Finite Group Databasebeta

← #202949

Subgroups — #202949

Order 512.

1000 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21803C211yesmaximal, characteristic
2#21362C211yesmaximal, characteristic, complemented
3#21260C211yesmaximal, characteristic
4#20550C211yesmaximal, characteristic, complemented
5C2×#1545C211yesmaximal, characteristic
6#9572C212yesmaximal, complemented
7#9571C212yesmaximal, complemented
8#9564C212yesmaximal, complemented
9#9563C212yesmaximal, complemented
10#8041C211yesmaximal, characteristic, complemented
11#7880C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×#423C2211yescharacteristic
17C2×#418—21no—
18C2×#418—21no—
19#2285C2211yescharacteristic
20C4×#117C2211yescharacteristic
21#1982—21no—
22#1982—21no—
23#1982—21no—
24#1982—21no—
25#1982C2212yes—
26#1982C2212yes—
27#1957C2212yes—
28#1957C2212yes—
29#1948C2211yescharacteristic, complemented
30#1948C2211yescharacteristic, complemented
31#1947C2211yescharacteristic, complemented
32#1947C2211yescharacteristic, complemented
33#1854—21no—
34#1854—21no—
35#1786—22no—
36C4×#107—21no—
37C4×#106—21no—
38#1770C2212yescomplemented
39C8×C4⋊C4C2212yescomplemented
40#1767—21no—
41C4×C4⋊C8—21no—
42#1753—22no—
43#1545C418yescomplemented
44C2×#331C2212yes—
45C2×#331C2212yes—
46#1296C2212yescomplemented
47#1296C2212yescomplemented
48#1277C2212yescomplemented
49#1277C2212yescomplemented
50C23×D4—21no—
51C23×D4—21no—
52C23×D4—21no—
53C22×M16—21no—
54C22×M16—21no—
55C2×#127—21no—
56C2×#127—21no—
57C2×#127—21no—
58C2×#127—22no—
59C2×#127—22no—
60C2×#127—22no—
61C2×#127C2×C414yes—
62C2×#127C2312yes—
63C2×#117—21no—
64C2×#117D411yescharacteristic
65C2×#117D411yescharacteristic
66C22×C42C2311yescharacteristic
67#431—22no—
68#431—22no—
69#431—22no—
70#431C2×C414yes—
71C2×#107—21no—
72C2×#107—21no—
73C2×#106—21no—
74C2×#106—21no—
75#423—22no—
76#423—22no—
77#423—22no—
78#423C2×C414yes—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8D411yescharacteristic
82C2×C4⋊C8D411yescharacteristic
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87#418—44no—
88#418—44no—
89C2×#102—22no—
90C2×#102—22no—
91C2×#102—22no—
92C2×#102—22no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—22no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8D412yes—
99C2×#97—22no—
100C2×C4×C8—21no—
101C2×C4×C8—22no—
102C2×C4×C8C2312yes—
103#384C2311yescharacteristic
104#384C2311yescharacteristic
105#382—21no—
106#380—21no—
107#380—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4D411yescharacteristic
111C4×C4⋊C4D411yescharacteristic
112C4×C4⋊C4C2311yescharacteristic
113C4×C4⋊C4C2311yescharacteristic
114C4×C22⋊C4—21no—
115C4×C22⋊C4—21no—
116#376—21no—
117C43—21no—
118#340—21no—
119#340—21no—
120#340—21no—
121#340—21no—
122#336—22no—
123#336—22no—
124#331C2×C418yes—
125#331C2×C418yes—
126C25—41no—
127C22×D4—24no—
128C22×D4—24no—
129C22×D4—24no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—42no—
134C22×D4—42no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4—41no—
138C22×D4—41no—
139C22×D4—41no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4—81no—
144C22×D4—81no—
145C22×D4—81no—
146C22×D4—81no—
147C22×D4—82no—
148C22×D4—82no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4#4111yescharacteristic
152C2×M16—24no—
153C2×M16—42no—
154C2×M16—42no—
155C2×M16—42no—
156C2×M16—81no—
157C2×M16—81no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8C2×D412yes—
163C22×C8C2×D412yes—
164#127—24no—
165#127—24no—
166#127—24no—
167#127—22no—
168#127—22no—
169#127—22no—
170#127—42no—
171#127—42no—
172#127—42no—
173#127C22×C414yes—
174#117—22no—
175#117—22no—
176#117—41no—
177#117—41no—
178#117—82no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—41no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—41no—
199C2×C42C22×C412yes—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42#4112yes—
204C2×C42#4112yes—
205C2×C42#4112yes—
206C2×C42C2411yescharacteristic
207#107—42no—
208#107—42no—
209#106—42no—
210#106—42no—
211C4⋊C8—24no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—42no—
220C4⋊C8C22×C414yes—
221C4⋊C8C22×C412yes—
222C4⋊C8C22×C412yes—
223#102—44no—
224#102—44no—
225#102—44no—
226#102—44no—
227#102—82no—
228#102—82no—
229#102—82no—
230#102—82no—
231C22⋊C8—24no—
232C22⋊C8—24no—
233#97—82no—
234C4×C8—24no—
235C4×C8—22no—
236C4×C8—22no—
237C4×C8—82no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—41no—
243#95—41no—
244C24—42no—
245C24—44no—
246C24—82no—
247C24—82no—
248C24—82no—
249C24—82no—
250C24—82no—
251C24—82no—
252C24#12111yescharacteristic
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—48no—
257C2×D4—48no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—84no—
273C2×D4—84no—
274C2×D4—82no—
275C2×D4—82no—
276C2×D4—82no—
277C2×D4—82no—
278C2×D4—84no—
279C2×D4—84no—
280C2×D4—82no—
281C2×D4—82no—
282C2×D4—84no—
283C2×D4—84no—
284C2×D4—88no—
285C2×D4—88no—
286C2×D4—82no—
287C2×D4—82no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4#11712yes—
305C22×C4C4×D412yes—
306C22×C4C4×D412yes—
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12312yes—
311C22×C4#12412yes—
312C22×C4#12612yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C2×#4111yescharacteristic
316M16—84no—
317M16—84no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—24no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—81no—
351C4⋊C4—81no—
352C4⋊C4AGL(1,ℤ/8)12yes—
353C4⋊C4AGL(1,ℤ/8)12yes—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—82no—
357C42—42no—
358C42—42no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—42no—
364C42—42no—
365C42—82no—
366C42#11712yes—
367C42C4×D412yes—
368C42C4×D412yes—
369C42#12312yes—
370C42#12412yes—
371C42#12612yes—
372C42C23×C411yescharacteristic
373C42C2×#4111yescharacteristic
374C42C2×#4111yescharacteristic
375C42C2×#4111yescharacteristic
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—44no—
383C23—88no—
384C23—88no—
385C23—88no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23—164no—
393C23—164no—
394C23—164no—
395C23—164no—
396C23#38812yes—
397C23C2×#12111yescharacteristic
398D4—44no—
399D4—44no—
400D4—44no—
401D4—44no—
402D4—84no—
403D4—88no—
404D4—88no—
405D4—84no—
406D4—82no—
407D4—82no—
408D4—82no—
409D4—82no—
410D4—88no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4#38812yes—
446C2×C4#38812yes—
447C2×C4#38812yes—
448C2×C4#47112yes—
449C2×C4#47212yes—
450C2×C4C2×#11711yescharacteristic
451C2×C4C2×C4×D411yescharacteristic
452C2×C4C2×C4×D411yescharacteristic
453C2×C4C2×#12311yescharacteristic
454C2×C4C2×#12411yescharacteristic
455C2×C4C2×#12611yescharacteristic
456C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
457C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
458C2×C4#57511yescharacteristic
459C2×C4#57511yescharacteristic
460C2×C4#57611yescharacteristic
461C2×C4#57611yescharacteristic
462C2×C4#57711yescharacteristic
463C2×C4#57711yescharacteristic
464C8—42no—
465C8—42no—
466C8—42no—
467C8—42no—
468C8—44no—
469C8—82no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—84no—
478C22—84no—
479C22—84no—
480C22—1616no—
481C22—162no—
482C22—162no—
483C22—162no—
484C22—162no—
485C22C2×#38811yescharacteristic
486C22#304511yescharacteristic
487C22#304511yescharacteristic
488C22#305511yescharacteristic
489C22#305511yescharacteristic
490C22C2×#47111yescharacteristic
491C22C2×#47211yescharacteristic
492C4—22no—
493C4—21no—
494C4—21no—
495C4—22no—
496C4—42no—
497C4—42no—
498C4—81no—
499C4—81no—
500C4—161no—
501C4—161no—
502C4#193114yes—
503C4C4×AGL(1,ℤ/8)11yescharacteristic
504C4C4×AGL(1,ℤ/8)11yescharacteristic
505C4#294811yescharacteristic
506C4#294911yescharacteristic
507C4#309611yescharacteristic
508C4#309611yescharacteristic
509C4#313811yescharacteristic
510C4#314011yescharacteristic
511C2—42no—
512C2—164no—
513C2#1948311yescharacteristic
514C2#1948411yescharacteristic
515C2#1949511yescharacteristic
516C2#1949611yescharacteristic
517C2#2106511yescharacteristic
518C2C2×#193111yescharacteristic
519C2#2109511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
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
2. #21362C2semidirect
4. #20550C2semidirect
6. #9572C2semidirect
7. #9571C2semidirect
8. #9564C2semidirect
9. #9563C2semidirect
10. #8041C2semidirect
11. #7880C2semidirect
29. #1948C22semidirect
30. #1948C22semidirect
31. #1947C22semidirect
32. #1947C22semidirect
38. #1770C22semidirect
39. C8×C4⋊C4C22semidirect
43. #1545C4semidirect
46. #1296C22semidirect
47. #1296C22semidirect
48. #1277C22semidirect
49. #1277C22semidirect

Maximal subgroups

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

subgroupindex
#218032
#213622
#212602
#205502
C2×#15452
#95722
#95712
#95642
#95632
#80412
#78802