Finite Group Databasebeta

← #239823

Subgroups — #239823

Order 512.

1036 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35320C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4#21171C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic
6C2×#1493C211yesmaximal, characteristic
7#10423C214yesmaximal, complemented
8#10422C214yesmaximal, complemented
9#7890C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#428—21no—
14C2×#423C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2862—21no—
17#2862—21no—
18#2642—22no—
19#2568—21no—
20#2568—21no—
21#2568C2212yes—
22#2568C2212yes—
23#2503—22no—
24#2503—22no—
25#2310—21no—
26#2310—21no—
27C2×#384C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1990—22no—
30#1990—21no—
31#1990C2214yes—
32#1956—21no—
33#1948—21no—
34#1948C2212yes—
35#1947—21no—
36#1947C2212yes—
37#1872—22no—
38#1803C2214yes—
39#1736—21no—
40C2×#340—22no—
41#1546—24no—
42C2×#332C2214yes—
43#1493—24no—
44C2×Q8⋊C8C2214yes—
45#1279C2212yes—
46#1279C2212yes—
47C23×Q8—21no—
48C22×M16—21no—
49C23×C8—21no—
50#533—22no—
51#533—22no—
52#533—22no—
53#533—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8D412yes—
63C2×C4⋊Q8C2312yes—
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8D412yes—
67C2×C4×Q8—21no—
68C2×C4×Q8C2312yes—
69C2×#117—21no—
70C2×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D412yes—
77C22×C42C2311yescharacteristic
78#431—22no—
79#431—22no—
80#428—44no—
81C2×#107—22no—
82C2×#106—22no—
83#423—22no—
84#423D414yes—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2312yes—
88C2×#103—24no—
89C2×C22⋊C8—21no—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8C2312yes—
95#400—24no—
96#400—24no—
97#400—24no—
98#400—24no—
99#391—22no—
100#391—22no—
101#391—22no—
102#391—22no—
103#391—22no—
104#391—22no—
105#391—22no—
106#391—22no—
107#391—22no—
108#391—22no—
109#391—22no—
110#391—22no—
111#384—21no—
112#384—21no—
113#384—21no—
114#384C2312yes—
115#382—22no—
116#382—22no—
117#382—22no—
118#382—22no—
119#381—24no—
120#381—24no—
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127#376—21no—
128C43—21no—
129#340—24no—
130#340—22no—
131#340—22no—
132#340—28no—
133#332—28no—
134Q8⋊C8—28no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8C2×D412yes—
145C23×C4—22no—
146C23×C4—41no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C2×M16—24no—
151C2×M16—42no—
152C2×M16—42no—
153C22×C8—24no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—42no—
158C22×C8—42no—
159C4⋊Q8—24no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—24no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173#125—42no—
174#125—42no—
175C22⋊Q8—24no—
176C22⋊Q8—24no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—42no—
184C4×Q8—42no—
185#117—22no—
186#117—22no—
187#117—22no—
188#117—22no—
189#117—42no—
190#117—42no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D412yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42#4112yes—
234C2×C42#4112yes—
235C2×C42#4112yes—
236C2×C42C2411yescharacteristic
237#107—44no—
238#106—44no—
239C4⋊C8—24no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8C2×D414yes—
244#103—84no—
245C22⋊C8—42no—
246C4×C8—22no—
247C4×C8—22no—
248C4×C8—42no—
249C4×C8—42no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262C24#12011yescharacteristic
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—44no—
275C2×Q8—44no—
276C2×Q8—44no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—84no—
280C2×Q8—84no—
281C2×Q8—84no—
282C2×Q8—84no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—42no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4#12011yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12112yes—
305C22×C4#12112yes—
306C22×C4#12112yes—
307C22×C4#12312yes—
308C22×C4#12312yes—
309C22×C4#12412yes—
310C22×C4#13512yes—
311C22×C4#13512yes—
312C22×C4AGL(1,ℤ/8)12yes—
313C22×C4#13712yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317M16—48no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—42no—
321C2×C8—48no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—44no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—44no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C22⋊C4—44no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42—42no—
367C42—42no—
368C42#12112yes—
369C42#12112yes—
370C42#12112yes—
371C42#12312yes—
372C42#12312yes—
373C42#12412yes—
374C42C22×D411yescharacteristic
375C42C2×#4111yescharacteristic
376C42C2×#4111yescharacteristic
377C42C2×#4111yescharacteristic
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23#39412yes—
383C23#44912yes—
384C23#45212yes—
385C23C2×#12011yescharacteristic
386Q8—44no—
387Q8—44no—
388Q8—82no—
389Q8—82no—
390Q8—82no—
391Q8—82no—
392Q8—84no—
393Q8—88no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—44no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—44no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—88no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—164no—
428C2×C4#39412yes—
429C2×C4#39412yes—
430C2×C4#39412yes—
431C2×C4#44912yes—
432C2×C4#45212yes—
433C2×C4#45412yes—
434C2×C4#45612yes—
435C2×C4#46314yes—
436C2×C4#46414yes—
437C2×C4#48814yes—
438C2×C4#48914yes—
439C2×C4C2×#12111yescharacteristic
440C2×C4C2×#12111yescharacteristic
441C2×C4C2×#12111yescharacteristic
442C2×C4C2×#12311yescharacteristic
443C2×C4C2×#12311yescharacteristic
444C2×C4C2×#12411yescharacteristic
445C2×C4C2×#13511yescharacteristic
446C2×C4C2×#13511yescharacteristic
447C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
448C2×C4C2×#13711yescharacteristic
449C2×C4#57712yes—
450C2×C4#57812yes—
451C8—44no—
452C8—44no—
453C8—82no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22#204214yes—
461C22C2×#39411yescharacteristic
462C22C2×#44911yescharacteristic
463C22C2×#45211yescharacteristic
464C22#301011yescharacteristic
465C22#301111yescharacteristic
466C22C2×#45411yescharacteristic
467C22C2×#45611yescharacteristic
468C4—22no—
469C4—21no—
470C4—21no—
471C4—82no—
472C4—82no—
473C4—84no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—162no—
479C4#203814yes—
480C4#204214yes—
481C4C2×#46311yescharacteristic
482C4C2×#46411yescharacteristic
483C4#306312yes—
484C4#309712yes—
485C4C2×#48811yescharacteristic
486C4C2×#48911yescharacteristic
487C2—24no—
488C2#1864411yescharacteristic
489C2#1864511yescharacteristic
490C2#1865411yescharacteristic
491C2#1865511yescharacteristic
492C2#2348411yescharacteristic
493C2C2×#203811yescharacteristic
494C2C2×#204211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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
7. #10423C2semidirect
8. #10422C2semidirect

Maximal subgroups

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

subgroupindex
#353202
#218602
C2×#19902
#211712
C2×#15462
C2×#14932
#104232
#104222
#78902