Finite Group Databasebeta

← #355170

Subgroups — #355170

Order 512.

1916 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2290C211yesmaximal, characteristic
2C2×#1799C212yesmaximal
3C4×#380C212yesmaximal, complemented
4#10960C214yesmaximal
5#10912C214yesmaximal, complemented
6#10846C212yesmaximal, complemented
7C22×C4×D4C2211yescharacteristic
8C22×#117C411yescharacteristic, complemented
9C23×C42C2211yescharacteristic
10C22×#99C2214yes—
11#2290—22no—
12#2290—22no—
13#2290—22no—
14#2290—22no—
15C2×#388C412yescomplemented
16C2×#387C412yescomplemented
17C2×#386C2212yes—
18C2×#385C2212yes—
19C2×#380C2214yes—
20C2×C4×C4⋊C4—22no—
21C2×C4×C4⋊C4C411yescharacteristic, complemented
22C2×C4×C22⋊C4C411yescharacteristic, complemented
23C2×C4×C22⋊C4C2214yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25C2×#376C411yescharacteristic, complemented
26#1799C4116yescomplemented
27C2×#342C2218yes—
28C2×#342C2214yes—
29C4⋊#95—22no—
30#1434—22no—
31C22⋊#95C2214yes—
32C4×#95—22no—
33C23×D4C2311yescharacteristic
34C24×C4C2×C411yescharacteristic
35C24×C4C2312yes—
36C24×C4C2311yescharacteristic
37C2×C4×D4—22no—
38C2×C4×D4—22no—
39C2×C4×D4—24no—
40C2×C4×D4C2×C418yes—
41C2×#117—22no—
42C2×#117—21no—
43C2×#117—21no—
44C2×#117—44no—
45C22×C4⋊C4—24no—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2×C412yes—
49C22×C22⋊C4C2×C411yescharacteristic
50C22×C22⋊C4C2×C412yes—
51C22×C22⋊C4C2×C412yes—
52C22×C22⋊C4C2312yes—
53C22×C22⋊C4C2318yes—
54C22×C22⋊C4C2311yescharacteristic
55C22×C42—24no—
56C22×C42—44no—
57C22×C42C2×C411yescharacteristic
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C412yes—
63C22×C42D412yes—
64C22×C42D412yes—
65C22×C42D412yes—
66C22×C42D412yes—
67C2×#99—48no—
68C2×#99C2×C4116yes—
69#388—22no—
70#388—22no—
71#388—22no—
72#388—22no—
73#387—22no—
74#387—22no—
75#387—22no—
76#387—22no—
77#386—22no—
78#386—22no—
79#386—22no—
80#386—22no—
81#385—22no—
82#385—22no—
83#385—22no—
84#385—22no—
85#380—28no—
86#380C2×C4116yes—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—48no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—21no—
95C4×C22⋊C4—21no—
96C4×C22⋊C4—48no—
97#376—21no—
98#376—22no—
99#376—21no—
100C2×#95—24no—
101C2×#95—24no—
102C2×#95—24no—
103C2×#95—24no—
104C2×#95C2×C412yes—
105C2×#95C2×C412yes—
106C2×#95C2×C412yes—
107C2×#95C2×C412yes—
108C2×#95C2×C412yes—
109#342—48no—
110#342C2×C4116yes—
111C25C22×C411yescharacteristic
112C25C2411yescharacteristic
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—24no—
116C22×D4—24no—
117C22×D4C22×C414yes—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—28no—
124C23×C4—44no—
125C23×C4—42no—
126C23×C4—44no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4C4214yescomplemented
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C412yes—
133C23×C4C22×C412yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C412yes—
136C23×C4C22×C412yes—
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C414yes—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4#4114yes—
148C4×D4—216no—
149C4×D4—216no—
150#117—48no—
151#117—44no—
152#117—44no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—216no—
160C2×C4⋊C4—48no—
161C2×C4⋊C4—48no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4C4214yescomplemented
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4C4214yescomplemented
185C2×C22⋊C4C22×C4116yes—
186C2×C22⋊C4C22×C414yes—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—24no—
191C2×C42—28no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—42no—
197C2×C42—44no—
198C2×C42—44no—
199C2×C42—44no—
200C2×C42—44no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—48no—
204C2×C42—48no—
205C2×C42—48no—
206C2×C42C4214yescomplemented
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22⋊C412yes—
209C2×C42C22⋊C412yes—
210C2×C42C22⋊C412yes—
211C2×C42C22⋊C412yes—
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C412yes—
215C2×C42C22⋊C412yes—
216C2×C42C22⋊C411yescharacteristic
217#99—416no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—216no—
229#95—48no—
230#95—48no—
231#95—48no—
232C24—21no—
233C24—21no—
234C24—24no—
235C24—41no—
236C24—41no—
237C24—42no—
238C24—41no—
239C24—42no—
240C24—41no—
241C24C2×C4212yes—
242C24C23×C411yescharacteristic
243C24C23×C412yes—
244C24C22×D412yes—
245C24C2×#4112yes—
246C24H(5,𝔽2)14yes—
247C242−1+414yes—
248C2×D4—28no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—28no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—44no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—44no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—216no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—48no—
291C22×C4—48no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—416no—
304C22×C4—416no—
305C22×C4—42no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—48no—
314C22×C4—48no—
315C22×C4—48no—
316C22×C4—48no—
317C22×C4—48no—
318C22×C4—48no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4C2×C4212yes—
324C22×C4C2×C4212yes—
325C22×C4C2×C4212yes—
326C22×C4C2×C4212yes—
327C22×C4C2×C4212yes—
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4#11712yes—
336C22×C4#11712yes—
337C22×C4C4×D418yes—
338C22×C4C4×D418yes—
339C4⋊C4—28no—
340C4⋊C4—432no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C22⋊C4—28no—
346C22⋊C4—28no—
347C22⋊C4—216no—
348C22⋊C4—216no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—44no—
354C22⋊C4—48no—
355C22⋊C4—48no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C42—24no—
359C42—24no—
360C42—24no—
361C42—24no—
362C42—24no—
363C42—24no—
364C42—22no—
365C42—41no—
366C42—41no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—42no—
372C42—42no—
373C42—816no—
374C42#9914yescomplemented
375C23—22no—
376C23—21no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—21no—
382C23—22no—
383C23—24no—
384C23—42no—
385C23—42no—
386C23—44no—
387C23—44no—
388C23—42no—
389C23—42no—
390C23—44no—
391C23—44no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—41no—
399C23—42no—
400C23—41no—
401C23—84no—
402C23—84no—
403C23C22×C4211yescharacteristic
404C23C22×C22⋊C411yescharacteristic
405C23C2×#11711yescharacteristic
406C23C2×C4×D414yes—
407C23#51812yes—
408C23#51912yes—
409C23#52014yes—
410C23#52014yes—
411C23#53512yes—
412C23#53612yes—
413C23#53714yes—
414C23#54012yes—
415C23#54112yes—
416C23#54214yes—
417D4—48no—
418D4—48no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—216no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—48no—
445C2×C4—48no—
446C2×C4—48no—
447C2×C4—48no—
448C2×C4—48no—
449C2×C4—48no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—48no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—416no—
463C2×C4—416no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—416no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—88no—
472C2×C4—88no—
473C2×C4—88no—
474C2×C4—88no—
475C2×C4—84no—
476C2×C4—84no—
477C2×C4C4×C22⋊C412yes—
478C2×C4C4×C22⋊C414yes—
479C2×C4C4×C22⋊C412yes—
480C2×C4C2×#9914yes—
481C2×C4C2×#9912yes—
482C2×C4#41014yes—
483C2×C4#41012yes—
484C22—22no—
485C22—22no—
486C22—24no—
487C22—24no—
488C22—22no—
489C22—24no—
490C22—24no—
491C22—22no—
492C22—24no—
493C22—21no—
494C22—21no—
495C22—24no—
496C22—21no—
497C22—21no—
498C22—42no—
499C22—42no—
500C22—44no—
501C22—42no—
502C22—42no—
503C22—44no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—84no—
508C22—84no—
509C22—82no—
510C22—82no—
511C22C2×C4×C22⋊C411yescharacteristic
512C22#227412yes—
513C22#231112yes—
514C22#231412yes—
515C22#232314yes—
516C22#232414yes—
517C22C22×#9911yescharacteristic
518C22C2×#41012yes—
519C22C2×#41011yescharacteristic
520C4—24no—
521C4—24no—
522C4—22no—
523C4—24no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—816no—
530C4—82no—
531C4—82no—
532C4C4×#9914yes—
533C2—24no—
534C2—22no—
535C2—22no—
536C2—24no—
537C2—82no—
538C2#1377711yescharacteristic
539C2C2×C4×#9911yescharacteristic
540C2C4×#41011yescharacteristic
541C2#2187412yes—
542C2#2187512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C4×#380C2semidirect
5. #10912C2semidirect
6. #10846C2semidirect
8. C22×#117C4semidirect
15. C2×#388C4semidirect
16. C2×#387C4semidirect
21. C2×C4×C4⋊C4C4semidirect
22. C2×C4×C22⋊C4C4semidirect
25. C2×#376C4semidirect
26. #1799C4semidirect
130. C23×C4C42semidirect
164. C2×C4⋊C4C42semidirect
184. C2×C22⋊C4C42semidirect
206. C2×C42C42semidirect
374. C42#99semidirect

Maximal subgroups

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

subgroupindex
C2×#22902
C2×#17992
C4×#3802
#109602
#109122
#108462