Finite Group Databasebeta

← #182011

Subgroups — #182011

Order 512.

2026 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#388C211yesmaximal, characteristic
2C4×#407C212yesmaximal, complemented
3C2×#1462C212yesmaximal
4#8155C218yesmaximal, complemented
5#7909C212yesmaximal, complemented
6C23×C4⋊C4C411yescharacteristic
7C23×C22⋊C4C411yescharacteristic
8C23×C22⋊C4C411yescharacteristic
9C23×C22⋊C4C2211yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#407C2214yes—
12C2×C4×M16—22no—
13C2×#388—22no—
14C2×#388—22no—
15C2×#388—22no—
16C2×#388—22no—
17C2×#388C414yes—
18C2×#388C2214yes—
19C22×#95C411yescharacteristic
20C22×#95C2211yescharacteristic
21#1749—22no—
22#1749C2214yescomplemented
23C4×C22⋊C8—22no—
24C4×C22⋊C8C2214yescomplemented
25#1728—22no—
26#1462C4116yescomplemented
27C2×#323C2218yes—
28#1280C2218yescomplemented
29C26C2×C411yescharacteristic
30C24×C4C2×C411yescharacteristic
31C24×C4C2×C411yescharacteristic
32C24×C4C2×C411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C24×C4C2×C411yescharacteristic
35C24×C4C2311yescharacteristic
36C22×M16—24no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—44no—
41C22×C4⋊C4C2×C414yes—
42C22×C22⋊C4—22no—
43C22×C22⋊C4—22no—
44C22×C22⋊C4—24no—
45C22×C22⋊C4—22no—
46C22×C22⋊C4—21no—
47C22×C22⋊C4—22no—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—44no—
52C22×C22⋊C4—44no—
53C22×C22⋊C4C2×C412yes—
54C22×C22⋊C4C2×C418yes—
55C22×C22⋊C4C2×C414yes—
56C22×C22⋊C4C2×C414yes—
57C22×C22⋊C4C2312yes—
58C22×C42—44no—
59C22×C42C2×C412yes—
60C22×C42D412yes—
61C22×C42D412yes—
62C22×C42D412yes—
63C22×C42D412yes—
64C22×C42C2312yes—
65#407—28no—
66#407C2×C4116yes—
67C2×C22⋊C8—24no—
68C2×C22⋊C8C2318yes—
69C4×M16—48no—
70C2×#97—24no—
71C2×C4×C8—24no—
72#388—24no—
73#388—24no—
74#388—24no—
75#388—24no—
76#388—24no—
77#388—24no—
78#388—24no—
79#388—24no—
80C2×#95—22no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95C2×C414yes—
86C2×#95C2×C412yes—
87C2×#95C2312yes—
88#336—24no—
89#336—24no—
90#323C2×C4132yescomplemented
91C25—21no—
92C25—21no—
93C25—22no—
94C25—42no—
95C25—42no—
96C25—42no—
97C25—42no—
98C25—42no—
99C25—42no—
100C25C4214yes—
101C25C22×C411yescharacteristic
102C25C22×C412yes—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—24no—
107C23×C4—24no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4—42no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4C4214yes—
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22×C412yes—
134C23×C4C22×C412yes—
135C23×C4C22×C412yes—
136C23×C4C22×C412yes—
137C23×C4C22×C412yes—
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C414yes—
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4#4114yes—
143C23×C4C2411yescharacteristic
144C2×M16—216no—
145C2×M16—48no—
146C2×M16—48no—
147C22×C8—28no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—48no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—48no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—216no—
157C2×C22⋊C4—216no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—44no—
166C2×C22⋊C4—48no—
167C2×C22⋊C4—48no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—48no—
171C2×C22⋊C4—48no—
172C2×C22⋊C4C4218yes—
173C2×C22⋊C4C22×C418yes—
174C2×C42—28no—
175C2×C42—44no—
176C2×C42—44no—
177C2×C42—44no—
178C2×C42—44no—
179C2×C42—44no—
180C2×C42C22⋊C412yes—
181C2×C42C22⋊C412yes—
182C2×C42C22⋊C412yes—
183C2×C42C22⋊C412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C22⋊C8—216no—
189C22⋊C8—28no—
190C22⋊C8C22×C4116yes—
191#97—48no—
192C4×C8—48no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95—24no—
197C24—22no—
198C24—22no—
199C24—22no—
200C24—22no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24—22no—
205C24—24no—
206C24—24no—
207C24—24no—
208C24—24no—
209C24—48no—
210C24—42no—
211C24—42no—
212C24—42no—
213C24—42no—
214C24—42no—
215C24—42no—
216C24—42no—
217C24—42no—
218C24—48no—
219C24—48no—
220C24—48no—
221C24—42no—
222C24—42no—
223C24—48no—
224C24—48no—
225C24—42no—
226C24—42no—
227C24—42no—
228C24—42no—
229C24—42no—
230C24—42no—
231C24—88no—
232C24—88no—
233C24—88no—
234C24—88no—
235C24C2×C4212yes—
236C24C2×C4214yes—
237C24C2×C22⋊C411yescharacteristic
238C24C2×C22⋊C411yescharacteristic
239C24#11712yes—
240C24C23×C411yescharacteristic
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—48no—
278C22×C4—42no—
279C22×C4—48no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—416no—
283C22×C4—416no—
284C22×C4—42no—
285C22×C4—48no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—48no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—88no—
308C22×C4—88no—
309C22×C4#9914yes—
310C22×C4#10014yes—
311C22×C4C2×C4214yes—
312C22×C4C2×C4212yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4#11712yes—
319C22×C4C4×D418yes—
320C22×C4C4×D418yes—
321C22×C4C23×C412yes—
322C22×C4C22×D412yes—
323C22×C4C2×#4112yes—
324M16—432no—
325C2×C8—216no—
326C2×C8—48no—
327C2×C8—48no—
328C4⋊C4—48no—
329C4⋊C4—48no—
330C22⋊C4—216no—
331C22⋊C4—216no—
332C22⋊C4—48no—
333C22⋊C4—48no—
334C22⋊C4—48no—
335C22⋊C4—48no—
336C42—24no—
337C42—24no—
338C42—24no—
339C42—24no—
340C42—42no—
341C42—42no—
342C42—42no—
343C42—42no—
344C23—28no—
345C23—28no—
346C23—28no—
347C23—28no—
348C23—24no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—24no—
362C23—24no—
363C23—24no—
364C23—24no—
365C23—24no—
366C23—24no—
367C23—28no—
368C23—48no—
369C23—48no—
370C23—44no—
371C23—48no—
372C23—44no—
373C23—48no—
374C23—44no—
375C23—44no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—44no—
383C23—48no—
384C23—48no—
385C23—84no—
386C23—84no—
387C23—84no—
388C23—84no—
389C23—832no—
390C23—832no—
391C23—84no—
392C23—84no—
393C23—84no—
394C23—84no—
395C23—84no—
396C23—84no—
397C23—84no—
398C23—84no—
399C23C4×C22⋊C412yes—
400C23C4×C22⋊C412yes—
401C23C2×#9912yes—
402C23#41012yes—
403C23C2×#10012yes—
404C23#41312yes—
405C23C22×C4211yescharacteristic
406C23C22×C22⋊C411yescharacteristic
407C23C2×#11711yescharacteristic
408C23#56812yes—
409C23#56812yes—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—216no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—48no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—48no—
434C2×C4—48no—
435C2×C4—48no—
436C2×C4—48no—
437C2×C4—44no—
438C2×C4—48no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—48no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4—84no—
454C2×C4—84no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4C4×C22⋊C414yes—
458C2×C4C2×#9914yes—
459C2×C4C2×#10014yes—
460C2×C4C2×C4×D414yes—
461C8—416no—
462C22—216no—
463C22—216no—
464C22—22no—
465C22—22no—
466C22—21no—
467C22—22no—
468C22—21no—
469C22—21no—
470C22—22no—
471C22—21no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—44no—
485C22—44no—
486C22—44no—
487C22—88no—
488C22—88no—
489C22—816no—
490C22—816no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22C4×#9914yes—
496C22C4×#10014yes—
497C22C2×C4×C22⋊C411yescharacteristic
498C22#287312yes—
499C22#287911yescharacteristic
500C22#287911yescharacteristic
501C22C22×#9911yescharacteristic
502C22C2×#41011yescharacteristic
503C22C22×#10011yescharacteristic
504C22C2×#41311yescharacteristic
505C22#292112yes—
506C4—24no—
507C4—24no—
508C4—24no—
509C4—24no—
510C4—44no—
511C4—42no—
512C4—42no—
513C4—82no—
514C4—82no—
515C2—22no—
516C2—22no—
517C2—24no—
518C2—24no—
519C2—84no—
520C2#1739211yescharacteristic
521C2#1739311yescharacteristic
522C2#1739611yescharacteristic
523C2#1739811yescharacteristic
524C2#2037711yescharacteristic
525C2C2×C4×#9911yescharacteristic
526C2C2×C4×#10011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
prime radicalC26C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#407C2semidirect
4. #8155C2semidirect
5. #7909C2semidirect
22. #1749C22semidirect
24. C4×C22⋊C8C22semidirect
26. #1462C4semidirect
28. #1280C22semidirect
90. #323C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3882
C4×#4072
C2×#14622
#81552
#79092