Finite Group Databasebeta

← #238114

Subgroups — #238114

Order 512.

1152 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2#21395C211yesmaximal, characteristic
3#21395C211yesmaximal, characteristic
4#21200C211yesmaximal, characteristic
5#10355C214yesmaximal, complemented
6#10354C214yesmaximal, complemented
7#8040C211yesmaximal, characteristic, complemented
8#7904C211yesmaximal, characteristic, complemented
9#7904C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×#125—21no—
12C2×#432C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#417—22no—
15C2×C4×M16C2211yescharacteristic
16#2552—22no—
17#2552—22no—
18#2552—22no—
19#2552C2214yes—
20C2×#398—22no—
21C2×#389—21no—
22C2×#389—21no—
23C2×#384—21no—
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1987C2212yes—
28#1987C2212yes—
29#1982C2212yes—
30#1982C2212yes—
31#1959—22no—
32#1957C2212yes—
33#1956C2212yes—
34#1874—21no—
35#1874—21no—
36#1874—21no—
37#1874—21no—
38#1762—22no—
39#1762—22no—
40#1296C2214yes—
41#1279C2214yes—
42#1279C2214yes—
43C22×#41—21no—
44C22×M16—21no—
45#532—22no—
46#532—22no—
47#532—22no—
48#532Q814yes—
49C2×C4⋊Q8C2311yescharacteristic
50C2×#127C2311yescharacteristic
51C2×#125—24no—
52C2×#125—24no—
53C2×#125—48no—
54C2×#124—21no—
55C2×#121—21no—
56C2×#121—21no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D412yes—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C42—21no—
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72#432—22no—
73#432—22no—
74#432—22no—
75#432—22no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78#417—48no—
79C2×#103—24no—
80C2×#102—24no—
81C2×#102—22no—
82C2×#102—22no—
83C2×#102—22no—
84C2×#102—22no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C4×M16—22no—
88C4×M16—22no—
89C2×#97—21no—
90C2×#97—21no—
91C2×#97—21no—
92C2×C4×C8C2312yes—
93#398—28no—
94#398—28no—
95#389—24no—
96#389—24no—
97#389—24no—
98#389—24no—
99#384—24no—
100#384—24no—
101#384—21no—
102#384—21no—
103#384D412yes—
104#384C2312yes—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4D412yes—
110C4×C4⋊C4D412yes—
111C4×C4⋊C4C2312yes—
112C4×C4⋊C4C2312yes—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—21no—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120C2×#41—24no—
121C2×#41—44no—
122C2×#41—44no—
123C2×#41—42no—
124C22×Q8—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—41no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—41no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×M16—24no—
139C2×M16—42no—
140C2×M16—42no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8C2×Q812yes—
148#127—21no—
149#127—21no—
150#127—21no—
151#127C2×Q812yes—
152#125—416no—
153#125—48no—
154#125—48no—
155#124—24no—
156#124—42no—
157#121—24no—
158#121—24no—
159#121—42no—
160#121—42no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—42no—
164C4×D4—42no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—48no—
185C2×C4⋊C4—48no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—84no—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D414yes—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—44no—
201C2×C42—44no—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×Q812yes—
206C2×C42#4112yes—
207C2×C42#4112yes—
208C2×C42#4112yes—
209C2×C42C2411yescharacteristic
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214#103—48no—
215#102—44no—
216#102—44no—
217#102—48no—
218#102—44no—
219#102—44no—
220C22⋊C8—42no—
221C22⋊C8—42no—
222#97—42no—
223#97—42no—
224#97—42no—
225C4×C8—22no—
226C4×C8—22no—
227#95—28no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232C24—41no—
233C24#12011yescharacteristic
234#41—416no—
235#41—88no—
236C2×Q8—22no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—41no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—44no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—82no—
254C2×D4—82no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—41no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4#12011yescharacteristic
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4C22⋊Q812yes—
289C22×C4C22⋊Q812yes—
290C22×C4C22⋊Q812yes—
291C22×C4#12412yes—
292C22×C4#12412yes—
293C22×C4#12412yes—
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297M16—48no—
298C2×C8—24no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—44no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—416no—
323C4⋊C4—416no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—82no—
329C4⋊C4—88no—
330C4⋊C4AGL(1,ℤ/8)14yes—
331C4⋊C4#13714yes—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C42—44no—
335C42—44no—
336C42—44no—
337C42—44no—
338C42C22⋊Q812yes—
339C42C22⋊Q812yes—
340C42C22⋊Q812yes—
341C42#12412yes—
342C42#12412yes—
343C42#12412yes—
344C42C22×Q811yescharacteristic
345C42C2×#4111yescharacteristic
346C42C2×#4111yescharacteristic
347C42C2×#4111yescharacteristic
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23#39312yes—
360C23#45312yes—
361C23#45412yes—
362C23C2×#12011yescharacteristic
363Q8—44no—
364Q8—82no—
365D4—44no—
366D4—44no—
367D4—44no—
368D4—82no—
369D4—88no—
370D4—82no—
371D4—82no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—48no—
389C2×C4—48no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—88no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4—88no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4#39312yes—
411C2×C4#39312yes—
412C2×C4#39312yes—
413C2×C4C2×C22⋊Q811yescharacteristic
414C2×C4C2×C22⋊Q811yescharacteristic
415C2×C4C2×C22⋊Q811yescharacteristic
416C2×C4C2×#12411yescharacteristic
417C2×C4C2×#12411yescharacteristic
418C2×C4C2×#12411yescharacteristic
419C2×C4C2×AGL(1,ℤ/8)12yes—
420C2×C4C2×#13712yes—
421C2×C4#57512yes—
422C2×C4#57512yes—
423C8—44no—
424C8—82no—
425C8—82no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—82no—
433C22—82no—
434C22—82no—
435C22—161no—
436C22—161no—
437C22—161no—
438C22—161no—
439C22#202714yes—
440C22C2×#39311yescharacteristic
441C22#300311yescharacteristic
442C22#300411yescharacteristic
443C22#300511yescharacteristic
444C22#300511yescharacteristic
445C22C2×#45311yescharacteristic
446C22C2×#45411yescharacteristic
447C4—22no—
448C4—22no—
449C4—21no—
450C4—22no—
451C4—21no—
452C4—44no—
453C4—44no—
454C4—82no—
455C4—82no—
456C4—84no—
457C4—82no—
458C4#307912yes—
459C4#308012yes—
460C4#314112yes—
461C4#314212yes—
462C2—24no—
463C2—162no—
464C2#1852511yescharacteristic
465C2#1853311yescharacteristic
466C2#1853511yescharacteristic
467C2#1854511yescharacteristic
468C2#2345211yescharacteristic
469C2#2345311yescharacteristic
470C2C2×#202711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
1. C2×#2552C2semidirect
5. #10355C2semidirect
6. #10354C2semidirect
7. #8040C2semidirect
8. #7904C2semidirect
9. #7904C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
#213952
#212002
#103552
#103542
#80402
#79042