Finite Group Databasebeta

← #361838

Subgroups — #361838

Order 512.

3046 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#380C211yesmaximal, characteristic
2C2×#1819C214yesmaximal
3#14680C211yesmaximal, characteristic
4C4×#379C211yesmaximal, characteristic
5#11353C214yesmaximal, complemented
6#10854C214yesmaximal, complemented
7C25×C4C2211yescharacteristic
8C23×C4⋊C4C412yescomplemented
9C23×C22⋊C4C2212yes—
10C2×#380—22no—
11C2×#380—22no—
12C2×#380—22no—
13C2×#380—22no—
14C2×#380—22no—
15C2×#380C412yescomplemented
16C2×#380C412yescomplemented
17C2×#380C2212yes—
18C2×#380C2212yes—
19C2×#380C2212yes—
20C2×#379C2212yes—
21C2×#379C2212yes—
22C2×C4×C22⋊C4—22no—
23C2×C4×C22⋊C4C2212yes—
24C22×#95C412yescomplemented
25#1819—216no—
26C2×#342C22116yes—
27C22⋊#95—22no—
28C22⋊#95—22no—
29C22⋊#95C2212yes—
30C22⋊#95C2212yes—
31C26C2311yescharacteristic
32C24×C4—28no—
33C24×C4—48no—
34C24×C4C2×C411yescharacteristic
35C24×C4C2×C414yes—
36C24×C4C2×C411yescharacteristic
37C24×C4C2×C411yescharacteristic
38C24×C4C2×C411yescharacteristic
39C24×C4D412yes—
40C24×C4D412yes—
41C24×C4D412yes—
42C24×C4D412yes—
43C24×C4C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—48no—
51C22×C4⋊C4C2×C414yes—
52C22×C4⋊C4C2×C414yes—
53C22×C22⋊C4—28no—
54C22×C22⋊C4—24no—
55C22×C22⋊C4—24no—
56C22×C22⋊C4—24no—
57C22×C22⋊C4—28no—
58C22×C22⋊C4—24no—
59C22×C22⋊C4C2×C414yes—
60C22×C22⋊C4C2314yes—
61C22×C22⋊C4C2314yes—
62C22×C22⋊C4C2314yes—
63C22×C42—22no—
64#380—24no—
65#380—24no—
66#380—24no—
67#380—24no—
68#380—24no—
69#380—24no—
70#380—22no—
71#380—22no—
72#380—22no—
73#380—22no—
74#380—28no—
75#380—48no—
76#379—24no—
77#379—28no—
78#379D418yes—
79C4×C22⋊C4—28no—
80C4×C22⋊C4—48no—
81C2×#95—24no—
82C2×#95—24no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—24no—
86C2×#95—24no—
87C2×#95C2×C414yes—
88C2×#95C2×C414yes—
89#342—232no—
90C25—24no—
91C25—24no—
92C25—44no—
93C25—44no—
94C25C22×C411yescharacteristic
95C25C22×C411yescharacteristic
96C25C2×D412yes—
97C25C2×D412yes—
98C25C2×D414yes—
99C25#4114yes—
100C25C2411yescharacteristic
101C23×C4—28no—
102C23×C4—24no—
103C23×C4—24no—
104C23×C4—28no—
105C23×C4—28no—
106C23×C4—24no—
107C23×C4—24no—
108C23×C4—24no—
109C23×C4—28no—
110C23×C4—28no—
111C23×C4—28no—
112C23×C4—28no—
113C23×C4—24no—
114C23×C4—24no—
115C23×C4—24no—
116C23×C4—216no—
117C23×C4—24no—
118C23×C4—44no—
119C23×C4—44no—
120C23×C4—44no—
121C23×C4—44no—
122C23×C4—44no—
123C23×C4—44no—
124C23×C4—44no—
125C23×C4—44no—
126C23×C4—44no—
127C23×C4—44no—
128C23×C4—44no—
129C23×C4—464no—
130C23×C4—48no—
131C23×C4—48no—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C412yes—
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C412yes—
139C23×C4C22⋊C412yes—
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C418yes—
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4#4118yes—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4—48no—
160C2×C4⋊C4—48no—
161C2×C4⋊C4—48no—
162C2×C4⋊C4—48no—
163C2×C4⋊C4—44no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—216no—
168C2×C22⋊C4—216no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—48no—
178C2×C22⋊C4—48no—
179C2×C22⋊C4—416no—
180C2×C22⋊C4—48no—
181C2×C22⋊C4—48no—
182C2×C22⋊C4—48no—
183C2×C22⋊C4—48no—
184C2×C22⋊C4—48no—
185C2×C22⋊C4C2×D418yes—
186C2×C42—48no—
187C2×C42—44no—
188#95—28no—
189#95—24no—
190#95—24no—
191#95—48no—
192#95—48no—
193#95—44no—
194#95—44no—
195C24—22no—
196C24—22no—
197C24—24no—
198C24—24no—
199C24—22no—
200C24—22no—
201C24—24no—
202C24—24no—
203C24—24no—
204C24—24no—
205C24—216no—
206C24—216no—
207C24—24no—
208C24—24no—
209C24—24no—
210C24—24no—
211C24—42no—
212C24—42no—
213C24—42no—
214C24—42no—
215C24—42no—
216C24—42no—
217C24—42no—
218C24—416no—
219C24—416no—
220C24—42no—
221C24—416no—
222C24—416no—
223C24—432no—
224C24C2×C22⋊C412yes—
225C24C2×C22⋊C412yes—
226C24C4×D414yes—
227C24C4×D414yes—
228C24#12118yes—
229C24#12414yes—
230C24#12714yes—
231C24C23×C411yescharacteristic
232C24C22×D411yescharacteristic
233C24C22×D411yescharacteristic
234C24C22×D412yes—
235C24C2×#4114yes—
236C24C2×#4112yes—
237C242−1+414yes—
238C22×C4—232no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—28no—
268C22×C4—216no—
269C22×C4—416no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—432no—
283C22×C4—432no—
284C22×C4—416no—
285C22×C4—432no—
286C22×C4—432no—
287C22×C4—416no—
288C22×C4—416no—
289C22×C4—416no—
290C22×C4—48no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—48no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—48no—
297C22×C4—48no—
298C22×C4—48no—
299C22×C4—48no—
300C22×C4—48no—
301C22×C4—48no—
302C22×C4—48no—
303C22×C4—48no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—832no—
308C22×C4#9918yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4#11714yes—
316C22×C4#11714yes—
317C22×C4#12314yes—
318C22×C4#12314yes—
319C22×C4#12314yes—
320C22×C4#12314yes—
321C4⋊C4—48no—
322C4⋊C4—48no—
323C4⋊C4—48no—
324C4⋊C4—48no—
325C4⋊C4—88no—
326C22⋊C4—216no—
327C22⋊C4—432no—
328C22⋊C4—48no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—48no—
332C22⋊C4—416no—
333C22⋊C4—416no—
334C22⋊C4—816no—
335C42—88no—
336C23—28no—
337C23—216no—
338C23—28no—
339C23—22no—
340C23—28no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—216no—
350C23—22no—
351C23—22no—
352C23—28no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—216no—
358C23—28no—
359C23—28no—
360C23—216no—
361C23—28no—
362C23—28no—
363C23—22no—
364C23—44no—
365C23—48no—
366C23—48no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23—48no—
373C23—48no—
374C23—48no—
375C23—48no—
376C23—48no—
377C23—48no—
378C23—48no—
379C23—48no—
380C23—48no—
381C23—44no—
382C23—464no—
383C23—464no—
384C23—48no—
385C23—44no—
386C23#39014yes—
387C23#39014yes—
388C23C2×#9914yes—
389C23#41014yes—
390C23C22×C22⋊C411yescharacteristic
391C23C2×#11712yes—
392C23C2×C4×D412yes—
393C23#51912yes—
394C23C2×#12112yes—
395C23C2×#12312yes—
396C23C2×#12312yes—
397C23C2×#12411yescharacteristic
398C23C2×#12711yescharacteristic
399C23#53612yes—
400C23#53612yes—
401C23#54718yes—
402C23#55218yes—
403C2×C4—216no—
404C2×C4—28no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—28no—
411C2×C4—28no—
412C2×C4—216no—
413C2×C4—28no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—48no—
421C2×C4—48no—
422C2×C4—48no—
423C2×C4—48no—
424C2×C4—48no—
425C2×C4—48no—
426C2×C4—48no—
427C2×C4—44no—
428C2×C4—48no—
429C2×C4—416no—
430C2×C4—48no—
431C2×C4—48no—
432C2×C4—48no—
433C2×C4—48no—
434C2×C4—84no—
435C2×C4—88no—
436C2×C4—84no—
437C2×C4—816no—
438C2×C4—816no—
439C2×C4—816no—
440C2×C4—88no—
441C2×C4—84no—
442C2×C4—88no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4#38112yes—
446C2×C4#38112yes—
447C2×C4#38112yes—
448C2×C4#38112yes—
449C2×C4C2×#9914yes—
450C2×C4C2×#9914yes—
451C22—232no—
452C22—24no—
453C22—24no—
454C22—24no—
455C22—24no—
456C22—24no—
457C22—24no—
458C22—22no—
459C22—24no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—24no—
466C22—24no—
467C22—24no—
468C22—22no—
469C22—24no—
470C22—22no—
471C22—24no—
472C22—24no—
473C22—24no—
474C22—48no—
475C22—48no—
476C22—432no—
477C22—416no—
478C22—416no—
479C22—416no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22#196418yes—
485C22C2×#38111yescharacteristic
486C22C2×#39011yescharacteristic
487C22#231011yescharacteristic
488C22#234414yes—
489C22#248614yes—
490C22#250112yes—
491C22#250312yes—
492C22C22×#9912yes—
493C22C2×#41012yes—
494C4—22no—
495C4—24no—
496C4—22no—
497C4—22no—
498C4—42no—
499C4—88no—
500C4—88no—
501C4—164no—
502C4#181914yescomplemented
503C2—24no—
504C2—22no—
505C2—24no—
506C2—28no—
507C2—22no—
508C2—44no—
509C2#1390411yescharacteristic
510C2C2×#181911yescharacteristic
511C2#2068811yescharacteristic
512C2C2×#196412yes—
513C2#2182512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC26C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11353C2semidirect
6. #10854C2semidirect
8. C23×C4⋊C4C4semidirect
15. C2×#380C4semidirect
16. C2×#380C4semidirect
24. C22×#95C4semidirect
502. C4#1819semidirect

Maximal subgroups

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

subgroupindex
C22×#3802
C2×#18192
#146802
C4×#3792
#113532
#108542