Finite Group Databasebeta

← #243213

Subgroups — #243213

Order 512.

672 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21365C211yesmaximal, characteristic
2#21170C211yesmaximal, characteristic
3#21146C211yesmaximal, characteristic
4#20824C211yesmaximal, characteristic
5#10658C211yesmaximal, characteristic, complemented
6#10657C211yesmaximal, characteristic, complemented
7#10656C211yesmaximal, characteristic, complemented
8#10655C211yesmaximal, characteristic, complemented
9#10653C211yesmaximal, characteristic, complemented
10#10653C211yesmaximal, characteristic, complemented
11#10652C211yesmaximal, characteristic, complemented
12#10651C211yesmaximal, characteristic, complemented
13#8042C211yesmaximal, characteristic
14#8041C211yesmaximal, characteristic
15#7876C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#428—21no—
18C2×#423—21no—
19C2×#423C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2285C2211yescharacteristic
23#2285C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1949—21no—
26#1949—21no—
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1947C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1940—21no—
36#1940—21no—
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1939C2211yescharacteristic
41#1853—21no—
42#1852—21no—
43#1845C2211yescharacteristic
44#1844C2211yescharacteristic
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1809—21no—
48#1809—21no—
49#1776—21no—
50#1776—21no—
51#1776—21no—
52#1776—21no—
53#1736—21no—
54#1728—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—
68C22×M16—21no—
69C22×M16D411yescharacteristic
70C22×M16D411yescharacteristic
71C2×#117—21no—
72C2×#117—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77#432—22no—
78#432Q814yes—
79#428—44no—
80C2×#107—21no—
81C2×#107—21no—
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—
92#423—22no—
93#423—22no—
94#423—22no—
95#423—44no—
96#423Q814yes—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C22⋊C8—21no—
110C4×M16—22no—
111C4×M16—22no—
112C2×#97—21no—
113C2×#97D411yescharacteristic
114C2×#97D411yescharacteristic
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—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4#4111yescharacteristic
159C2×M16—24no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—41no—
165C2×M16—41no—
166C2×M16—42no—
167C2×M16—42no—
168C2×M16—42no—
169C2×M16—42no—
170C2×M16—42no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181#117—42no—
182#117—42no—
183#117—82no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
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—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42—41no—
215C2×C42—41no—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×Q812yes—
219C2×C42C2×Q812yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42#4111yescharacteristic
223C2×C42C2411yescharacteristic
224#107—42no—
225#107—42no—
226#107—42no—
227#107—42no—
228#107—42no—
229#107—42no—
230#106—42no—
231#106—42no—
232#106—42no—
233#106—42no—
234#106—42no—
235#106—42no—
236C4⋊C8—24no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8C2×Q812yes—
251C4⋊C8C2×Q812yes—
252C4⋊C8C2×Q812yes—
253C4⋊C8C2×Q812yes—
254C22⋊C8—24no—
255#97—24no—
256#97—22no—
257#97—22no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—41no—
267#95—41no—
268#95—41no—
269#95—41no—
270C24#12111yescharacteristic
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—21no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—81no—
294C22×C4—81no—
295C22×C4#12111yescharacteristic
296C22×C4#12111yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4C22⋊Q812yes—
299C22×C4C22⋊Q812yes—
300C22×C4#12312yes—
301C22×C4#12512yes—
302C22×C4#12512yes—
303C22×C4C4⋊Q812yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4111yescharacteristic
307M16—44no—
308M16—44no—
309M16—48no—
310M16—84no—
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—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
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×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8AGL(1,ℤ/8)12yes—
340C2×C8#13712yes—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—42no—
348C4⋊C4—42no—
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—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—81no—
366C4⋊C4—81no—
367C4⋊C4—81no—
368C4⋊C4—81no—
369C4⋊C4AGL(1,ℤ/8)12yes—
370C4⋊C4#13712yes—
371C22⋊C4—82no—
372C22⋊C4—82no—
373C42—42no—
374C42—42no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—82no—
380C42—82no—
381C42C22⋊Q812yes—
382C42C22⋊Q812yes—
383C42#12312yes—
384C42#12512yes—
385C42#12512yes—
386C42C4⋊Q812yes—
387C42C22×Q811yescharacteristic
388C42C22×Q811yescharacteristic
389C42C2×#4111yescharacteristic
390C42C2×#4111yescharacteristic
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23#39812yes—
398C23C2×#12111yescharacteristic
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—84no—
430C2×C4—84no—
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—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4#39812yes—
444C2×C4#39812yes—
445C2×C4#39812yes—
446C2×C4C2×C22⋊Q811yescharacteristic
447C2×C4C2×C22⋊Q811yescharacteristic
448C2×C4C2×#12311yescharacteristic
449C2×C4C2×#12511yescharacteristic
450C2×C4C2×#12511yescharacteristic
451C2×C4C2×C4⋊Q811yescharacteristic
452C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
453C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
454C2×C4C2×#13711yescharacteristic
455C2×C4C2×#13711yescharacteristic
456C2×C4#57511yescharacteristic
457C2×C4#57511yescharacteristic
458C2×C4#57511yescharacteristic
459C2×C4#57511yescharacteristic
460C8—22no—
461C8—22no—
462C8—42no—
463C8—42no—
464C8—42no—
465C8—42no—
466C8—41no—
467C8—41no—
468C8—81no—
469C8—81no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22C2×#39811yescharacteristic
478C22#303611yescharacteristic
479C22#303711yescharacteristic
480C22#305511yescharacteristic
481C22#305611yescharacteristic
482C22#306811yescharacteristic
483C22#306911yescharacteristic
484C4—22no—
485C4—22no—
486C4—21no—
487C4—21no—
488C4—22no—
489C4—42no—
490C4—42no—
491C4—81no—
492C4—81no—
493C4—161no—
494C4—161no—
495C4—161no—
496C4—161no—
497C4#307711yescharacteristic
498C4#307811yescharacteristic
499C4#307911yescharacteristic
500C4#308011yescharacteristic
501C4#309411yescharacteristic
502C4#309511yescharacteristic
503C4#316511yescharacteristic
504C4#316611yescharacteristic
505C2—42no—
506C2#1930411yescharacteristic
507C2#1930511yescharacteristic
508C2#1931011yescharacteristic
509C2#1931111yescharacteristic
510C2#2365711yescharacteristic
511C2#2367011yescharacteristic
512C2#2367511yescharacteristic
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. #10657C2semidirect
7. #10656C2semidirect
8. #10655C2semidirect
9. #10653C2semidirect
10. #10653C2semidirect
11. #10652C2semidirect
12. #10651C2semidirect

Maximal subgroups

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

subgroupindex
#213652
#211702
#211462
#208242
#106582
#106572
#106562
#106552
#106532
#106522
#106512
#80422
#80412
#78762