Finite Group Databasebeta

← #127389

Subgroups — #127389

Order 512.

887 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20898C211yesmaximal, characteristic
2#20726C211yesmaximal, characteristic
3C2×#1799C211yesmaximal, characteristic
4#7236C214yesmaximal, complemented
5C23×M16C411yescharacteristic, complemented
6C22×C4×D4C2211yescharacteristic
7#2920—21no—
8C22×#99—21no—
9C2×#407—21no—
10C2×#407C411yescharacteristic, complemented
11C2×#380—21no—
12#1870—21no—
13#1870—21no—
14#1869C412yescomplemented
15#1869C2212yes—
16#1821C412yescomplemented
17#1800—21no—
18#1800C2212yes—
19#1799—21no—
20#1799—21no—
21#1799C412yes—
22#1799C2212yes—
23#1783—21no—
24C4×#100—21no—
25C4×#100C412yescomplemented
26C2×#342C412yes—
27#1728—21no—
28C23×D4D411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—21no—
32C22×M16—21no—
33C22×M16—21no—
34C22×M16—44no—
35C22×M16—41no—
36C22×M16—41no—
37C23×C8C2×C412yescomplemented
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
50C22×C4⋊C4—41no—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C22⋊C4—22no—
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4Q811yescharacteristic
55C22×C42D411yescharacteristic
56#432—41no—
57#432—41no—
58#431—41no—
59#431—41no—
60#423—22no—
61C2×#100—22no—
62C2×#100—22no—
63C2×#99—22no—
64C2×#99—22no—
65C2×#99—44no—
66#407—22no—
67#407—44no—
68#407—41no—
69#407—41no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8C2×C412yescomplemented
74C4×M16—22no—
75#380—22no—
76#380—22no—
77#380—41no—
78#380—41no—
79C2×#95—41no—
80#343—22no—
81#343C2×C414yescomplemented
82#342—22no—
83#342C2×C414yes—
84#336—22no—
85#336—22no—
86C25C22⋊C411yescharacteristic
87C25C4⋊C411yescharacteristic
88C22×D4—22no—
89C22×D4—22no—
90C22×D4—41no—
91C22×D4—41no—
92C22×D4—42no—
93C22×D4C22⋊C411yescharacteristic
94C22×D4C22⋊C411yescharacteristic
95C22×D4C4⋊C411yescharacteristic
96C22×D4C2×D411yescharacteristic
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4—42no—
109C23×C4—42no—
110C23×C4C4211yescharacteristic
111C23×C4C4211yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C22⋊C411yescharacteristic
116C23×C4C22⋊C411yescharacteristic
117C23×C4C4⋊C411yescharacteristic
118C23×C4C4⋊C411yescharacteristic
119C23×C4C4⋊C411yescharacteristic
120C23×C4C4⋊C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C2×M16—24no—
124C2×M16—24no—
125C2×M16—44no—
126C2×M16—42no—
127C2×M16—42no—
128C2×M16—44no—
129C2×M16—48no—
130C2×M16—41no—
131C2×M16—41no—
132C2×M16—44no—
133C2×M16—42no—
134C2×M16—42no—
135C2×M16—41no—
136C2×M16—41no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—44no—
140C22×C8—42no—
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—
154C4×D4—41no—
155C4×D4—41no—
156C4×D4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4C4211yescharacteristic
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C4⋊C411yescharacteristic
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C22⋊C4—24no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
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×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C4⋊C411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C4⋊C8—42no—
192C4⋊C8—42no—
193#100—44no—
194#100—42no—
195#100—42no—
196#99—44no—
197#99—44no—
198C22⋊C8—44no—
199C22⋊C8—44no—
200C22⋊C8—42no—
201C22⋊C8—42no—
202C22⋊C8—42no—
203C22⋊C8—42no—
204#97—42no—
205C4×C8—42no—
206#95—42no—
207#95—42no—
208C24—21no—
209C24—22no—
210C24—21no—
211C24—21no—
212C24—21no—
213C24—21no—
214C24—21no—
215C24—22no—
216C24—41no—
217C24—42no—
218C24—42no—
219C24—81no—
220C24—81no—
221C24#9511yescharacteristic
222C24#9511yescharacteristic
223C24#9511yescharacteristic
224C24C2×C22⋊C411yescharacteristic
225C24C2×C4⋊C411yescharacteristic
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—44no—
239C2×D4—44no—
240C2×D4—44no—
241C2×D4—44no—
242C2×D4AGL(1,ℤ/8)12yes—
243C2×D4#13712yes—
244C22×C4—22no—
245C22×C4—22no—
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—21no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—81no—
300C22×C4—84no—
301C22×C4—84no—
302C22×C4—84no—
303C22×C4#9511yescharacteristic
304C22×C4#9511yescharacteristic
305C22×C4#9511yescharacteristic
306C22×C4#9511yescharacteristic
307C22×C4#9511yescharacteristic
308C22×C4#9911yescharacteristic
309C22×C4#9911yescharacteristic
310C22×C4#9911yescharacteristic
311C22×C4#9911yescharacteristic
312C22×C4C2×C4211yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C4⋊C411yescharacteristic
316C22×C4C2×C4⋊C411yescharacteristic
317M16—44no—
318M16—48no—
319M16—44no—
320M16—44no—
321M16—44no—
322M16—82no—
323M16—82no—
324C2×C8—24no—
325C2×C8—42no—
326C2×C8—44no—
327C2×C8—42no—
328C2×C8—44no—
329C2×C8—44no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—84no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—84no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—41no—
376C23—42no—
377C23—42no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—42no—
389C23—42no—
390C23—81no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23—84no—
395C23—82no—
396C23—82no—
397C23—84no—
398C23—84no—
399C23H(3,ℤ/4)12yes—
400C23#34211yescharacteristic
401C23#34211yescharacteristic
402C23C2×#9511yescharacteristic
403C23#41811yescharacteristic
404C23#41911yescharacteristic
405C23#42812yes—
406D4—42no—
407D4—42no—
408D4—44no—
409D4—44no—
410D4—42no—
411D4—42no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—84no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4—168no—
468C2×C4#34211yescharacteristic
469C2×C4#34211yescharacteristic
470C2×C4AΣL(1,𝔽16)12yes—
471C2×C4#35212yes—
472C2×C4C2×#9911yescharacteristic
473C2×C4C2×#9911yescharacteristic
474C2×C4#41712yes—
475C8—44no—
476C8—82no—
477C8—84no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—22no—
487C22—21no—
488C22—22no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—41no—
493C22—41no—
494C22—42no—
495C22—44no—
496C22—44no—
497C22—42no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—81no—
508C22—81no—
509C22—84no—
510C22—81no—
511C22—81no—
512C22—82no—
513C22—82no—
514C22—84no—
515C22#139012yes—
516C22C2×H(3,ℤ/4)11yescharacteristic
517C22#173611yescharacteristic
518C22C2×#34211yescharacteristic
519C4—21no—
520C4—21no—
521C4—22no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—164no—
530C4#139012yes—
531C4C2×AΣL(1,𝔽16)11yescharacteristic
532C4C2×#35211yescharacteristic
533C2—22no—
534C2—21no—
535C2—21no—
536C2—41no—
537C2—41no—
538C2—42no—
539C2—41no—
540C2—42no—
541C2—81no—
542C2#798311yescharacteristic
543C2C2×#139011yescharacteristic
544C2#1258911yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. #7236C2semidirect
5. C23×M16C4semidirect
10. C2×#407C4semidirect
14. #1869C4semidirect
16. #1821C4semidirect
25. C4×#100C4semidirect
37. C23×C8C2×C4semidirect
73. C2×C22⋊C8C2×C4semidirect
81. #343C2×C4semidirect

Maximal subgroups

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

subgroupindex
#208982
#207262
C2×#17992
#72362