Finite Group Databasebeta

← #127819

Subgroups — #127819

Order 512.

906 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20898C211yesmaximal, characteristic
2#20722C211yesmaximal, characteristic
3C2×C4×#100C211yesmaximal, characteristic
4#7282C214yesmaximal, complemented
5C23×M16C411yescharacteristic, complemented
6C22×C4×D4C2211yescharacteristic
7C22×#100—21no—
8C2×#407—21no—
9C2×#407C411yescharacteristic, complemented
10C2×C4×M16—21no—
11#2280—21no—
12#1870C412yescomplemented
13#1870C2212yes—
14#1869—21no—
15#1869—21no—
16#1819C412yes—
17#1799—21no—
18#1799C412yes—
19#1781—21no—
20C4×#100—21no—
21C4×#100—21no—
22C4×#100C412yescomplemented
23C4×#100C2212yes—
24C4×#99—21no—
25C4×#99C2212yes—
26C2×#343C412yescomplemented
27#1728—21no—
28C23×D4D411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—21no—
32C22×M16—21no—
33C22×M16—22no—
34C22×M16—44no—
35C22×M16—41no—
36C22×M16—41no—
37C22×M16C2×C412yes—
38C23×C8—21no—
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
51C2×#117—41no—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4Q811yescharacteristic
55C22×C42D411yescharacteristic
56C2×#100—28no—
57C2×#100—22no—
58C2×#100—22no—
59C2×#99—22no—
60C2×#99—22no—
61#407—44no—
62#407—41no—
63#407—41no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8C2×C412yes—
68C4×M16—22no—
69C4×M16—22no—
70C4×M16—41no—
71C4×M16—41no—
72C2×#97—41no—
73C2×C4×C8—41no—
74#381—41no—
75#380—22no—
76#380—41no—
77#379—22no—
78C4×C22⋊C4—22no—
79C4×C22⋊C4—41no—
80#343—22no—
81#343C2×C414yes—
82#342—22no—
83#342C2×C414yes—
84#336—22no—
85#336—22no—
86C25C22⋊C411yescharacteristic
87C25C4⋊C411yescharacteristic
88C22×D4—24no—
89C22×D4—42no—
90C22×D4—41no—
91C22×D4—41no—
92C22×D4C22⋊C411yescharacteristic
93C22×D4C22⋊C411yescharacteristic
94C22×D4C4⋊C411yescharacteristic
95C22×D4C2×D411yescharacteristic
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4C4211yescharacteristic
109C23×C4C4211yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C4⋊C411yescharacteristic
116C23×C4C4⋊C411yescharacteristic
117C23×C4C4⋊C411yescharacteristic
118C23×C4C4⋊C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C2×M16—24no—
122C2×M16—28no—
123C2×M16—24no—
124C2×M16—44no—
125C2×M16—44no—
126C2×M16—42no—
127C2×M16—42no—
128C2×M16—48no—
129C2×M16—44no—
130C2×M16—41no—
131C2×M16—41no—
132C2×M16—44no—
133C2×M16—42no—
134C2×M16—42no—
135C2×M16—41no—
136C2×M16—41no—
137C2×M16—42no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—44no—
141C22×C8—42no—
142C22×C8—42no—
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—
159#117—82no—
160#117—82no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4C4211yescharacteristic
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C4⋊C411yescharacteristic
167C2×C4⋊C4C22×C411yescharacteristic
168C2×C22⋊C4—24no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4C4211yescharacteristic
176C2×C22⋊C4C22⋊C411yescharacteristic
177C2×C22⋊C4C4⋊C411yescharacteristic
178C2×C22⋊C4C4⋊C411yescharacteristic
179C2×C22⋊C4C4⋊C411yescharacteristic
180C2×C22⋊C4C4⋊C411yescharacteristic
181C2×C22⋊C4C22×C411yescharacteristic
182C2×C22⋊C4C2×Q811yescharacteristic
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—42no—
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C4⋊C411yescharacteristic
191C2×C42C2×D411yescharacteristic
192#100—28no—
193#100—28no—
194#99—44no—
195#99—44no—
196C22⋊C8—44no—
197C22⋊C8—44no—
198C22⋊C8—42no—
199C22⋊C8—42no—
200#97—42no—
201#97—42no—
202C4×C8—42no—
203C4×C8—42no—
204#95—42no—
205#95—42no—
206C24—21no—
207C24—21no—
208C24—21no—
209C24—21no—
210C24—22no—
211C24—22no—
212C24—41no—
213C24—41no—
214C24—42no—
215C24—42no—
216C24—81no—
217C24—81no—
218C24#9511yescharacteristic
219C24#9511yescharacteristic
220C24#9511yescharacteristic
221C24C2×C22⋊C411yescharacteristic
222C24C2×C4⋊C411yescharacteristic
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—48no—
235C2×D4—48no—
236C2×D4#10414yes—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
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—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—81no—
293C22×C4#9511yescharacteristic
294C22×C4#9511yescharacteristic
295C22×C4#9511yescharacteristic
296C22×C4#9511yescharacteristic
297C22×C4#9511yescharacteristic
298C22×C4#9911yescharacteristic
299C22×C4#9911yescharacteristic
300C22×C4#9911yescharacteristic
301C22×C4#9911yescharacteristic
302C22×C4C2×C4211yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C4⋊C411yescharacteristic
306C22×C4C2×C4⋊C411yescharacteristic
307M16—28no—
308M16—48no—
309M16—44no—
310M16—44no—
311M16—44no—
312M16—44no—
313M16—44no—
314M16—82no—
315M16—82no—
316C2×C8—24no—
317C2×C8—44no—
318C2×C8—42no—
319C2×C8—48no—
320C2×C8—48no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—84no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—84no—
341C22⋊C4—84no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—82no—
351C42—82no—
352C23—22no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—41no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—42no—
377C23—42no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—42no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—84no—
387C23—84no—
388C23—84no—
389C23—84no—
390C23#33912yes—
391C23#34211yescharacteristic
392C23#34211yescharacteristic
393C23C2×#9511yescharacteristic
394C23C2×#10412yes—
395D4—44no—
396D4—44no—
397D4—44no—
398D4—44no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4#33912yes—
454C2×C4#34211yescharacteristic
455C2×C4#34211yescharacteristic
456C2×C4AΣL(1,𝔽16)12yes—
457C2×C4#35212yes—
458C2×C4C2×#9911yescharacteristic
459C2×C4C2×#9911yescharacteristic
460C2×C4#42112yes—
461C8—48no—
462C8—44no—
463C8—82no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—22no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—81no—
490C22—84no—
491C22—84no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—84no—
496C22—84no—
497C22#139012yes—
498C22C2×#33911yescharacteristic
499C22#173311yescharacteristic
500C22C2×#34211yescharacteristic
501C4—21no—
502C4—21no—
503C4—22no—
504C4—22no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—164no—
512C4#139012yes—
513C4C2×AΣL(1,𝔽16)11yescharacteristic
514C4C2×#35211yescharacteristic
515C2—21no—
516C2—21no—
517C2—22no—
518C2—42no—
519C2—41no—
520C2—42no—
521C2—81no—
522C2—81no—
523C2#805111yescharacteristic
524C2C2×#139011yescharacteristic
525C2#1258911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ 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. #7282C2semidirect
5. C23×M16C4semidirect
9. C2×#407C4semidirect
12. #1870C4semidirect
22. C4×#100C4semidirect
26. C2×#343C4semidirect

Maximal subgroups

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

subgroupindex
#208982
#207222
C2×C4×#1002
#72822