Finite Group Databasebeta

← #127398

Subgroups — #127398

Order 512.

874 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20898C211yesmaximal, characteristic
2#20722C211yesmaximal, characteristic
3C2×#1800C211yesmaximal, characteristic
4#7236C214yesmaximal, complemented
5C23×M16C411yescharacteristic, complemented
6C22×C4×D4C2211yescharacteristic
7C2×#423—21no—
8C22×#100—21no—
9C2×#407—21no—
10C2×#407C411yescharacteristic, complemented
11#2280—21no—
12#1870—21no—
13#1870—21no—
14#1869C412yescomplemented
15#1869C2212yes—
16#1819C412yes—
17#1800—21no—
18#1800—21no—
19#1800C412yescomplemented
20#1800C2212yes—
21#1799—21no—
22#1799C2212yes—
23#1781—21no—
24C4×#99—21no—
25C4×#99C412yes—
26C2×#343C412yescomplemented
27#1728—21no—
28C23×D4D411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—21no—
32C22×M16—21no—
33C22×M16—21no—
34C22×M16—22no—
35C22×M16—44no—
36C22×M16—41no—
37C22×M16—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
51C2×#117—41no—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4Q811yescharacteristic
55C22×C42D411yescharacteristic
56#423—22no—
57#423—22no—
58#423—41no—
59#423—41no—
60C2×C4⋊C8—41no—
61C2×C4⋊C8—41no—
62C2×#100—22no—
63C2×#100—22no—
64C2×#100—44no—
65C2×#99—22no—
66C2×#99—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×C412yes—
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—22no—
89C22×D4—22no—
90C22×D4—42no—
91C22×D4—41no—
92C22×D4—41no—
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×C4C4211yescharacteristic
110C23×C4C4211yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C22⋊C411yescharacteristic
116C23×C4C4⋊C411yescharacteristic
117C23×C4C4⋊C411yescharacteristic
118C23×C4C4⋊C411yescharacteristic
119C23×C4C4⋊C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C2×M16—24no—
123C2×M16—24no—
124C2×M16—44no—
125C2×M16—42no—
126C2×M16—42no—
127C2×M16—44no—
128C2×M16—48no—
129C2×M16—41no—
130C2×M16—41no—
131C2×M16—44no—
132C2×M16—42no—
133C2×M16—42no—
134C2×M16—42no—
135C2×M16—41no—
136C2×M16—41no—
137C2×M16—44no—
138C2×M16—44no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—44no—
142C22×C8—42no—
143C22×C8—42no—
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—
159C4×D4—41no—
160#117—82no—
161#117—82no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4C4211yescharacteristic
166C2×C4⋊C4C22⋊C411yescharacteristic
167C2×C4⋊C4C4⋊C411yescharacteristic
168C2×C4⋊C4C22×C411yescharacteristic
169C2×C22⋊C4—24no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4C4211yescharacteristic
177C2×C22⋊C4C22⋊C411yescharacteristic
178C2×C22⋊C4C4⋊C411yescharacteristic
179C2×C22⋊C4C4⋊C411yescharacteristic
180C2×C22⋊C4C4⋊C411yescharacteristic
181C2×C22⋊C4C4⋊C411yescharacteristic
182C2×C22⋊C4C22×C411yescharacteristic
183C2×C22⋊C4C2×Q811yescharacteristic
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—42no—
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C4⋊C411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C4⋊C8—42no—
194C4⋊C8—42no—
195C4⋊C8—42no—
196C4⋊C8—42no—
197#100—44no—
198#100—44no—
199#99—44no—
200#99—42no—
201#99—42no—
202C22⋊C8—44no—
203C22⋊C8—44no—
204C22⋊C8—42no—
205C22⋊C8—42no—
206#95—42no—
207#95—42no—
208C24—21no—
209C24—21no—
210C24—21no—
211C24—21no—
212C24—22no—
213C24—22no—
214C24—41no—
215C24—41no—
216C24—42no—
217C24—42no—
218C24—81no—
219C24—81no—
220C24#9511yescharacteristic
221C24#9511yescharacteristic
222C24#9511yescharacteristic
223C24C2×C22⋊C411yescharacteristic
224C24C2×C4⋊C411yescharacteristic
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4—44no—
240C2×D4—44no—
241C2×D4AGL(1,ℤ/8)12yes—
242C2×D4#13712yes—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
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—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
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#9511yescharacteristic
301C22×C4#9511yescharacteristic
302C22×C4#9511yescharacteristic
303C22×C4#9511yescharacteristic
304C22×C4#9511yescharacteristic
305C22×C4#9911yescharacteristic
306C22×C4#9911yescharacteristic
307C22×C4#9911yescharacteristic
308C22×C4#9911yescharacteristic
309C22×C4C2×C4211yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C4⋊C411yescharacteristic
313C22×C4C2×C4⋊C411yescharacteristic
314M16—44no—
315M16—48no—
316M16—44no—
317M16—44no—
318M16—44no—
319M16—82no—
320M16—82no—
321M16—84no—
322C2×C8—24no—
323C2×C8—42no—
324C2×C8—44no—
325C2×C8—42no—
326C2×C8—44no—
327C2×C8—42no—
328C2×C8—44no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—84no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—84no—
349C22⋊C4—84no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—84no—
359C23—22no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—41no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—42no—
384C23—42no—
385C23—41no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23—81no—
393C23—84no—
394C23—84no—
395C23—84no—
396C23—84no—
397C23#34211yescharacteristic
398C23#34211yescharacteristic
399C23C2×#9511yescharacteristic
400C23#41712yes—
401C23#42812yes—
402D4—42no—
403D4—42no—
404D4—44no—
405D4—44no—
406D4—42no—
407D4—42no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—84no—
462C2×C4—84no—
463C2×C4—84no—
464C2×C4H(3,ℤ/4)12yes—
465C2×C4#34211yescharacteristic
466C2×C4#34211yescharacteristic
467C2×C4AΣL(1,𝔽16)12yes—
468C2×C4#35212yes—
469C2×C4C2×#9911yescharacteristic
470C2×C4C2×#9911yescharacteristic
471C2×C4#41811yescharacteristic
472C2×C4#41911yescharacteristic
473C8—44no—
474C8—82no—
475C8—84no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—22no—
484C22—22no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—81no—
502C22—84no—
503C22—84no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—84no—
508C22—84no—
509C22#139012yes—
510C22C2×H(3,ℤ/4)11yescharacteristic
511C22#173611yescharacteristic
512C22C2×#34211yescharacteristic
513C4—21no—
514C4—21no—
515C4—21no—
516C4—22no—
517C4—21no—
518C4—41no—
519C4—41no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—164no—
525C4#139012yes—
526C4C2×AΣL(1,𝔽16)11yescharacteristic
527C4C2×#35211yescharacteristic
528C2—21no—
529C2—21no—
530C2—22no—
531C2—42no—
532C2—41no—
533C2—42no—
534C2—81no—
535C2—81no—
536C2#798311yescharacteristic
537C2C2×#139011yescharacteristic
538C2#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
19. #1800C4semidirect
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×#18002
#72362