Finite Group Databasebeta

← #127101

Subgroups — #127101

Order 512.

831 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#20898C211yesmaximal, characteristic
2#20726C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4#7203C214yesmaximal, complemented
5C23×M16C411yescharacteristic
6C22×C4×D4C2211yescharacteristic
7#2920—21no—
8C2×#407—21no—
9C2×#407C411yescharacteristic
10C22×C22⋊C8—21no—
11#1870—21no—
12#1870—21no—
13#1869C412yes—
14#1869C2212yes—
15#1821—21no—
16#1800—21no—
17#1800C412yes—
18#1783C412yes—
19C4×#100—21no—
20C4×#100C2212yes—
21#1728—21no—
22#1459—21no—
23#1459—21no—
24#1459C412yes—
25#1459C2212yes—
26C2×#324—21no—
27C2×#323C412yes—
28C23×D4C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—21no—
32C22×M16—21no—
33C22×M16—21no—
34C22×M16—44no—
35C22×M16—41no—
36C22×M16—41no—
37C23×C8—41no—
38C23×C8C2×C412yes—
39C2×C4×D4—41no—
40C2×C4×D4—41no—
41C2×C4×D4—41no—
42C2×C4×D4—41no—
43C2×C4×D4C2×C411yescharacteristic
44C2×C4×D4C2×C411yescharacteristic
45C2×C4×D4C2×C411yescharacteristic
46C2×C4×D4D411yescharacteristic
47C2×C4×D4D411yescharacteristic
48C2×C4×D4D411yescharacteristic
49C2×C4×D4Q811yescharacteristic
50C2×C4×D4C2311yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4Q811yescharacteristic
54C22×C42D411yescharacteristic
55#432—41no—
56#432—41no—
57#431—41no—
58#431—41no—
59#423—22no—
60C2×#100—22no—
61C2×#100C2×C414yes—
62#407—22no—
63#407—44no—
64#407—41no—
65#407—41no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—41no—
73C2×C22⋊C8—41no—
74C2×C22⋊C8—41no—
75C2×C22⋊C8C2×C412yes—
76C4×M16—22no—
77#343—22no—
78#343—22no—
79#336—22no—
80#336—22no—
81#324—22no—
82#324—22no—
83#323—22no—
84#323C2×C414yes—
85C25C4211yescharacteristic
86C25C4⋊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×C4C22⋊C411yescharacteristic
114C23×C4C4⋊C411yescharacteristic
115C23×C4C4⋊C411yescharacteristic
116C23×C4C4⋊C411yescharacteristic
117C23×C4C4⋊C411yescharacteristic
118C23×C4C22×C411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C2×M16—24no—
121C2×M16—24no—
122C2×M16—44no—
123C2×M16—42no—
124C2×M16—42no—
125C2×M16—44no—
126C2×M16—48no—
127C2×M16—41no—
128C2×M16—41no—
129C2×M16—44no—
130C2×M16—42no—
131C2×M16—42no—
132C2×M16—41no—
133C2×M16—41no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—44no—
137C22×C8—42no—
138C22×C8—42no—
139C22×C8—42no—
140C22×C8—42no—
141C22×C8—82no—
142C22×C8—82no—
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—
154C4×D4—41no—
155C4×D4—41no—
156C4×D4—41no—
157C4×D4—41no—
158C4×D4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4C22⋊C411yescharacteristic
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C4⋊C411yescharacteristic
164C2×C4⋊C4C2×D411yescharacteristic
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4C4211yescharacteristic
170C2×C22⋊C4C22⋊C411yescharacteristic
171C2×C22⋊C4C4⋊C411yescharacteristic
172C2×C22⋊C4C4⋊C411yescharacteristic
173C2×C22⋊C4C4⋊C411yescharacteristic
174C2×C22⋊C4C4⋊C411yescharacteristic
175C2×C22⋊C4C22×C411yescharacteristic
176C2×C22⋊C4C2×Q811yescharacteristic
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C4⋊C411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C4⋊C8—42no—
186C4⋊C8—42no—
187#100—28no—
188#100—44no—
189C22⋊C8—24no—
190C22⋊C8—44no—
191C22⋊C8—44no—
192C22⋊C8—42no—
193C22⋊C8—42no—
194C22⋊C8—42no—
195C22⋊C8—42no—
196C22⋊C8—42no—
197C22⋊C8—42no—
198C22⋊C8—42no—
199#97—42no—
200C4×C8—42no—
201C24—21no—
202C24—21no—
203C24—22no—
204C24—22no—
205C24—21no—
206C24—21no—
207C24—22no—
208C24—41no—
209C24—42no—
210C24—42no—
211C24—81no—
212C24—81no—
213C24#9511yescharacteristic
214C24#9511yescharacteristic
215C24#9511yescharacteristic
216C24C2×C4211yescharacteristic
217C24C2×C4⋊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—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
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—81no—
283C22×C4—81no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—81no—
287C22×C4#9511yescharacteristic
288C22×C4#9511yescharacteristic
289C22×C4#9511yescharacteristic
290C22×C4#9511yescharacteristic
291C22×C4#9511yescharacteristic
292C22×C4#9911yescharacteristic
293C22×C4#9911yescharacteristic
294C22×C4#9911yescharacteristic
295C22×C4#9911yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C4⋊C411yescharacteristic
300C22×C4C2×C4⋊C411yescharacteristic
301M16—44no—
302M16—48no—
303M16—44no—
304M16—44no—
305M16—44no—
306M16—82no—
307M16—82no—
308C2×C8—24no—
309C2×C8—42no—
310C2×C8—44no—
311C2×C8—42no—
312C2×C8—44no—
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—84no—
319C2×C8—82no—
320C2×C8—82no—
321C2×C8—84no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—21no—
342C23—24no—
343C23—24no—
344C23—22no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—41no—
357C23—42no—
358C23—42no—
359C23—41no—
360C23—41no—
361C23—42no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23—42no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—84no—
376C23—84no—
377C23—84no—
378C23—84no—
379C23#34211yescharacteristic
380C23#34211yescharacteristic
381C23#34412yes—
382C23C2×#9511yescharacteristic
383C23#40512yes—
384C23#42412yes—
385D4—44no—
386D4—44no—
387D4—44no—
388D4—44no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
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—42no—
408C2×C4—42no—
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—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4#34211yescharacteristic
442C2×C4#34211yescharacteristic
443C2×C4AΣL(1,𝔽16)12yes—
444C2×C4#35212yes—
445C2×C4#40812yes—
446C2×C4C2×#9911yescharacteristic
447C2×C4C2×#9911yescharacteristic
448C8—44no—
449C8—82no—
450C8—84no—
451C8—84no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—48no—
468C22—42no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—81no—
479C22—81no—
480C22—84no—
481C22—81no—
482C22—81no—
483C22—84no—
484C22—84no—
485C22#139012yes—
486C22#172911yescharacteristic
487C22C2×#34211yescharacteristic
488C22C2×#34411yescharacteristic
489C4—21no—
490C4—21no—
491C4—22no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4#139012yes—
500C4C2×AΣL(1,𝔽16)11yescharacteristic
501C4C2×#35211yescharacteristic
502C2—22no—
503C2—21no—
504C2—21no—
505C2—41no—
506C2—42no—
507C2—42no—
508C2—42no—
509C2—81no—
510C2#792511yescharacteristic
511C2C2×#139011yescharacteristic
512C2#1257911yescharacteristic
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. #7203C2semidirect

Maximal subgroups

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

subgroupindex
#208982
#207262
C2×#14592
#72032