Finite Group Databasebeta

← #203875

Subgroups — #203875

Order 512.

690 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21984C211yesmaximal, characteristic
2#21359C211yesmaximal, characteristic
3#21170C211yesmaximal, characteristic
4#20535C211yesmaximal, characteristic
5C2×#1574C211yesmaximal, characteristic
6#9622C211yesmaximal, characteristic, complemented
7#9621C211yesmaximal, characteristic, complemented
8#9619C211yesmaximal, characteristic, complemented
9#9618C211yesmaximal, characteristic, complemented
10#9616C211yesmaximal, characteristic, complemented
11#9615C211yesmaximal, characteristic, complemented
12#9613C211yesmaximal, characteristic, complemented
13#9612C211yesmaximal, characteristic, complemented
14#8046C211yesmaximal, characteristic
15#8042C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#428—21no—
20C2×#423C2211yescharacteristic
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1949—21no—
25#1949—21no—
26#1949—21no—
27#1949—21no—
28#1948C2211yescharacteristic
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1854—21no—
37#1854—21no—
38#1853—21no—
39#1853—21no—
40#1845C2211yescharacteristic
41#1844C2211yescharacteristic
42#1834C2211yescharacteristic
43#1834C2211yescharacteristic
44C4×#106—21no—
45C4×#106—21no—
46#1771—21no—
47#1771—21no—
48#1770C2211yescharacteristic
49#1770C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1749—21no—
53C4×C22⋊C8—21no—
54#1736—21no—
55#1574C418yescomplemented
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#334C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68C22×M16—21no—
69C2×#117—21no—
70C2×#117—21no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75#432—22no—
76#432Q814yes—
77#431—22no—
78#431C2×C414yes—
79#431C2×C414yes—
80#431Q814yes—
81#428—44no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94#423—22no—
95#423C2×C414yes—
96C2×C4⋊C8—21no—
97C2×C4⋊C8D411yescharacteristic
98C2×C4⋊C8D411yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8D411yescharacteristic
108C2×C22⋊C8D411yescharacteristic
109C2×#97—21no—
110C2×#97—21no—
111C2×#97—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#382—21no—
122#382—21no—
123#380—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C2×#95—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#340—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#335C2×C414yescomplemented
155#335C2×C414yescomplemented
156#334C2×C414yescomplemented
157#334C2×C414yescomplemented
158C23×C4—21no—
159C23×C4—21no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4#4111yescharacteristic
163C2×M16—24no—
164C2×M16—42no—
165C2×M16—42no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176#117—42no—
177#117—42no—
178#117—81no—
179#117—81no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
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⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—41no—
209C2×C42—41no—
210C2×C42—81no—
211C2×C42—81no—
212C2×C42C22×C412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×Q812yes—
216C2×C42C2×Q812yes—
217C2×C42#4112yes—
218C2×C42#4111yescharacteristic
219C2×C42C2411yescharacteristic
220#107—42no—
221#107—42no—
222#107—81no—
223#107—81no—
224#106—42no—
225#106—42no—
226#106—42no—
227#106—42no—
228#106—42no—
229#106—42no—
230#106—81no—
231#106—81no—
232C4⋊C8—24no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8C4⋊C414yes—
240C4⋊C8C4⋊C414yes—
241C4⋊C8C22×C412yes—
242C4⋊C8C22×C412yes—
243C4⋊C8C22×C412yes—
244C4⋊C8C22×C412yes—
245C4⋊C8C2×Q812yes—
246C4⋊C8C2×Q812yes—
247C22⋊C8—24no—
248C22⋊C8—24no—
249C22⋊C8C4⋊C414yes—
250C22⋊C8C4⋊C414yes—
251#97—24no—
252#97—81no—
253#97—81no—
254C4×C8—24no—
255C4×C8—81no—
256C4×C8—81no—
257C4×C8C22×C412yes—
258C4×C8C22×C412yes—
259C4×C8C2×Q812yes—
260C4×C8C2×Q812yes—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—41no—
266#95—41no—
267#95—41no—
268#95—41no—
269C24#12111yescharacteristic
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4C2×C4⋊C412yes—
295C22×C4C4×D412yes—
296C22×C4C4×Q812yes—
297C22×C4#12111yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4C22⋊Q812yes—
301C22×C4#12512yes—
302C22×C4C4⋊Q812yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C2×#4111yescharacteristic
306M16—48no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8C2×C4⋊C412yes—
334C2×C8C2×C4⋊C412yes—
335C2×C8C2×C4⋊C412yes—
336C2×C8C2×C4⋊C412yes—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—81no—
362C4⋊C4—81no—
363C4⋊C4—81no—
364C4⋊C4—81no—
365C4⋊C4AGL(1,ℤ/8)12yes—
366C4⋊C4#13712yes—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—81no—
374C42—81no—
375C42—82no—
376C42—82no—
377C42C2×C4⋊C412yes—
378C42C4×D412yes—
379C42C4×Q812yes—
380C42C22⋊Q812yes—
381C42#12512yes—
382C42C4⋊Q812yes—
383C42C23×C411yescharacteristic
384C42C22×Q811yescharacteristic
385C42C22×Q811yescharacteristic
386C42C2×#4111yescharacteristic
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23#38912yes—
394C23C2×#12111yescharacteristic
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—84no—
426C2×C4—84no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4#38912yes—
440C2×C4#38912yes—
441C2×C4#38912yes—
442C2×C4C22×C4⋊C411yescharacteristic
443C2×C4C2×C4×D411yescharacteristic
444C2×C4C2×C4×Q811yescharacteristic
445C2×C4C2×C22⋊Q811yescharacteristic
446C2×C4C2×#12511yescharacteristic
447C2×C4C2×C4⋊Q811yescharacteristic
448C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
449C2×C4C2×#13711yescharacteristic
450C2×C4#57511yescharacteristic
451C2×C4#57511yescharacteristic
452C2×C4#57711yescharacteristic
453C2×C4#57711yescharacteristic
454C2×C4#57711yescharacteristic
455C2×C4#57711yescharacteristic
456C8—22no—
457C8—22no—
458C8—22no—
459C8—22no—
460C8—42no—
461C8—42no—
462C8—42no—
463C8—42no—
464C8—42no—
465C8—42no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22C2×#38911yescharacteristic
474C22#304211yescharacteristic
475C22#304311yescharacteristic
476C22#305511yescharacteristic
477C22#305611yescharacteristic
478C22#306711yescharacteristic
479C22#306711yescharacteristic
480C4—22no—
481C4—22no—
482C4—21no—
483C4—21no—
484C4—22no—
485C4—42no—
486C4—42no—
487C4—81no—
488C4—81no—
489C4—161no—
490C4—161no—
491C4—161no—
492C4—161no—
493C4#291211yescharacteristic
494C4#291211yescharacteristic
495C4C4×AGL(1,ℤ/8)11yescharacteristic
496C4C4×#13711yescharacteristic
497C4#308411yescharacteristic
498C4#308411yescharacteristic
499C4#316511yescharacteristic
500C4#316611yescharacteristic
501C2—42no—
502C2#1946611yescharacteristic
503C2#1946711yescharacteristic
504C2#1947211yescharacteristic
505C2#1947311yescharacteristic
506C2#2115911yescharacteristic
507C2#2117411yescharacteristic
508C2#2118111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ 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
6. #9622C2semidirect
7. #9621C2semidirect
8. #9619C2semidirect
9. #9618C2semidirect
10. #9616C2semidirect
11. #9615C2semidirect
12. #9613C2semidirect
13. #9612C2semidirect
55. #1574C4semidirect
154. #335C2×C4semidirect
155. #335C2×C4semidirect
156. #334C2×C4semidirect
157. #334C2×C4semidirect

Maximal subgroups

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

subgroupindex
#219842
#213592
#211702
#205352
C2×#15742
#96222
#96212
#96192
#96182
#96162
#96152
#96132
#96122
#80462
#80422