Finite Group Databasebeta

← #242890

Subgroups — #242890

Order 512.

872 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21981C211yesmaximal, characteristic, complemented
2#21788C211yesmaximal, characteristic
3#20738C211yesmaximal, characteristic
4C2×#1802C211yesmaximal, characteristic
5#10622C212yesmaximal, complemented
6#10621C212yesmaximal, complemented
7#10619C212yesmaximal, complemented
8#10619C212yesmaximal, complemented
9#8044C211yesmaximal, characteristic, complemented
10#8040C211yesmaximal, characteristic, complemented
11#7879C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×#102—21no—
18#2285C2211yescharacteristic
19C2×#384C2211yescharacteristic
20#1955—22no—
21#1954C2212yes—
22#1954C2212yes—
23#1948—22no—
24#1853—21no—
25#1853—21no—
26#1844C2212yescomplemented
27#1834C2212yescomplemented
28#1824—21no—
29#1802—21no—
30#1802—21no—
31#1802C2211yescharacteristic
32#1802C2211yescharacteristic
33#1802C2211yescharacteristic
34#1802C2211yescharacteristic
35#1802C2212yes—
36#1802C2212yes—
37#1786—21no—
38#1779—21no—
39#1779—21no—
40#1762—21no—
41#1762—21no—
42C2×#340—21no—
43#1296C2212yescomplemented
44#1296C2212yescomplemented
45#1296C2212yescomplemented
46#1296C2212yescomplemented
47#1277C2212yescomplemented
48#1277C2212yescomplemented
49C23×D4D411yescharacteristic
50C24×C4—21no—
51C23×C8—21no—
52C2×C4×D4—21no—
53C2×C4×D4—21no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4D411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4C2311yescharacteristic
61C2×C4×D4C2311yescharacteristic
62C2×#117—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C22⋊C4—21no—
68C22×C42C2311yescharacteristic
69#432—22no—
70#432—22no—
71#432—22no—
72#432—22no—
73#431—22no—
74#431—22no—
75C2×#106—22no—
76C2×#106—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8D412yes—
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2312yes—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—21no—
88C2×#102—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—44no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8D411yescharacteristic
99C2×C22⋊C8D411yescharacteristic
100C2×#97—21no—
101C2×#97—21no—
102C2×C4×C8C2312yes—
103#384—21no—
104#384—21no—
105#384—21no—
106#384C2311yescharacteristic
107#384C2312yes—
108#384C2311yescharacteristic
109#382—21no—
110#380—21no—
111#380—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—22no—
118#340—21no—
119#340—21no—
120#340—21no—
121#340—21no—
122C25—21no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—41no—
129C22×D4—41no—
130C22×D4C2×D411yescharacteristic
131C22×D4C2×D411yescharacteristic
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—41no—
140C23×C4—82no—
141C23×C4—82no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4#4111yescharacteristic
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—42no—
150C22×C8—42no—
151C22×C8—42no—
152C22×C8C2×D412yes—
153C22×C8C2×D412yes—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—21no—
159C4×D4—21no—
160C4×D4—21no—
161C4×D4—21no—
162C4×D4—21no—
163C4×D4—21no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—42no—
167C4×D4—42no—
168C4×D4—42no—
169C4×D4—42no—
170#117—82no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—41no—
199C2×C42—41no—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42#4112yes—
203C2×C42#4112yes—
204C2×C42#4112yes—
205C2×C42#4112yes—
206C2×C42#4111yescharacteristic
207C2×C42C2411yescharacteristic
208#106—82no—
209#106—82no—
210C4⋊C8—24no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—42no—
220#102—42no—
221#102—42no—
222#102—44no—
223#102—42no—
224#102—44no—
225#102—42no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—42no—
229C22⋊C8—42no—
230C22⋊C8—82no—
231C22⋊C8—82no—
232C22⋊C8—82no—
233C22⋊C8—82no—
234#97—42no—
235#97—42no—
236C4×C8—22no—
237C4×C8—22no—
238#95—41no—
239#95—41no—
240C24—21no—
241C24—21no—
242C24—42no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—81no—
248C24—81no—
249C24#12111yescharacteristic
250C2×D4—28no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—44no—
262C2×D4—44no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—44no—
266C2×D4—42no—
267C2×D4AGL(1,ℤ/8)12yes—
268C2×D4#13712yes—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—81no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4—88no—
298C22×C4—88no—
299C22×C4—81no—
300C22×C4—81no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4#12111yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12312yes—
308C22×C4#12312yes—
309C22×C4#12312yes—
310C22×C4#12412yes—
311C22×C4#12612yes—
312C22×C4#12612yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—84no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—82no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—82no—
363C42—82no—
364C42#12312yes—
365C42#12312yes—
366C42#12312yes—
367C42#12412yes—
368C42#12612yes—
369C42#12612yes—
370C42C2×#4111yescharacteristic
371C42C2×#4111yescharacteristic
372C42C2×#4111yescharacteristic
373C42C2×#4111yescharacteristic
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—22no—
379C23—22no—
380C23—48no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—84no—
394C23—84no—
395C23—84no—
396C23—84no—
397C23—84no—
398C23—84no—
399C23#39712yes—
400C23#47112yes—
401C23#47112yes—
402C23C2×#12111yescharacteristic
403D4—22no—
404D4—22no—
405D4—22no—
406D4—22no—
407D4—48no—
408D4—41no—
409D4—41no—
410D4—41no—
411D4—41no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—164no—
451C2×C4#39712yes—
452C2×C4#39712yes—
453C2×C4#39712yes—
454C2×C4C2×#12311yescharacteristic
455C2×C4C2×#12311yescharacteristic
456C2×C4C2×#12311yescharacteristic
457C2×C4C2×#12411yescharacteristic
458C2×C4C2×#12611yescharacteristic
459C2×C4C2×#12611yescharacteristic
460C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
461C2×C4C2×#13711yescharacteristic
462C2×C4#57511yescharacteristic
463C2×C4#57511yescharacteristic
464C2×C4#57612yes—
465C2×C4#57712yes—
466C8—42no—
467C8—42no—
468C8—81no—
469C8—81no—
470C8—82no—
471C8—82no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—816no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22#207914yes—
494C22C2×#39711yescharacteristic
495C22#304111yescharacteristic
496C22#304211yescharacteristic
497C22#305511yescharacteristic
498C22#305611yescharacteristic
499C22C2×#47111yescharacteristic
500C22C2×#47111yescharacteristic
501C4—22no—
502C4—22no—
503C4—21no—
504C4—21no—
505C4—22no—
506C4—81no—
507C4—81no—
508C4—162no—
509C4—161no—
510C4—161no—
511C4#309211yescharacteristic
512C4#309311yescharacteristic
513C4#309411yescharacteristic
514C4#309511yescharacteristic
515C4#309612yes—
516C4#313612yes—
517C2—24no—
518C2—82no—
519C2—82no—
520C2#1945611yescharacteristic
521C2#1945711yescharacteristic
522C2#1946211yescharacteristic
523C2#1946311yescharacteristic
524C2#2363211yescharacteristic
525C2#2364211yescharacteristic
526C2C2×#207911yescharacteristic
subgroupG/H
center, socleC23C2×#121
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
1. #21981C2semidirect
5. #10622C2semidirect
6. #10621C2semidirect
7. #10619C2semidirect
8. #10619C2semidirect
9. #8044C2semidirect
10. #8040C2semidirect
11. #7879C2semidirect
26. #1844C22semidirect
27. #1834C22semidirect
43. #1296C22semidirect
44. #1296C22semidirect
45. #1296C22semidirect
46. #1296C22semidirect
47. #1277C22semidirect
48. #1277C22semidirect

Maximal subgroups

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

subgroupindex
#219812
#217882
#207382
C2×#18022
#106222
#106212
#106192
#80442
#80402
#78792