Finite Group Databasebeta

← #242899

Subgroups — #242899

Order 512.

770 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21982C211yesmaximal, characteristic, complemented
2#21849C211yesmaximal, characteristic
3#20737C211yesmaximal, characteristic
4#20648C211yesmaximal, characteristic
5#10626C212yesmaximal, complemented
6#10625C212yesmaximal, complemented
7#10617C212yesmaximal, complemented
8#10616C212yesmaximal, complemented
9#8043C211yesmaximal, characteristic, complemented
10#8041C211yesmaximal, characteristic, complemented
11#7880C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×#423C2211yescharacteristic
17C2×#417—21no—
18#2285C2211yescharacteristic
19C2×#384C2211yescharacteristic
20#1984C2212yes—
21#1954C2212yes—
22#1947—21no—
23#1947—21no—
24#1870—22no—
25#1852—22no—
26#1845C2212yes—
27#1834C2212yes—
28#1825—21no—
29#1803—21no—
30#1803C2211yescharacteristic
31#1803C2211yescharacteristic
32#1803C2212yes—
33#1802—21no—
34#1802C2211yescharacteristic
35#1802C2211yescharacteristic
36#1802C2212yes—
37#1797—21no—
38#1797—21no—
39#1788—21no—
40C4×#103—21no—
41C4×#102—21no—
42#1736—21no—
43#1296C2212yes—
44#1296C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1277C2212yes—
48#1277C2212yes—
49C22×#41D411yescharacteristic
50C22×M16—21no—
51C22×M16—21no—
52C4×#41—22no—
53C4×#41—22no—
54C4×#41—22no—
55C4×#41—22no—
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8C2311yescharacteristic
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4D411yescharacteristic
61C2×C4×D4C2311yescharacteristic
62C2×#117—21no—
63C2×#117—21no—
64C2×#117D411yescharacteristic
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70#431—22no—
71#431—22no—
72#431—22no—
73#431—22no—
74C2×#107—22no—
75C2×#107—22no—
76#423—22no—
77#423—22no—
78#423—22no—
79#423—22no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8D411yescharacteristic
83C2×C4⋊C8D411yescharacteristic
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2312yes—
88#417—44no—
89C2×#103—22no—
90C2×#103—21no—
91C2×#103—21no—
92C2×#102—22no—
93C2×#102—21no—
94C2×#102—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8D412yes—
99C2×C4×C8—21no—
100C2×C4×C8C2312yes—
101#384—21no—
102#384—21no—
103#384—21no—
104#384C2311yescharacteristic
105#384C2312yes—
106#384C2311yescharacteristic
107#382—21no—
108#380—21no—
109#380—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112#340—22no—
113#340—22no—
114#340—22no—
115#340—22no—
116#340—21no—
117#340—21no—
118#340—21no—
119#340—21no—
120#336—22no—
121#336—22no—
122C2×#41—22no—
123C2×#41—22no—
124C2×#41—22no—
125C2×#41—22no—
126C2×#41—41no—
127C2×#41—41no—
128C22×Q8C2×D411yescharacteristic
129C22×D4—21no—
130C22×D4C2×D411yescharacteristic
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—41no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4#4111yescharacteristic
138C2×M16—42no—
139C2×M16—42no—
140C2×M16—42no—
141C2×M16—42no—
142C2×M16—81no—
143C2×M16—81no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8C2×D412yes—
149C22×C8C2×D412yes—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—21no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—21no—
159C4×D4—21no—
160C4×D4—21no—
161C4×D4—21no—
162C4×D4—42no—
163C4×D4—42no—
164C4×D4—42no—
165C4×D4—42no—
166#117—22no—
167#117—22no—
168#117—42no—
169#117—42no—
170#117—82no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—81no—
188C2×C4⋊C4—81no—
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—41no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—41no—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42#4112yes—
205C2×C42#4112yes—
206C2×C42#4112yes—
207C2×C42#4112yes—
208C2×C42#4111yescharacteristic
209C2×C42C2411yescharacteristic
210#107—82no—
211#107—82no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—42no—
223C4⋊C8—42no—
224#103—42no—
225#103—42no—
226#103—44no—
227#102—42no—
228#102—44no—
229#102—42no—
230C22⋊C8—24no—
231C22⋊C8—42no—
232C22⋊C8—82no—
233C22⋊C8—82no—
234C4×C8—22no—
235C4×C8—22no—
236C4×C8—42no—
237#95—41no—
238#95—41no—
239C24—21no—
240C24#12111yescharacteristic
241#41—28no—
242#41—44no—
243#41—44no—
244#41—44no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8AGL(1,ℤ/8)12yes—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4#13712yes—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4#12312yes—
295C22×C4#12312yes—
296C22×C4#12312yes—
297C22×C4#12412yes—
298C22×C4#12612yes—
299C22×C4#12612yes—
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C2×#4111yescharacteristic
303M16—84no—
304M16—84no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—82no—
343C4⋊C4—81no—
344C4⋊C4—81no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C22⋊C4—42no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—42no—
352C22⋊C4—82no—
353C42—41no—
354C42—41no—
355C42—42no—
356C42—42no—
357C42—82no—
358C42#12312yes—
359C42#12312yes—
360C42#12312yes—
361C42#12412yes—
362C42#12612yes—
363C42#12612yes—
364C42C2×#4111yescharacteristic
365C42C2×#4111yescharacteristic
366C42C2×#4111yescharacteristic
367C42C2×#4111yescharacteristic
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—22no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23#39712yes—
382C23C2×#12111yescharacteristic
383Q8—22no—
384Q8—22no—
385Q8—41no—
386Q8—41no—
387D4—22no—
388D4—22no—
389D4—48no—
390D4—41no—
391D4—41no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—48no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—84no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—82no—
432C2×C4—164no—
433C2×C4#39712yes—
434C2×C4#39712yes—
435C2×C4#39712yes—
436C2×C4#47212yes—
437C2×C4#47312yes—
438C2×C4C2×#12311yescharacteristic
439C2×C4C2×#12311yescharacteristic
440C2×C4C2×#12311yescharacteristic
441C2×C4C2×#12411yescharacteristic
442C2×C4C2×#12611yescharacteristic
443C2×C4C2×#12611yescharacteristic
444C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
445C2×C4C2×#13711yescharacteristic
446C2×C4#57511yescharacteristic
447C2×C4#57511yescharacteristic
448C2×C4#57611yescharacteristic
449C2×C4#57711yescharacteristic
450C2×C4#57711yescharacteristic
451C2×C4#57811yescharacteristic
452C8—42no—
453C8—42no—
454C8—81no—
455C8—81no—
456C8—82no—
457C8—82no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22C2×#39711yescharacteristic
473C22#304511yescharacteristic
474C22#304611yescharacteristic
475C22#305511yescharacteristic
476C22#305611yescharacteristic
477C22C2×#47211yescharacteristic
478C22C2×#47311yescharacteristic
479C4—22no—
480C4—22no—
481C4—21no—
482C4—21no—
483C4—42no—
484C4—81no—
485C4—81no—
486C4—161no—
487C4—161no—
488C4—162no—
489C4#207814yes—
490C4#309211yescharacteristic
491C4#309311yescharacteristic
492C4#309411yescharacteristic
493C4#309511yescharacteristic
494C4#309811yescharacteristic
495C4#309911yescharacteristic
496C4#313611yescharacteristic
497C4#313711yescharacteristic
498C2—42no—
499C2—82no—
500C2#1948511yescharacteristic
501C2#1949111yescharacteristic
502C2#1949611yescharacteristic
503C2#1950211yescharacteristic
504C2#2363411yescharacteristic
505C2#2364211yescharacteristic
506C2C2×#207811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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. #21982C2semidirect
5. #10626C2semidirect
6. #10625C2semidirect
7. #10617C2semidirect
8. #10616C2semidirect
9. #8043C2semidirect
10. #8041C2semidirect
11. #7880C2semidirect

Maximal subgroups

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

subgroupindex
#219822
#218492
#207372
#206482
#106262
#106252
#106172
#106162
#80432
#80412
#78802