Finite Group Databasebeta

← #374466

Subgroups — #374466

Order 512.

1762 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic
3C2×#1992C212yesmaximal, complemented
4C2×#1749C211yesmaximal, characteristic
5#11200C212yesmaximal
6#11199C212yesmaximal
7#10885C212yesmaximal, complemented
8#10884C212yesmaximal, complemented
9#10843C212yesmaximal, complemented
10C22×#125C2211yescharacteristic
11C22×#117—21no—
12C23×C42C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C22×#97C2212yes—
15#2851—24no—
16#2851C418yescomplemented
17C2×#389C2211yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#387—21no—
20C2×#387—21no—
21C2×#385—21no—
22C2×#385—21no—
23#2285—28no—
24C2×#384C2212yes—
25C2×#382—21no—
26C2×#382C2211yescharacteristic
27C2×#380—21no—
28C2×#380C2212yes—
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2212yes—
33#1992—28no—
34#1749—24no—
35C2×#340C2214yes—
36C2×#340C2214yes—
37C2×#340C2214yes—
38C2×#340C2214yes—
39C2×#336C2212yes—
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8C2314yes—
44C2×#125—22no—
45C2×#125—24no—
46C2×#125—24no—
47C2×#125C2×C414yes—
48C2×#117—28no—
49C2×#117—44no—
50C2×#117—44no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2314yes—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C42—24no—
66C22×C42—24no—
67C22×C42—22no—
68C22×C42C2×C414yes—
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D414yes—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×C22⋊C8—24no—
75C2×C22⋊C8—28no—
76C2×#97—24no—
77C2×#97—24no—
78C2×#97—24no—
79#389—22no—
80#389—22no—
81#389C2×C414yes—
82#389C2×C414yes—
83#387—28no—
84#387—28no—
85#385—28no—
86#385—28no—
87#384—24no—
88#384D418yes—
89#382—28no—
90#382—22no—
91#382C2×C414yes—
92#380—28no—
93#380—22no—
94#380—22no—
95#380—28no—
96#380C2×C414yes—
97#380C2×C414yes—
98C4×C4⋊C4—28no—
99C4×C4⋊C4—28no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108#340—28no—
109#340—28no—
110#340—28no—
111#340—28no—
112#336—24no—
113C25C2×D411yescharacteristic
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—24no—
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—24no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4C22⋊C414yes—
134C23×C4C22⋊C414yes—
135C23×C4C22×C412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—24no—
144C22×C8—24no—
145C22×C8—24no—
146C22×C8—24no—
147C22×C8—24no—
148C22×C8—24no—
149C22×C8#4118yes—
150#125—216no—
151#125—48no—
152#117—432no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C2×D418yes—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—42no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—216no—
205C2×C42—216no—
206C2×C42—28no—
207C2×C42—44no—
208C2×C42—44no—
209C2×C42C22⋊C414yes—
210C2×C42C22⋊C414yes—
211C2×C42C22×C412yes—
212C2×C42C22×C412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D414yes—
215C2×C42C2×D412yes—
216C2×C42#4114yes—
217C22⋊C8—216no—
218#97—48no—
219#95—28no—
220#95—28no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227C24—21no—
228C24—21no—
229C24—22no—
230C24—22no—
231C24—21no—
232C24—21no—
233C24—22no—
234C24—22no—
235C24C2×C22⋊C412yes—
236C24AGL(1,ℤ/8)12yes—
237C24#13712yes—
238C24C22×D411yescharacteristic
239C22×C4—28no—
240C22×C4—28no—
241C22×C4—28no—
242C22×C4—28no—
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—22no—
252C22×C4—22no—
253C22×C4—28no—
254C22×C4—28no—
255C22×C4—28no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—28no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—28no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—48no—
279C22×C4—48no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4#12012yes—
314C22×C4#12012yes—
315C22×C4#12312yes—
316C22×C4#12312yes—
317C22×C4AGL(1,ℤ/8)12yes—
318C22×C4#13712yes—
319C22×C4C23×C411yescharacteristic
320C22×C4C22×D412yes—
321C22×C4C22×D411yescharacteristic
322C22×C4C2×#4112yes—
323C22×C4C2×#4114yes—
324C22×C42−1+414yes—
325C2×C8—28no—
326C2×C8—28no—
327C2×C8—28no—
328C2×C8—44no—
329C2×C8—44no—
330C2×C8—44no—
331C2×C8—44no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—28no—
337C4⋊C4—28no—
338C4⋊C4—416no—
339C4⋊C4—48no—
340C4⋊C4—416no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—48no—
348C4⋊C4—48no—
349C4⋊C4—48no—
350C22⋊C4—48no—
351C22⋊C4—416no—
352C22⋊C4—48no—
353C42—24no—
354C42—24no—
355C42—28no—
356C42—28no—
357C42—416no—
358C42C4×D418yes—
359C42#12118yes—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—24no—
367C23—24no—
368C23—24no—
369C23—24no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23#41714yes—
383C23C22×C22⋊C411yescharacteristic
384C23C2×#12011yescharacteristic
385C23C2×#12311yescharacteristic
386C23#53212yes—
387C23#53612yes—
388C23C2×AGL(1,ℤ/8)11yescharacteristic
389C23C2×AGL(1,ℤ/8)11yescharacteristic
390C23C2×#13711yescharacteristic
391C23C2×#13711yescharacteristic
392C23#57512yes—
393C23#57512yes—
394C2×C4—232no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—416no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—48no—
423C2×C4—48no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—48no—
433C2×C4—48no—
434C2×C4—48no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—82no—
442C2×C4—84no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—84no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4#38614yes—
454C2×C4#38614yes—
455C2×C4#41714yes—
456C2×C4C2×C4×D412yes—
457C2×C4C4×#4112yes—
458C2×C4#51912yes—
459C2×C4C2×#12112yes—
460C2×C4#52512yes—
461C2×C4#53612yes—
462C2×C4#54714yes—
463C2×C4#54814yes—
464C2×C4#55214yes—
465C2×C4#55614yes—
466C8—48no—
467C22—24no—
468C22—24no—
469C22—24no—
470C22—42no—
471C22—44no—
472C22—44no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—44no—
477C22—44no—
478C22C2×#38611yescharacteristic
479C22#229711yescharacteristic
480C22#231011yescharacteristic
481C22#243212yes—
482C22#249912yes—
483C22C2×#41711yescharacteristic
484C22C2×#41711yescharacteristic
485C22#289611yescharacteristic
486C22#289611yescharacteristic
487C22#300311yescharacteristic
488C22#300411yescharacteristic
489C22#300512yes—
490C22#309211yescharacteristic
491C22#309211yescharacteristic
492C22#309312yes—
493C22#317112yes—
494C22#317212yes—
495C22#318012yes—
496C22#318112yes—
497C22#323412yes—
498C22#323512yes—
499C22#323612yes—
500C22#323712yes—
501C4—24no—
502C4—24no—
503C4—24no—
504C4—42no—
505C4—42no—
506C4—88no—
507C4—82no—
508C4—82no—
509C4—82no—
510C4—82no—
511C4#236214yes—
512C4#241514yes—
513C2—44no—
514C2#1413811yescharacteristic
515C2#2090911yescharacteristic
516C2#2091011yescharacteristic
517C2#2140311yescharacteristic
518C2#2140611yescharacteristic
519C2#2183811yescharacteristic
520C2#2183911yescharacteristic
521C2#2567612yes—
522C2#2567712yes—
523C2#2788012yes—
524C2#2788112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1992C2semidirect
7. #10885C2semidirect
8. #10884C2semidirect
9. #10843C2semidirect
16. #2851C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#22852
C2×#19922
C2×#17492
#112002
#111992
#108852
#108842
#108432