Finite Group Databasebeta

← #404572

Subgroups — #404572

Order 512.

1144 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1948C211yesmaximal, characteristic, complemented
2C2×#1925C211yesmaximal, characteristic
3#15136C211yesmaximal, characteristic
4#13810C211yesmaximal, characteristic, complemented
5#11371C211yesmaximal, characteristic, complemented
6#11257C212yesmaximal, complemented
7#11255C211yesmaximal, characteristic, complemented
8#11121C212yesmaximal
9#11076C211yesmaximal, characteristic
10#10931C212yesmaximal
11#10907C212yesmaximal, complemented
12C22×#107C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2212yes—
18C2×#397—22no—
19C2×#390C2211yescharacteristic
20C2×#390C2212yes—
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×#376—21no—
25#1948—24no—
26#1925—24no—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1439—21no—
36#1439—21no—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40#1434—22no—
41#1434—21no—
42C4×#95C2211yescharacteristic
43C23×D4C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C42—22no—
59C22×C42—21no—
60C22×C42C2312yes—
61C22×C42C2311yescharacteristic
62C2×#107—22no—
63C2×#107—24no—
64C2×#106—24no—
65C2×#106—42no—
66C2×#106—42no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—22no—
69C2×#102—22no—
70C2×#102—22no—
71C2×#102—22no—
72C2×#102—24no—
73C2×#102—22no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—42no—
77#397—48no—
78#390—24no—
79#390—22no—
80#390—22no—
81#390—42no—
82#384—22no—
83#384—22no—
84C4×C4⋊C4—24no—
85C4×C4⋊C4—24no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4D414yes—
88#376—44no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95D412yes—
98C2×#95D411yescharacteristic
99C2×#95D411yescharacteristic
100#340—24no—
101#340—22no—
102#340—24no—
103#340—24no—
104#340—42no—
105#336—24no—
106#336—22no—
107#336—42no—
108C25—21no—
109C22×D4—21no—
110C22×D4—21no—
111C22×D4—22no—
112C22×D4—21no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4—22no—
118C22×D4#4112yes—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2411yescharacteristic
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8#4114yes—
145C22×C8#4112yes—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4#4114yes—
175C2×C4⋊C4#4114yes—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4—82no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—28no—
190C2×C42—24no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—44no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—44no—
201C2×C42C2×D412yes—
202C2×C42#4112yes—
203#107—44no—
204#106—48no—
205C4⋊C8—28no—
206#102—28no—
207#102—48no—
208#102—44no—
209#95—28no—
210#95—24no—
211#95—22no—
212#95—22no—
213#95—24no—
214#95—44no—
215#95—44no—
216#95—42no—
217#95—44no—
218#95—81no—
219#95—82no—
220C24—22no—
221C24—41no—
222C24—42no—
223C24—42no—
224C24—41no—
225C24—41no—
226C24—42no—
227C24—41no—
228C24—42no—
229C24—41no—
230C24—41no—
231C24C22×D411yescharacteristic
232C2×D4—22no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—41no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4#12112yes—
301C22×C4#12112yes—
302C22×C4#13512yes—
303C22×C4AGL(1,ℤ/8)14yes—
304C22×C4AGL(1,ℤ/8)12yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4112yes—
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4112yes—
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4112yes—
312C22×C4H(5,𝔽2)11yescharacteristic
313C22×C4H(5,𝔽2)11yescharacteristic
314C22×C42−1+412yes—
315C2×C8—22no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—42no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—84no—
350C4⋊C4—84no—
351C4⋊C4#12414yes—
352C22⋊C4—88no—
353C22⋊C4—84no—
354C22⋊C4—88no—
355C22⋊C4—84no—
356C42—22no—
357C42—22no—
358C42—24no—
359C42—416no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—42no—
364C42—48no—
365C23—42no—
366C23—42no—
367C23—44no—
368C23—42no—
369C23—44no—
370C23—82no—
371C23—82no—
372C23—82no—
373C23—82no—
374C23—81no—
375C23—82no—
376C23—82no—
377C23—82no—
378C23—82no—
379C23—81no—
380C23—81no—
381C23—82no—
382C23—81no—
383C23—82no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23C2×#12111yescharacteristic
389C23#52512yes—
390C23#54611yescharacteristic
391C23#54712yes—
392C23#55011yescharacteristic
393C23C2×#13511yescharacteristic
394C23C2×AGL(1,ℤ/8)11yescharacteristic
395C23C2×AGL(1,ℤ/8)12yes—
396C23#57712yes—
397C23#57711yescharacteristic
398C23#57811yescharacteristic
399D4—42no—
400D4—42no—
401D4—44no—
402D4—81no—
403D4—81no—
404D4—82no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—81no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4#46314yes—
460C2×C4#52511yescharacteristic
461C2×C4#52511yescharacteristic
462C2×C4C2×#12411yescharacteristic
463C2×C4#52912yes—
464C2×C4#52912yes—
465C2×C4#52912yes—
466C2×C4#53211yescharacteristic
467C2×C4#53511yescharacteristic
468C2×C4#53611yescharacteristic
469C2×C4#54212yes—
470C2×C4#54212yes—
471C2×C4#55112yes—
472C2×C4#55212yes—
473C2×C4#55212yes—
474C2×C4#55212yes—
475C2×C4#55312yes—
476C2×C4#55311yescharacteristic
477C2×C4#55511yescharacteristic
478C8—44no—
479C8—42no—
480C8—81no—
481C22—84no—
482C22—84no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—161no—
487C22—161no—
488C22—161no—
489C22—161no—
490C22—162no—
491C22—162no—
492C22#241311yescharacteristic
493C22#241411yescharacteristic
494C22#247012yes—
495C22#247912yes—
496C22#285011yescharacteristic
497C22C2×#46311yescharacteristic
498C22#303811yescharacteristic
499C22#304311yescharacteristic
500C22#304611yescharacteristic
501C22#310612yes—
502C22#310912yes—
503C22#312112yes—
504C22#312512yes—
505C22#330311yescharacteristic
506C22#331012yes—
507C22#333011yescharacteristic
508C22#333712yes—
509C22#334312yes—
510C22#334512yes—
511C22#335412yes—
512C22#339311yescharacteristic
513C22#339711yescharacteristic
514C22#339712yes—
515C4—24no—
516C4—44no—
517C4—42no—
518C4—44no—
519C4—82no—
520C4—82no—
521C4—84no—
522C4—161no—
523C4—162no—
524C4#239811yescharacteristic
525C4#252912yes—
526C4#253012yes—
527C4#260311yescharacteristic
528C4#261111yescharacteristic
529C4#264011yescharacteristic
530C2—162no—
531C2#1489411yescharacteristic
532C2#2458111yescharacteristic
533C2#2459411yescharacteristic
534C2#2462711yescharacteristic
535C2#2464411yescharacteristic
536C2#2717112yes—
537C2#2717912yes—
538C2#2749912yes—
539C2#2750712yes—
540C2#3303011yescharacteristic
541C2#3303711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1948C2semidirect
4. #13810C2semidirect
5. #11371C2semidirect
6. #11257C2semidirect
7. #11255C2semidirect
11. #10907C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19482
C2×#19252
#151362
#138102
#113712
#112572
#112552
#111212
#110762
#109312
#109072