Finite Group Databasebeta

← #127121

Subgroups — #127121

Order 512.

881 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1869C211yesmaximal, characteristic
2#20722C211yesmaximal, characteristic
3#17364C211yesmaximal, characteristic
4#7204C214yesmaximal, complemented
5C24×C8C411yescharacteristic, complemented
6C22×C4×D4C2211yescharacteristic
7#2920—21no—
8C22×C22⋊C8—21no—
9C22×C22⋊C8C411yescharacteristic, complemented
10#2280—21no—
11#1869—21no—
12#1869—21no—
13#1869C412yescomplemented
14#1869C2212yes—
15#1819—21no—
16#1799—21no—
17#1799C412yes—
18#1781C412yes—
19C4×#99—21no—
20C4×#99C2212yes—
21C2×#336—21no—
22#1466—21no—
23#1466C412yescomplemented
24#1462—21no—
25#1462C412yescomplemented
26#1460—21no—
27#1460C2212yes—
28C23×D4C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C23×C8—21no—
32C23×C8—21no—
33C23×C8—21no—
34C23×C8—44no—
35C23×C8—41no—
36C23×C8—41no—
37C23×C8C2×C412yes—
38C2×C4×D4—41no—
39C2×C4×D4—41no—
40C2×C4×D4—41no—
41C2×C4×D4—41no—
42C2×C4×D4C2×C411yescharacteristic
43C2×C4×D4C2×C411yescharacteristic
44C2×C4×D4C2×C411yescharacteristic
45C2×C4×D4D411yescharacteristic
46C2×C4×D4D411yescharacteristic
47C2×C4×D4D411yescharacteristic
48C2×C4×D4Q811yescharacteristic
49C2×C4×D4C2311yescharacteristic
50C2×#117—41no—
51C22×C4⋊C4Q811yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4D411yescharacteristic
54C22×C42D411yescharacteristic
55#432—22no—
56#432—41no—
57#431—22no—
58#431—41no—
59#423—41no—
60C2×#99—22no—
61C2×#99C2×C414yes—
62#407—22no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—44no—
67C2×C22⋊C8—41no—
68C2×C22⋊C8—41no—
69C2×C22⋊C8C2×C412yes—
70C4×M16—41no—
71#381—41no—
72#380—22no—
73#380—41no—
74#379—22no—
75C4×C22⋊C4—22no—
76C4×C22⋊C4—41no—
77#342—22no—
78#342—22no—
79#336—22no—
80#336—22no—
81#323—22no—
82#323—22no—
83#323—22no—
84#323C2×C414yes—
85C25C4211yescharacteristic
86C25C22⋊C411yescharacteristic
87C22×D4—24no—
88C22×D4—41no—
89C22×D4—41no—
90C22×D4—42no—
91C22×D4C4211yescharacteristic
92C22×D4C22⋊C411yescharacteristic
93C22×D4C4⋊C411yescharacteristic
94C22×D4C22×C411yescharacteristic
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—42no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4C4211yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C4⋊C411yescharacteristic
114C23×C4C4⋊C411yescharacteristic
115C23×C4C4⋊C411yescharacteristic
116C23×C4C4⋊C411yescharacteristic
117C23×C4C4⋊C411yescharacteristic
118C23×C4C22×C411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C2×M16—42no—
121C22×C8—24no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—44no—
125C22×C8—44no—
126C22×C8—42no—
127C22×C8—42no—
128C22×C8—48no—
129C22×C8—44no—
130C22×C8—41no—
131C22×C8—41no—
132C22×C8—42no—
133C22×C8—44no—
134C22×C8—42no—
135C22×C8—42no—
136C22×C8—41no—
137C22×C8—41no—
138C4×D4—41no—
139C4×D4—41no—
140C4×D4—41no—
141C4×D4—41no—
142C4×D4—41no—
143C4×D4—41no—
144C4×D4—41no—
145C4×D4—41no—
146C4×D4—41no—
147C4×D4—41no—
148C4×D4—41no—
149C4×D4—41no—
150C4×D4—41no—
151C4×D4—41no—
152C4×D4—41no—
153C4×D4—41no—
154#117—82no—
155#117—82no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4C4⋊C411yescharacteristic
160C2×C4⋊C4C4⋊C411yescharacteristic
161C2×C4⋊C4C4⋊C411yescharacteristic
162C2×C4⋊C4C2×Q811yescharacteristic
163C2×C22⋊C4—24no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4C4211yescharacteristic
171C2×C22⋊C4C22⋊C411yescharacteristic
172C2×C22⋊C4C22⋊C411yescharacteristic
173C2×C22⋊C4C22⋊C411yescharacteristic
174C2×C22⋊C4C4⋊C411yescharacteristic
175C2×C22⋊C4C4⋊C411yescharacteristic
176C2×C22⋊C4C22×C411yescharacteristic
177C2×C22⋊C4C2×D411yescharacteristic
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—42no—
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C4⋊C411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C4⋊C8—42no—
188C4⋊C8—42no—
189#99—28no—
190#99—44no—
191C22⋊C8—24no—
192C22⋊C8—44no—
193C22⋊C8—44no—
194C22⋊C8—42no—
195C22⋊C8—42no—
196C22⋊C8—42no—
197#97—42no—
198C4×C8—42no—
199#95—42no—
200#95—42no—
201C24—21no—
202C24—21no—
203C24—22no—
204C24—22no—
205C24—22no—
206C24—21no—
207C24—21no—
208C24—41no—
209C24—42no—
210C24—42no—
211C24—42no—
212C24—42no—
213C24#9511yescharacteristic
214C24#9511yescharacteristic
215C24#9511yescharacteristic
216C24C2×C4211yescharacteristic
217C24C2×C22⋊C411yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—48no—
230C2×D4—48no—
231C2×D4#13114yes—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4#9511yescharacteristic
289C22×C4#9511yescharacteristic
290C22×C4#9511yescharacteristic
291C22×C4#9511yescharacteristic
292C22×C4#9511yescharacteristic
293C22×C4#9911yescharacteristic
294C22×C4#9911yescharacteristic
295C22×C4#9911yescharacteristic
296C22×C4#9911yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C4⋊C411yescharacteristic
300C22×C4C2×C4⋊C411yescharacteristic
301C22×C4C2×C4⋊C411yescharacteristic
302M16—84no—
303C2×C8—24no—
304C2×C8—42no—
305C2×C8—44no—
306C2×C8—48no—
307C2×C8—44no—
308C2×C8—44no—
309C2×C8—42no—
310C2×C8—44no—
311C2×C8—44no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—82no—
316C2×C8—82no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—84no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—84no—
333C22⋊C4—84no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—82no—
343C42—82no—
344C23—22no—
345C23—22no—
346C23—21no—
347C23—21no—
348C23—24no—
349C23—24no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—42no—
360C23—42no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—48no—
367C23—48no—
368C23—42no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—48no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—81no—
379C23—81no—
380C23—84no—
381C23#34211yescharacteristic
382C23#34211yescharacteristic
383C23#34412yes—
384C23C2×#9511yescharacteristic
385C23#40512yes—
386C23#40812yes—
387D4—44no—
388D4—44no—
389D4—44no—
390D4—44no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
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—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—84no—
446C2×C4#34211yescharacteristic
447C2×C4#34211yescharacteristic
448C2×C4AΣL(1,𝔽16)12yes—
449C2×C4#35212yes—
450C2×C4C2×#9911yescharacteristic
451C2×C4C2×#9911yescharacteristic
452C2×C4#42412yes—
453C8—44no—
454C8—82no—
455C8—84no—
456C22—24no—
457C22—24no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—48no—
472C22—48no—
473C22—42no—
474C22—42no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—81no—
485C22—84no—
486C22—84no—
487C22—81no—
488C22#139012yes—
489C22#139112yes—
490C22#172911yescharacteristic
491C22C2×#34211yescharacteristic
492C22C2×#34411yescharacteristic
493C4—21no—
494C4—21no—
495C4—22no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—164no—
504C4#139012yes—
505C4C2×AΣL(1,𝔽16)11yescharacteristic
506C4C2×#35211yescharacteristic
507C2—21no—
508C2—22no—
509C2—22no—
510C2—21no—
511C2—42no—
512C2—42no—
513C2—42no—
514C2—81no—
515C2#792711yescharacteristic
516C2C2×#139011yescharacteristic
517C2C2×#139111yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC24C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7204C2semidirect
5. C24×C8C4semidirect
9. C22×C22⋊C8C4semidirect
13. #1869C4semidirect
23. #1466C4semidirect
25. #1462C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18692
#207222
#173642
#72042