Finite Group Databasebeta

← #243264

Subgroups — #243264

Order 512.

622 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21984C211yesmaximal, characteristic
2#21974C211yesmaximal, characteristic
3#21364C211yesmaximal, characteristic
4#21169C211yesmaximal, characteristic
5#10658C211yesmaximal, characteristic, complemented
6#10658C211yesmaximal, characteristic, complemented
7#10656C211yesmaximal, characteristic, complemented
8#10655C211yesmaximal, characteristic, complemented
9#10654C211yesmaximal, characteristic, complemented
10#10653C211yesmaximal, characteristic, complemented
11#10652C211yesmaximal, characteristic, complemented
12#10649C211yesmaximal, characteristic, complemented
13#8045C211yesmaximal, characteristic
14#8042C211yesmaximal, characteristic
15#7879C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#427—21no—
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23C4×#117C2211yescharacteristic
24#1949—21no—
25#1949—21no—
26#1949—21no—
27#1949—21no—
28#1948C2211yescharacteristic
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1854—21no—
41#1854—21no—
42#1854—21no—
43#1854—21no—
44#1851—21no—
45#1851—21no—
46#1845C2211yescharacteristic
47#1844C2211yescharacteristic
48#1842—21no—
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1831—21no—
52#1776—21no—
53#1776—21no—
54#1735—21no—
55#1296C2211yescharacteristic
56#1296C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C23×C8—21no—
69C2×#117—21no—
70C2×#117—21no—
71C2×#117D411yescharacteristic
72C2×#117D411yescharacteristic
73C22×C42C2311yescharacteristic
74#432—22no—
75#432Q814yes—
76#431—22no—
77#431—22no—
78#427—44no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91#423—22no—
92#423—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8D411yescharacteristic
101C2×C4⋊C8D411yescharacteristic
102C2×C4⋊C8Q814yes—
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C22⋊C8—21no—
110C2×C22⋊C8D411yescharacteristic
111C2×C22⋊C8D411yescharacteristic
112C2×#97—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#382—21no—
120#382—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135#376—21no—
136C43—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4#4111yescharacteristic
156C2×M16—42no—
157C2×M16—81no—
158C2×M16—81no—
159C22×C8—24no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—42no—
167C22×C8—42no—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172#117—22no—
173#117—22no—
174#117—41no—
175#117—41no—
176#117—42no—
177#117—42no—
178#117—81no—
179#117—81no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—41no—
207C2×C42—41no—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×Q812yes—
211C2×C42C2×Q812yes—
212C2×C42#4112yes—
213C2×C42#4112yes—
214C2×C42#4111yescharacteristic
215C2×C42C2411yescharacteristic
216#107—42no—
217#107—42no—
218#107—42no—
219#107—42no—
220#107—81no—
221#107—81no—
222#106—42no—
223#106—42no—
224#106—42no—
225#106—42no—
226#106—81no—
227#106—81no—
228C4⋊C8—24no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—42no—
240C4⋊C8—81no—
241C4⋊C8—81no—
242C4⋊C8—81no—
243C4⋊C8—81no—
244C4⋊C8C2×Q812yes—
245C4⋊C8C2×Q812yes—
246C4⋊C8C2×Q812yes—
247C4⋊C8C2×Q812yes—
248C22⋊C8—24no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251#97—24no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—41no—
261#95—41no—
262#95—41no—
263#95—41no—
264C24#12111yescharacteristic
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4#12111yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4C22⋊Q812yes—
287C22×C4C22⋊Q812yes—
288C22×C4#12312yes—
289C22×C4#12512yes—
290C22×C4#12512yes—
291C22×C4C4⋊Q812yes—
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C2×#4111yescharacteristic
295M16—84no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—48no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—81no—
350C4⋊C4—81no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C4⋊C4AGL(1,ℤ/8)12yes—
354C4⋊C4#13712yes—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C42—42no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—82no—
368C42—82no—
369C42C22⋊Q812yes—
370C42C22⋊Q812yes—
371C42#12312yes—
372C42#12512yes—
373C42#12512yes—
374C42C4⋊Q812yes—
375C42C22×Q811yescharacteristic
376C42C22×Q811yescharacteristic
377C42C2×#4111yescharacteristic
378C42C2×#4111yescharacteristic
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23#39812yes—
386C23C2×#12111yescharacteristic
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4#39812yes—
430C2×C4#39812yes—
431C2×C4#39812yes—
432C2×C4C2×C22⋊Q811yescharacteristic
433C2×C4C2×C22⋊Q811yescharacteristic
434C2×C4C2×#12311yescharacteristic
435C2×C4C2×#12511yescharacteristic
436C2×C4C2×#12511yescharacteristic
437C2×C4C2×C4⋊Q811yescharacteristic
438C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
439C2×C4C2×#13711yescharacteristic
440C2×C4#57511yescharacteristic
441C2×C4#57511yescharacteristic
442C2×C4#57611yescharacteristic
443C2×C4#57711yescharacteristic
444C2×C4#57711yescharacteristic
445C2×C4#57811yescharacteristic
446C8—42no—
447C8—42no—
448C8—42no—
449C8—42no—
450C8—41no—
451C8—41no—
452C8—41no—
453C8—41no—
454C8—81no—
455C8—81no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22C2×#39811yescharacteristic
464C22#304511yescharacteristic
465C22#304611yescharacteristic
466C22#305511yescharacteristic
467C22#305611yescharacteristic
468C22#306711yescharacteristic
469C22#306711yescharacteristic
470C4—22no—
471C4—22no—
472C4—21no—
473C4—21no—
474C4—22no—
475C4—42no—
476C4—42no—
477C4—81no—
478C4—81no—
479C4—161no—
480C4—161no—
481C4—161no—
482C4—161no—
483C4#307711yescharacteristic
484C4#307811yescharacteristic
485C4#307911yescharacteristic
486C4#308011yescharacteristic
487C4#309611yescharacteristic
488C4#309711yescharacteristic
489C4#316311yescharacteristic
490C4#316411yescharacteristic
491C2—42no—
492C2#1948011yescharacteristic
493C2#1948711yescharacteristic
494C2#1949311yescharacteristic
495C2#1949811yescharacteristic
496C2#2366211yescharacteristic
497C2#2367011yescharacteristic
498C2#2367411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
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
5. #10658C2semidirect
6. #10658C2semidirect
7. #10656C2semidirect
8. #10655C2semidirect
9. #10654C2semidirect
10. #10653C2semidirect
11. #10652C2semidirect
12. #10649C2semidirect

Maximal subgroups

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

subgroupindex
#219842
#219742
#213642
#211692
#106582
#106562
#106552
#106542
#106532
#106522
#106492
#80452
#80422
#78792