×Q8—84no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—24no—
253C2×D4—28no—
254C2×D4—28no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4C2×SD1614yes—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—42no—
265C22×C4—48no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—48no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—88no—
280C22×C4—88no—
281C22×C4#12011yescharacteristic
282C22×C4#12011yescharacteristic
283C22×C4#12011yescharacteristic
284C22×C4#12112yes—
285C22×C4#12112yes—
286C22×C4#12112yes—
287C22×C4#12312yes—
288C22×C4#12312yes—
289C22×C4#12412yes—
290C22×C4C2×SD1612yes—
291C22×C4#13512yes—
292C22×C4AGL(1,ℤ/8)12yes—
293C22×C4AGL(1,ℤ/8)12yes—
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297M16—84no—
298C2×C8—24no—
299C2×C8—44no—
300C2×C8—44no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C4⋊C4—22no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C22⋊C4—432no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—44no—
325C22⋊C4—82no—
326C42—42no—
327C42—42no—
328C42—44no—
329C42—82no—
330C42#12112yes—
331C42#12112yes—
332C42#12112yes—
333C42#12312yes—
334C42#12312yes—
335C42#12412yes—
336C42C22×D411yescharacteristic
337C42C2×#4111yescharacteristic
338C42C2×#4111yescharacteristic
339C42C2×#4111yescharacteristic
340C23—22no—
341C23—22no—
342C23—24no—
343C23—416no—
344C23—42no—
345C23—42no—
346C23—44no—
347C23—44no—
348C23—48no—
349C23—84no—
350C23—88no—
351C23—84no—
352C23#39412yes—
353C23#44912yes—
354C23#45012yes—
355C23C2×#12011yescharacteristic
356Q8—82no—
357Q8—82no—
358Q8—82no—
359Q8—82no—
360Q8—88no—
361D4—24no—
362D4—24no—
363D4—42no—
364D4—42no—
365D4—48no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—88no—
382C2×C4—88no—
383C2×C4—84no—
384C2×C4—88no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4#39412yes—
391C2×C4#39412yes—
392C2×C4#39412yes—
393C2×C4#44912yes—
394C2×C4#45012yes—
395C2×C4#46514yes—
396C2×C4#46914yes—
397C2×C4#48114yes—
398C2×C4#48214yes—
399C2×C4C2×#12111yescharacteristic
400C2×C4C2×#12111yescharacteristic
401C2×C4C2×#12111yescharacteristic
402C2×C4C2×#12311yescharacteristic
403C2×C4C2×#12311yescharacteristic
404C2×C4C2×#12411yescharacteristic
405C2×C4C22×SD1611yescharacteristic
406C2×C4C2×#13511yescharacteristic
407C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
408C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
409C2×C4#57514yes—
410C8—44no—
411C8—82no—
412C8—82no—
413C22—24no—
414C22—24no—
415C22—22no—
416C22—22no—
417C22—48no—
418C22—44no—
419C22—816no—
420C22—84no—
421C22—82no—
422C22—82no—
423C22#203414yes—
424C22C2×#39411yescharacteristic
425C22C2×#45011yescharacteristic
426C22C2×#44911yescharacteristic
427C22#300512yes—
428C22#302012yes—
429C4—21no—
430C4—21no—
431C4—22no—
432C4—22no—
433C4—84no—
434C4—84no—
435C4—82no—
436C4—162no—
437C4#203414yes—
438C4#304012yes—
439C4C2×#46511yescharacteristic
440C4C2×#46911yescharacteristic
441C4C2×#48211yescharacteristic
442C4C2×#48111yescharacteristic
443C4#314012yes—
444C2—24no—
445C2—84no—
446C2#1875712yes—
447C2#1878012yes—
448C2C2×#203411yescharacteristic
449C2#2349411yescharacteristic
450C2#2350611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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. #21833C2semidirect
3. #21359C2semidirect
5. C2×#1546C2semidirect
7. #10417C2semidirect
8. #7901C2semidirect
25. #1990C22semidirect
27. #1948C22semidirect
35. C2×#332C22semidirect
38. #1279C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
#218332
#213592
#206462
C2×#15462
C2×#14782
#104172
#79012