Finite Group Databasebeta

← #120019

Subgroups — #120019

Order 512.

1364 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14353C211yesmaximal, characteristic
2C2×#1444C211yesmaximal, characteristic
3#10730C212yesmaximal
4C2×#1294C212yesmaximal
5C2×#1280C211yesmaximal, characteristic
6C23×C4⋊C4—21no—
7C23×C4⋊C4C2211yescharacteristic
8C22×C22⋊C8C2212yes—
9C2×#380—21no—
10C2×#380—21no—
11C2×#380C2212yes—
12C22×#95—21no—
13C22×#95C2211yescharacteristic
14C22×#95C2211yescharacteristic
15C2×#340—24no—
16#1444—22no—
17#1444—24no—
18#1444C414yescomplemented
19#1294—24no—
20#1280—22no—
21C24×C4—21no—
22C24×C4—21no—
23C24×C4D411yescharacteristic
24C24×C4D411yescharacteristic
25C24×C4C2311yescharacteristic
26C23×C8—22no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4—24no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—44no—
34C22×C4⋊C4—44no—
35C22×C4⋊C4—44no—
36C22×C4⋊C4—42no—
37C22×C4⋊C4—42no—
38C22×C4⋊C4C2×C412yes—
39C22×C4⋊C4D414yes—
40C22×C22⋊C4—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8Q814yes—
47#380—22no—
48#380—22no—
49#380—22no—
50#380—22no—
51#380—42no—
52#380—42no—
53#380—42no—
54#380—42no—
55C2×#95—22no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—24no—
63C2×#95—22no—
64C2×#95—44no—
65C2×#95C2×C412yes—
66C2×#95C2×C412yes—
67C2×#95D412yes—
68C2×#95D412yes—
69C2×#95Q812yes—
70#340—48no—
71C25C2×D411yescharacteristic
72C23×C4—24no—
73C23×C4—24no—
74C23×C4—22no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—24no—
86C23×C4—22no—
87C23×C4—24no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4—41no—
93C23×C4—41no—
94C23×C4—41no—
95C23×C4—41no—
96C23×C4—42no—
97C23×C4—42no—
98C23×C4—42no—
99C23×C4—42no—
100C23×C4—42no—
101C23×C4—42no—
102C23×C4—42no—
103C23×C4—41no—
104C23×C4—41no—
105C23×C4—41no—
106C23×C4—41no—
107C23×C4—41no—
108C23×C4—41no—
109C23×C4C22⋊C412yes—
110C23×C4C22⋊C412yes—
111C23×C4D812yes—
112C23×C4SD1612yes—
113C23×C4SD1612yes—
114C23×C4Q1612yes—
115C23×C4C22×C411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×Q811yescharacteristic
120C23×C4C2×Q812yes—
121C23×C4#4111yescharacteristic
122C23×C4#4111yescharacteristic
123C23×C4#4112yes—
124C23×C4#4112yes—
125C22×C8—24no—
126C22×C8—44no—
127C22×C8—44no—
128C22×C8—42no—
129C22×C8—42no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—48no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—48no—
139C2×C4⋊C4—48no—
140C2×C4⋊C4—416no—
141C2×C4⋊C4—416no—
142C2×C4⋊C4—48no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—82no—
149C2×C4⋊C4—82no—
150C2×C4⋊C4—82no—
151C2×C4⋊C4—82no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—42no—
158C22⋊C8—24no—
159C22⋊C8—24no—
160C22⋊C8—42no—
161C22⋊C8—42no—
162#95—22no—
163#95—24no—
164#95—22no—
165#95—24no—
166#95—22no—
167#95—48no—
168#95—44no—
169#95—44no—
170#95—42no—
171#95—42no—
172#95C4⋊C414yescomplemented
173C24—22no—
174C24—22no—
175C24—22no—
176C24—22no—
177C24C2×C22⋊C411yescharacteristic
178C24#12011yescharacteristic
179C24#12111yescharacteristic
180C24C22⋊Q811yescharacteristic
181C24#12311yescharacteristic
182C24#12411yescharacteristic
183C24C4⋊Q811yescharacteristic
184C24C2×D811yescharacteristic
185C24C2×SD1611yescharacteristic
186C24C2×SD1611yescharacteristic
187C24C2×Q1611yescharacteristic
188C24AGL(1,ℤ/8)11yescharacteristic
189C24AGL(1,ℤ/8)11yescharacteristic
190C24#13711yescharacteristic
191C24#13711yescharacteristic
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—28no—
196C22×C4—24no—
197C22×C4—28no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—44no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—81no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—82no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4#10214yes—
292C22×C4#10314yes—
293C22×C4C2×C4⋊C411yescharacteristic
294C22×C4#11711yescharacteristic
295C22×C4C4×D412yes—
296C22×C4C4×Q812yes—
297C22×C4#12112yes—
298C22×C4#12112yes—
299C22×C4#12111yescharacteristic
300C22×C4C22⋊Q812yes—
301C22×C4C22⋊Q812yes—
302C22×C4C22⋊Q812yes—
303C22×C4C22⋊Q811yescharacteristic
304C22×C4#12312yes—
305C22×C4#12512yes—
306C22×C4#12512yes—
307C22×C4#12612yes—
308C22×C4C4⋊Q812yes—
309C2×C8—48no—
310C2×C8—44no—
311C2×C8—82no—
312C2×C8—82no—
313C2×C8—84no—
314C4⋊C4—28no—
315C4⋊C4—44no—
316C4⋊C4—48no—
317C4⋊C4—48no—
318C4⋊C4—432no—
319C4⋊C4—84no—
320C4⋊C4—88no—
321C4⋊C4—88no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23#38011yescharacteristic
359C23#38611yescharacteristic
360C23#39111yescharacteristic
361C23#39411yescharacteristic
362C23#39511yescharacteristic
363C23#39811yescharacteristic
364C23#40011yescharacteristic
365C23C2×#10211yescharacteristic
366C23C2×#10311yescharacteristic
367C23#41711yescharacteristic
368C23#41711yescharacteristic
369C23#44711yescharacteristic
370C23#44811yescharacteristic
371C23#45011yescharacteristic
372C23#45111yescharacteristic
373C23#45312yes—
374C23#45612yes—
375C23B(4,𝔽2)12yes—
376C23#45812yes—
377C23#45911yescharacteristic
378C23#46011yescharacteristic
379C23#46111yescharacteristic
380C23#46211yescharacteristic
381C23#47411yescharacteristic
382C23#47511yescharacteristic
383C23#47611yescharacteristic
384C23#47711yescharacteristic
385C23#48011yescharacteristic
386C23#48111yescharacteristic
387C23#48311yescharacteristic
388C23#48411yescharacteristic
389C23#48611yescharacteristic
390C23#48711yescharacteristic
391C23#49011yescharacteristic
392C23#49111yescharacteristic
393C23#49811yescharacteristic
394C23#50011yescharacteristic
395C23#50111yescharacteristic
396C23#50111yescharacteristic
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—28no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—48no—
423C2×C4—48no—
424C2×C4—48no—
425C2×C4—48no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4—84no—
458C2×C4—84no—
459C2×C4—84no—
460C2×C4—84no—
461C2×C4—84no—
462C2×C4#38712yes—
463C2×C4#38912yes—
464C2×C4#39612yes—
465C2×C4#39812yes—
466C8—84no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22#144111yescharacteristic
508C22#159512yes—
509C22#159612yes—
510C22#160412yes—
511C22#160512yes—
512C22#161012yes—
513C22#161112yes—
514C22#161312yes—
515C22#161412yes—
516C22#180211yescharacteristic
517C22#180311yescharacteristic
518C22#187211yescharacteristic
519C22#187411yescharacteristic
520C22#188611yescharacteristic
521C22#188711yescharacteristic
522C22#188911yescharacteristic
523C22#189011yescharacteristic
524C22#199011yescharacteristic
525C22#199211yescharacteristic
526C22#203912yes—
527C22#204111yescharacteristic
528C22#204311yescharacteristic
529C22#204412yes—
530C22#205711yescharacteristic
531C22#205811yescharacteristic
532C22#206012yes—
533C22#206212yes—
534C22#208311yescharacteristic
535C22#208411yescharacteristic
536C22#209911yescharacteristic
537C22#210011yescharacteristic
538C4—22no—
539C4—22no—
540C4—22no—
541C4—22no—
542C4—48no—
543C4—44no—
544C4—44no—
545C4—84no—
546C4—164no—
547C2—21no—
548C2—21no—
549C2—21no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2—21no—
554C2—21no—
555C2#901511yescharacteristic
556C2#901711yescharacteristic
557C2#902111yescharacteristic
558C2#902711yescharacteristic
559C2#1040212yes—
560C2#1040612yes—
561C2#1056712yes—
562C2#1057012yes—
563C2#1120711yescharacteristic
564C2#1121011yescharacteristic
565C2#1122211yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #1444C4semidirect
172. #95C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#143532
C2×#14442
#107302
C2×#12942
C2×#12802