Finite Group Databasebeta

← #409993

Subgroups — #409993

Order 512.

2178 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#481C211yesmaximal, characteristic
2C23×#106C211yesmaximal, characteristic, complemented
3C2×#2501C211yesmaximal, characteristic
4C2×#1893C211yesmaximal, characteristic
5C2×#1823C211yesmaximal, characteristic, complemented
6#13924C211yesmaximal, characteristic, complemented
7#11323C214yesmaximal, complemented
8#11121C214yesmaximal, complemented
9#10714C211yesmaximal, characteristic, complemented
10C24×C8C2211yescharacteristic
11C22×#123—22no—
12C22×#121C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#481—24no—
16C2×#481—24no—
17C2×#481—24no—
18C22×#106—24no—
19C22×#106—24no—
20C22×#106—24no—
21C22×#106C2212yes—
22C22×#106C2214yes—
23C22×#102C2212yes—
24C22×#102C2212yes—
25C22×C22⋊C8C2211yescharacteristic
26#2501—24no—
27#2501—24no—
28C2×#399—22no—
29C2×#390C2212yes—
30C2×#384C2212yes—
31C2×#380C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#1893—24no—
34#1823—24no—
35C2×#340C2212yes—
36C2×#340C2214yes—
37C2×#336C2214yes—
38#1445—22no—
39C4⋊#95C2214yes—
40C22⋊#95—22no—
41C23×D4—21no—
42C23×D4C2311yescharacteristic
43C24×C4—21no—
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C23×C8—22no—
47C23×C8—22no—
48C23×C8—28no—
49C23×C8D414yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#123—216no—
53C2×#123—48no—
54C2×#123—48no—
55C2×#121—22no—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121D414yes—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4D414yes—
75C22×C4⋊C4D414yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—24no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4D412yes—
84C22×C42C2312yes—
85#481—232no—
86C2×#106—232no—
87C2×#106—24no—
88C2×#106—28no—
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—28no—
92C2×#106—28no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—28no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8D414yes—
101#399—216no—
102#390—24no—
103#390—24no—
104#384—24no—
105#384—24no—
106#380—22no—
107#380—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C2×#95—21no—
111C2×#95—28no—
112C2×#95D412yes—
113#340—28no—
114#340—24no—
115#336—28no—
116C25—21no—
117C25C2×D411yescharacteristic
118C22×D4—24no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4—24no—
125C22×D4—24no—
126C22×D4—42no—
127C22×D4—42no—
128C22×D4—44no—
129C22×D4—44no—
130C22×D4C2×D412yes—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4—84no—
146C23×C4C2×D414yes—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4#4112yes—
154C23×C4C2411yescharacteristic
155C22×C8—216no—
156C22×C8—216no—
157C22×C8—28no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—24no—
161C22×C8—216no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—24no—
165C22×C8—24no—
166C22×C8C2×D414yes—
167C22×C8C2×D414yes—
168C22×C8#4118yes—
169#123—464no—
170#121—28no—
171#121—28no—
172#121—44no—
173#121—44no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—216no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—48no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4C2×D412yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—216no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—48no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—48no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4SD1618yes—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—24no—
227C2×C42—24no—
228#106—232no—
229#106—216no—
230#102—216no—
231#102—48no—
232C22⋊C8—28no—
233#95—24no—
234#95—42no—
235#95—88no—
236C24—24no—
237C24—24no—
238C24—22no—
239C24—41no—
240C24—41no—
241C24—44no—
242C24—42no—
243C24—42no—
244C24—44no—
245C24#12012yes—
246C24#12112yes—
247C24#12712yes—
248C24C2×SD1614yes—
249C24AGL(1,ℤ/8)14yes—
250C24C22×D411yescharacteristic
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—28no—
254C2×D4—48no—
255C2×D4—416no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—88no—
267C2×D4—84no—
268C2×D4—84no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—28no—
279C22×C4—24no—
280C22×C4—28no—
281C22×C4—24no—
282C22×C4—28no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—48no—
297C22×C4—48no—
298C22×C4—48no—
299C22×C4—416no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—88no—
305C22×C4—88no—
306C22×C4—88no—
307C22×C4—88no—
308C22×C4—84no—
309C22×C4—84no—
310C22×C4#12014yes—
311C22×C4#12014yes—
312C22×C4#12012yes—
313C22×C4#12114yes—
314C22×C4#12114yes—
315C22×C4#12112yes—
316C22×C4#12712yes—
317C22×C4C2×SD1618yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4114yes—
325C22×C42−1+414yes—
326C2×C8—232no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—28no—
330C2×C8—28no—
331C2×C8—24no—
332C2×C8—28no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—44no—
336C2×C8#12318yescomplemented
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—432no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—48no—
351C4⋊C4—48no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—44no—
355C22⋊C4—216no—
356C22⋊C4—432no—
357C22⋊C4—48no—
358C22⋊C4—84no—
359C42—42no—
360C42—42no—
361C42—44no—
362C23—216no—
363C23—216no—
364C23—24no—
365C23—24no—
366C23—24no—
367C23—24no—
368C23—24no—
369C23—24no—
370C23—42no—
371C23—44no—
372C23—44no—
373C23—44no—
374C23—88no—
375C23—88no—
376C23—81no—
377C23—81no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23#39214yes—
384C23#45018yes—
385C23#46018yes—
386C23#49214yes—
387C23#49614yes—
388C23C2×#12011yescharacteristic
389C23C2×#12111yescharacteristic
390C23C2×#12711yescharacteristic
391C23#54714yes—
392C23C22×SD1612yes—
393C23C2×AGL(1,ℤ/8)12yes—
394C23#57714yes—
395D4—44no—
396D4—44no—
397D4—816no—
398D4—81no—
399D4—81no—
400D4—82no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—28no—
409C2×C4—216no—
410C2×C4—28no—
411C2×C4—48no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—48no—
418C2×C4—44no—
419C2×C4—48no—
420C2×C4—816no—
421C2×C4—82no—
422C2×C4—81no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4—84no—
426C2×C4—84no—
427C2×C4—84no—
428C2×C4—82no—
429C2×C4—816no—
430C2×C4—88no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4#39214yes—
435C2×C4C2×#12011yescharacteristic
436C2×C4C2×#12011yescharacteristic
437C2×C4C2×#12111yescharacteristic
438C2×C4C2×#12111yescharacteristic
439C2×C4C2×#12312yes—
440C2×C4#52512yes—
441C2×C4#52512yes—
442C2×C4#53612yes—
443C2×C4#53612yes—
444C2×C4#53612yes—
445C2×C4#54714yes—
446C2×C4#54714yes—
447C2×C4#55214yes—
448C8—28no—
449C8—44no—
450C22—216no—
451C22—28no—
452C22—216no—
453C22—28no—
454C22—24no—
455C22—24no—
456C22—88no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—164no—
461C22—162no—
462C22—162no—
463C22#201418yes—
464C22C2×#39211yescharacteristic
465C22#240612yes—
466C22#241312yes—
467C22#250112yes—
468C22C2×#45012yes—
469C22#301012yes—
470C22C2×#46012yes—
471C22#304212yes—
472C22C2×#49211yescharacteristic
473C22C2×#49611yescharacteristic
474C22#315412yes—
475C22#333518yes—
476C22#334518yes—
477C4—24no—
478C4—48no—
479C4—84no—
480C4—82no—
481C4—84no—
482C4—162no—
483C4#240612yes—
484C4#241312yes—
485C4#248612yes—
486C4#261411yescharacteristic
487C4#284911yescharacteristic
488C2—28no—
489C2—162no—
490C2#1506911yescharacteristic
491C2C2×#201411yescharacteristic
492C2#2342411yescharacteristic
493C2#2408914yes—
494C2#2457814yes—
495C2#2792412yes—
496C2#2793412yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C23×#106C2semidirect
5. C2×#1823C2semidirect
6. #13924C2semidirect
7. #11323C2semidirect
8. #11121C2semidirect
9. #10714C2semidirect
336. C2×C8#123semidirect

Maximal subgroups

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

subgroupindex
C22×#4812
C23×#1062
C2×#25012
C2×#18932
C2×#18232
#139242
#113232
#111212
#107142