Finite Group Databasebeta

← #410222

Subgroups — #410222

Order 512.

1814 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#389C211yesmaximal, characteristic
2C2×#1852C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic
4C2×#1824C211yesmaximal, characteristic
5C2×#1817C211yesmaximal, characteristic
6#14685C211yesmaximal, characteristic
7#11293C214yesmaximal, complemented
8#11219C214yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10C23×C4⋊C4—21no—
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#107C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C2×#389—22no—
18C2×#389—24no—
19C2×#389—24no—
20C2×#389C2214yes—
21C2×#381—21no—
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C22×#95—21no—
25#1852—24no—
26#1824—24no—
27#1824—24no—
28#1817—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2214yes—
32C2×#340C2214yes—
33C2×#340C2212yes—
34C2×#336C2214yes—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37#1434—22no—
38C22⋊#95—21no—
39C24×C4—21no—
40C24×C4—21no—
41C24×C4D411yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—28no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—28no—
54C22×C4⋊C4—24no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—48no—
58C22×C4⋊C4D414yes—
59C22×C4⋊C4D414yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C42—22no—
68C22×C42—22no—
69C22×C42—22no—
70C22×C42—22no—
71C22×C42—22no—
72C22×C42—22no—
73C22×C42D412yes—
74C22×C42C2312yes—
75C2×#107—24no—
76C2×#107—28no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8D414yes—
83#389—216no—
84#389—28no—
85#389—28no—
86#381—44no—
87#380—22no—
88#380—22no—
89#380—22no—
90#380—22no—
91C2×#95—28no—
92C2×#95—24no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99#340—28no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—28no—
104#336—28no—
105C25C2×D411yescharacteristic
106C23×C4—24no—
107C23×C4—24no—
108C23×C4—28no—
109C23×C4—22no—
110C23×C4—24no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—42no—
127C23×C4—44no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4—44no—
131C23×C4—44no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4Q1614yes—
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4#4112yes—
142C23×C4#4112yes—
143C23×C4#4112yes—
144C23×C4C2411yescharacteristic
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—24no—
148C22×C8—24no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—24no—
152C22×C8—22no—
153C22×C8—24no—
154C22×C8C2×D414yes—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—216no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—216no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—216no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—432no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—48no—
187C2×C4⋊C4C2×D414yes—
188C2×C4⋊C4C2×D414yes—
189C2×C4⋊C4#4118yes—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—44no—
206C2×C42—44no—
207C2×C42—44no—
208C2×C42—44no—
209#107—48no—
210C22⋊C8—28no—
211C22⋊C8—44no—
212#95—216no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—42no—
217#95—42no—
218#95—44no—
219C24—22no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24#12012yes—
224C24#12112yes—
225C24#12312yes—
226C24C2×Q1612yes—
227C24#13512yes—
228C24AGL(1,ℤ/8)12yes—
229C24#13712yes—
230C24C22×D411yescharacteristic
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—28no—
238C22×C4—28no—
239C22×C4—28no—
240C22×C4—28no—
241C22×C4—216no—
242C22×C4—28no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—28no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—24no—
261C22×C4—48no—
262C22×C4—48no—
263C22×C4—48no—
264C22×C4—416no—
265C22×C4—416no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—48no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—48no—
277C22×C4—48no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—48no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—88no—
301C22×C4—88no—
302C22×C4#12012yes—
303C22×C4#12114yes—
304C22×C4#12114yes—
305C22×C4#12112yes—
306C22×C4#12314yes—
307C22×C4#12312yes—
308C22×C4#12414yes—
309C22×C4C2×Q1614yes—
310C22×C4AGL(1,ℤ/8)14yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4114yes—
317C22×C4C2×#4111yescharacteristic
318C22×C42−1+414yes—
319C2×C8—24no—
320C2×C8—28no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—44no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—44no—
328C2×C8—44no—
329C4⋊C4—216no—
330C4⋊C4—24no—
331C4⋊C4—28no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—432no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—48no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4#12318yes—
345C22⋊C4—84no—
346C22⋊C4—84no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42—48no—
352C23—24no—
353C23—24no—
354C23—22no—
355C23—24no—
356C23—24no—
357C23—24no—
358C23—22no—
359C23—24no—
360C23—24no—
361C23—24no—
362C23—24no—
363C23—24no—
364C23—24no—
365C23—22no—
366C23—22no—
367C23—24no—
368C23#39414yes—
369C23#44914yes—
370C23#45114yes—
371C23#46714yes—
372C23#46814yes—
373C23#48414yes—
374C23#48514yes—
375C23C2×#12011yescharacteristic
376C23C2×#12111yescharacteristic
377C23C2×#12311yescharacteristic
378C23#52512yes—
379C23#53612yes—
380C23C22×Q1611yescharacteristic
381C23C2×#13511yescharacteristic
382C23C2×AGL(1,ℤ/8)11yescharacteristic
383C23C2×AGL(1,ℤ/8)12yes—
384C23C2×#13711yescharacteristic
385C23#57512yes—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—28no—
391C2×C4—28no—
392C2×C4—28no—
393C2×C4—28no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—28no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—416no—
402C2×C4—432no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—416no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—816no—
420C2×C4—84no—
421C2×C4—84no—
422C2×C4—81no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4#39414yes—
440C2×C4C2×#12111yescharacteristic
441C2×C4C2×#12111yescharacteristic
442C2×C4C2×#12311yescharacteristic
443C2×C4C2×#12312yes—
444C2×C4#52512yes—
445C2×C4C2×#12411yescharacteristic
446C2×C4#53612yes—
447C2×C4#53612yes—
448C2×C4#54714yes—
449C2×C4#55214yes—
450C2×C4#55214yes—
451C2×C4#55214yes—
452C8—44no—
453C8—82no—
454C22—28no—
455C22—28no—
456C22—24no—
457C22—24no—
458C22—24no—
459C22—28no—
460C22—24no—
461C22—28no—
462C22—24no—
463C22—24no—
464C22#203618yes—
465C22C2×#39411yescharacteristic
466C22#241312yes—
467C22#248412yes—
468C22#261411yescharacteristic
469C22#284911yescharacteristic
470C22C2×#45111yescharacteristic
471C22C2×#44911yescharacteristic
472C22#300311yescharacteristic
473C22#300511yescharacteristic
474C22C2×#46711yescharacteristic
475C22C2×#46811yescharacteristic
476C22#305512yes—
477C22C2×#48511yescharacteristic
478C22C2×#48411yescharacteristic
479C22#309412yes—
480C22#310412yes—
481C22#310512yes—
482C22#310912yes—
483C22#311712yes—
484C22#323012yes—
485C22#323212yes—
486C22#323412yes—
487C22#323912yes—
488C4—24no—
489C4—44no—
490C4—44no—
491C4—48no—
492C4—84no—
493C4—162no—
494C4—161no—
495C4—161no—
496C4#241312yes—
497C4#248412yes—
498C4#248612yes—
499C4#250312yes—
500C2—28no—
501C2#1507311yescharacteristic
502C2C2×#203611yescharacteristic
503C2#2348011yescharacteristic
504C2#2460212yes—
505C2#2460712yes—
506C2#2761412yes—
507C2#2761712yes—
508C2#2964011yescharacteristic
509C2#2964211yescharacteristic
510C2#3300411yescharacteristic
511C2#3301011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
7. #11293C2semidirect
8. #11219C2semidirect

Maximal subgroups

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

subgroupindex
C22×#3892
C2×#18522
C2×#18242
C2×#18172
#146852
#112932
#112192
#107302