Finite Group Databasebeta

← #120020

Subgroups — #120020

Order 512.

1212 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14353C211yesmaximal, characteristic
2C2×#1442C211yesmaximal, characteristic
3#10730C212yesmaximal
4C2×#1294C212yesmaximal
5C2×#1280C211yesmaximal, characteristic
6C23×C4⋊C4C2211yescharacteristic
7C22×C22⋊C8C2212yes—
8C2×#380—21no—
9C2×#380—21no—
10C2×#380C2212yes—
11C22×#95—21no—
12C22×#95—21no—
13C22×#95C2211yescharacteristic
14C22×#95C2211yescharacteristic
15C2×#340—24no—
16#1444—24no—
17#1442—22no—
18#1442C414yescomplemented
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—21no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—44no—
33C22×C4⋊C4C2×C412yes—
34C22×C4⋊C4D414yes—
35C22×C22⋊C4—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8Q814yes—
42#380—22no—
43#380—22no—
44#380—22no—
45#380—22no—
46#380—42no—
47#380—42no—
48#380—42no—
49#380—42no—
50C2×#95—21no—
51C2×#95—22no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95—24no—
58C2×#95—24no—
59C2×#95—22no—
60C2×#95—44no—
61C2×#95—42no—
62C2×#95—42no—
63C2×#95C2×C412yes—
64C2×#95C2×C412yes—
65C2×#95D412yes—
66C2×#95D412yes—
67C2×#95Q812yes—
68#340—48no—
69C25C2×D411yescharacteristic
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—24no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—24no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4—42no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4—42no—
96C23×C4—42no—
97C23×C4—41no—
98C23×C4—41no—
99C23×C4—41no—
100C23×C4—41no—
101C23×C4—41no—
102C23×C4—41no—
103C23×C4—41no—
104C23×C4—41no—
105C23×C4—41no—
106C23×C4—41no—
107C23×C4C22⋊C412yes—
108C23×C4C22⋊C412yes—
109C23×C4D812yes—
110C23×C4SD1612yes—
111C23×C4SD1612yes—
112C23×C4Q1612yes—
113C23×C4C22×C411yescharacteristic
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×Q811yescharacteristic
118C23×C4C2×Q812yes—
119C23×C4#4111yescharacteristic
120C23×C4#4111yescharacteristic
121C23×C4#4112yes—
122C23×C4#4112yes—
123C22×C8—24no—
124C22×C8—44no—
125C22×C8—44no—
126C22×C8—42no—
127C22×C8—42no—
128C2×C4⋊C4—28no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—48no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—82no—
142C2×C4⋊C4—82no—
143C2×C4⋊C4—82no—
144C2×C4⋊C4—82no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—42no—
151C22⋊C8—24no—
152C22⋊C8—24no—
153C22⋊C8—42no—
154C22⋊C8—42no—
155#95—22no—
156#95—24no—
157#95—24no—
158#95—22no—
159#95—22no—
160#95—48no—
161#95—44no—
162#95—44no—
163#95—48no—
164#95—42no—
165#95—42no—
166#95C4⋊C414yescomplemented
167C24—22no—
168C24—22no—
169C24—22no—
170C24—22no—
171C24C2×C22⋊C411yescharacteristic
172C24#12011yescharacteristic
173C24#12111yescharacteristic
174C24C22⋊Q811yescharacteristic
175C24#12311yescharacteristic
176C24#12411yescharacteristic
177C24C4⋊Q811yescharacteristic
178C24C2×D811yescharacteristic
179C24C2×SD1611yescharacteristic
180C24C2×SD1611yescharacteristic
181C24C2×Q1611yescharacteristic
182C24AGL(1,ℤ/8)11yescharacteristic
183C24AGL(1,ℤ/8)11yescharacteristic
184C24#13711yescharacteristic
185C24#13711yescharacteristic
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—44no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—82no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4#10214yes—
286C22×C4#10314yes—
287C22×C4C2×C4⋊C411yescharacteristic
288C22×C4#11711yescharacteristic
289C22×C4C4×D412yes—
290C22×C4C4×Q812yes—
291C22×C4#12112yes—
292C22×C4#12112yes—
293C22×C4#12111yescharacteristic
294C22×C4C22⋊Q812yes—
295C22×C4C22⋊Q812yes—
296C22×C4C22⋊Q812yes—
297C22×C4C22⋊Q811yescharacteristic
298C22×C4#12312yes—
299C22×C4#12512yes—
300C22×C4#12512yes—
301C22×C4#12612yes—
302C22×C4C4⋊Q812yes—
303C2×C8—48no—
304C2×C8—44no—
305C2×C8—82no—
306C2×C8—82no—
307C2×C8—84no—
308C4⋊C4—28no—
309C4⋊C4—44no—
310C4⋊C4—48no—
311C4⋊C4—48no—
312C4⋊C4—84no—
313C4⋊C4—88no—
314C4⋊C4—88no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
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—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23#38011yescharacteristic
352C23#38611yescharacteristic
353C23#39111yescharacteristic
354C23#39411yescharacteristic
355C23#39511yescharacteristic
356C23#39811yescharacteristic
357C23#40011yescharacteristic
358C23C2×#10211yescharacteristic
359C23C2×#10311yescharacteristic
360C23#41711yescharacteristic
361C23#41711yescharacteristic
362C23#44711yescharacteristic
363C23#44811yescharacteristic
364C23#45011yescharacteristic
365C23#45111yescharacteristic
366C23#45412yes—
367C23#45512yes—
368C23B(4,𝔽2)12yes—
369C23#45812yes—
370C23#45911yescharacteristic
371C23#46011yescharacteristic
372C23#46111yescharacteristic
373C23#46211yescharacteristic
374C23#47411yescharacteristic
375C23#47511yescharacteristic
376C23#47611yescharacteristic
377C23#47711yescharacteristic
378C23#48011yescharacteristic
379C23#48111yescharacteristic
380C23#48311yescharacteristic
381C23#48411yescharacteristic
382C23#48611yescharacteristic
383C23#48711yescharacteristic
384C23#49011yescharacteristic
385C23#49111yescharacteristic
386C23#49811yescharacteristic
387C23#50011yescharacteristic
388C23#50111yescharacteristic
389C23#50111yescharacteristic
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—84no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—84no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4—84no—
454C2×C4—84no—
455C2×C4#38712yes—
456C2×C4#38912yes—
457C2×C4#39612yes—
458C2×C4#39812yes—
459C8—84no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22#144111yescharacteristic
501C22#159812yes—
502C22#159912yes—
503C22#160112yes—
504C22#160212yes—
505C22#160712yes—
506C22#160812yes—
507C22#161612yes—
508C22#161712yes—
509C22#180211yescharacteristic
510C22#180311yescharacteristic
511C22#187211yescharacteristic
512C22#187411yescharacteristic
513C22#188511yescharacteristic
514C22#188811yescharacteristic
515C22#188911yescharacteristic
516C22#189011yescharacteristic
517C22#199011yescharacteristic
518C22#199211yescharacteristic
519C22#204012yes—
520C22#204111yescharacteristic
521C22#204311yescharacteristic
522C22#204412yes—
523C22#205711yescharacteristic
524C22#205811yescharacteristic
525C22#206112yes—
526C22#206212yes—
527C22#208311yescharacteristic
528C22#208411yescharacteristic
529C22#209911yescharacteristic
530C22#210011yescharacteristic
531C4—22no—
532C4—22no—
533C4—22no—
534C4—22no—
535C4—44no—
536C4—44no—
537C4—84no—
538C4—84no—
539C4—164no—
540C2—21no—
541C2—21no—
542C2—21no—
543C2—21no—
544C2—21no—
545C2—21no—
546C2—21no—
547C2—21no—
548C2#901311yescharacteristic
549C2#901911yescharacteristic
550C2#902311yescharacteristic
551C2#902511yescharacteristic
552C2#1040312yes—
553C2#1040712yes—
554C2#1056812yes—
555C2#1056912yes—
556C2#1120711yescharacteristic
557C2#1121111yescharacteristic
558C2#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. #1442C4semidirect
166. #95C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#143532
C2×#14422
#107302
C2×#12942
C2×#12802