Finite Group Databasebeta

← #192775

Subgroups — #192775

Order 512.

1108 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34186C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic, complemented
3#21964C211yesmaximal, characteristic, complemented
4#20738C211yesmaximal, characteristic
5C2×#1486C211yesmaximal, characteristic
6#8897C212yesmaximal, complemented
7#8893C212yesmaximal, complemented
8#8889C212yesmaximal, complemented
9#8885C212yesmaximal, complemented
10#8042C211yesmaximal, characteristic, complemented
11#7900C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16#2357—22no—
17#2357—22no—
18#2354—21no—
19#2354—21no—
20#2354C2212yes—
21#2354C2212yes—
22#2312—22no—
23#2309—21no—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27#2280—22no—
28#2274—21no—
29#1949—22no—
30#1948—21no—
31#1948C2212yescomplemented
32#1947—21no—
33#1947C2212yescomplemented
34#1854—22no—
35#1845C2211yescharacteristic, complemented
36#1844C2211yescharacteristic, complemented
37#1834C2211yescharacteristic, complemented
38#1834C2211yescharacteristic, complemented
39#1824—21no—
40#1809—22no—
41#1802C2212yes—
42#1802C2212yes—
43#1786—21no—
44#1762—22no—
45#1486C418yescomplemented
46C2×D4⋊C8C2212yes—
47C2×D4⋊C8C2212yes—
48#1296C2212yescomplemented
49#1296C2212yescomplemented
50#1279C2212yescomplemented
51#1279C2212yescomplemented
52C23×D4D411yescharacteristic
53C24×C4—21no—
54C22×M16—21no—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4C2×C414yes—
60C2×C4×D4D412yes—
61C2×C4×D4Q814yes—
62C2×C4×D4C2312yes—
63C2×#117—21no—
64C2×#117—21no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C22⋊C4—21no—
70C22×C42C2311yescharacteristic
71#432—22no—
72#432—22no—
73#431—22no—
74#431C2×C414yes—
75C2×#107—22no—
76C2×#107—22no—
77C2×#106—22no—
78C2×#106—22no—
79#423—22no—
80#423C2×C414yes—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85C2×#102—22no—
86C2×#102—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×#97—21no—
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#390—22no—
93#390—22no—
94#390—22no—
95#390—22no—
96#388—24no—
97#388—24no—
98#388—24no—
99#388—24no—
100#384D411yescharacteristic
101#384D411yescharacteristic
102#384C2311yescharacteristic
103#384C2311yescharacteristic
104#384C2311yescharacteristic
105#384C2311yescharacteristic
106#382—21no—
107#382—21no—
108#382—21no—
109#381—24no—
110#380—21no—
111#380—21no—
112#380—22no—
113#380—22no—
114#380—21no—
115#380—21no—
116#380D412yes—
117#380D412yes—
118#379—24no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4C2312yes—
129C4×C22⋊C4—24no—
130C4×C22⋊C4—24no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137D4⋊C8C2×C418yes—
138D4⋊C8C2×C418yes—
139C25—21no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4C4⋊C414yes—
145C22×D4C2×D412yes—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—81no—
152C23×C4—81no—
153C23×C4—81no—
154C23×C4—81no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C2×M16—24no—
159C2×M16—81no—
160C2×M16—81no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4C4⋊C418yes—
173C4×D4C22×C414yes—
174C4×D4C2×Q814yes—
175#117—44no—
176#117—44no—
177#117—82no—
178#117—82no—
179#117—82no—
180#117—82no—
181#117—82no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
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—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C4⋊C414yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—41no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—41no—
220C2×C42C22×C412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×Q812yes—
225C2×C42#4112yes—
226C2×C42#4112yes—
227C2×C42C2411yescharacteristic
228#107—82no—
229#107—82no—
230#106—82no—
231#106—82no—
232C4⋊C8—24no—
233C4⋊C8—24no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8C22×C414yes—
238#102—44no—
239#102—44no—
240C22⋊C8—24no—
241C22⋊C8—42no—
242#97—42no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8C22×C412yes—
246C4×C8C22×C412yes—
247#95—24no—
248#95—24no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—41no—
254#95—41no—
255C24—22no—
256C24—24no—
257C24—42no—
258C24—42no—
259C24—81no—
260C24—81no—
261C24—81no—
262C24—81no—
263C24#12011yescharacteristic
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—48no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—48no—
274C2×D4C2×C4⋊C414yes—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—28no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—84no—
299C22×C4—84no—
300C22×C4—84no—
301C22×C4—84no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—81no—
305C22×C4—81no—
306C22×C4C2×C4⋊C412yes—
307C22×C4C4×D412yes—
308C22×C4C4×D412yes—
309C22×C4#12011yescharacteristic
310C22×C4#12011yescharacteristic
311C22×C4#12011yescharacteristic
312C22×C4C22⋊Q812yes—
313C22×C4C22⋊Q812yes—
314C22×C4#12312yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318M16—84no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—24no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C4⋊C4C2×C4⋊C414yes—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—28no—
361C22⋊C4—44no—
362C22⋊C4—44no—
363C22⋊C4—84no—
364C22⋊C4—84no—
365C22⋊C4—84no—
366C22⋊C4—84no—
367C22⋊C4—82no—
368C42—44no—
369C42—44no—
370C42—82no—
371C42—84no—
372C42—84no—
373C42C2×C4⋊C412yes—
374C42C4×D412yes—
375C42C4×D412yes—
376C42C22⋊Q812yes—
377C42C22⋊Q812yes—
378C42#12312yes—
379C42C23×C411yescharacteristic
380C42C22×Q811yescharacteristic
381C42C2×#4111yescharacteristic
382C42C2×#4111yescharacteristic
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—24no—
390C23—48no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—84no—
398C23—84no—
399C23—84no—
400C23—84no—
401C23—84no—
402C23—84no—
403C23#38512yes—
404C23C2×#12011yescharacteristic
405D4—24no—
406D4—24no—
407D4—48no—
408D4—42no—
409D4—42no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—88no—
446C2×C4—84no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4#38512yes—
454C2×C4#38512yes—
455C2×C4#38512yes—
456C2×C4C22×C4⋊C411yescharacteristic
457C2×C4C2×C4×D411yescharacteristic
458C2×C4C2×C4×D411yescharacteristic
459C2×C4C2×C22⋊Q811yescharacteristic
460C2×C4C2×C22⋊Q811yescharacteristic
461C2×C4C2×#12311yescharacteristic
462C2×C4#57512yes—
463C2×C4#57512yes—
464C2×C4#57611yescharacteristic
465C2×C4#57611yescharacteristic
466C2×C4#57711yescharacteristic
467C2×C4#57711yescharacteristic
468C8—42no—
469C8—42no—
470C8—44no—
471C8—82no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—44no—
480C22—44no—
481C22—44no—
482C22—816no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22C2×#38511yescharacteristic
488C22#300511yescharacteristic
489C22#300511yescharacteristic
490C22#301211yescharacteristic
491C22#301211yescharacteristic
492C22#302111yescharacteristic
493C22#302211yescharacteristic
494C4—22no—
495C4—21no—
496C4—21no—
497C4—22no—
498C4—22no—
499C4—44no—
500C4—82no—
501C4—82no—
502C4—82no—
503C4—82no—
504C4—161no—
505C4—161no—
506C4#291112yes—
507C4#294811yescharacteristic
508C4#294911yescharacteristic
509C4#308111yescharacteristic
510C4#308111yescharacteristic
511C4#309312yes—
512C2—42no—
513C2—84no—
514C2#1885611yescharacteristic
515C2#1885711yescharacteristic
516C2#1886411yescharacteristic
517C2#1886511yescharacteristic
518C2#2084511yescharacteristic
519C2#2086511yescharacteristic
520C2#2087911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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. #21984C2semidirect
3. #21964C2semidirect
6. #8897C2semidirect
7. #8893C2semidirect
8. #8889C2semidirect
9. #8885C2semidirect
10. #8042C2semidirect
11. #7900C2semidirect
31. #1948C22semidirect
33. #1947C22semidirect
35. #1845C22semidirect
36. #1844C22semidirect
37. #1834C22semidirect
38. #1834C22semidirect
45. #1486C4semidirect
48. #1296C22semidirect
49. #1296C22semidirect
50. #1279C22semidirect
51. #1279C22semidirect

Maximal subgroups

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

subgroupindex
#341862
#219842
#219642
#207382
C2×#14862
#88972
#88932
#88892
#88852
#80422
#79002