Finite Group Databasebeta

← #470192

Subgroups — #470192

Order 512.

1062 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1948C211yesmaximal, characteristic, complemented
2C2×#1926C211yesmaximal, characteristic
3#15136C211yesmaximal, characteristic
4#13936C211yesmaximal, characteristic, complemented
5#11373C211yesmaximal, characteristic, complemented
6#11322C211yesmaximal, characteristic, complemented
7#11257C212yesmaximal, complemented
8#11231C212yesmaximal, complemented
9#11121C212yesmaximal
10#11076C212yesmaximal
11#11076C211yesmaximal, characteristic
12C22×#106C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C2×#397—21no—
20C2×#397—21no—
21C2×#390C2211yescharacteristic
22C2×#390C2212yes—
23C2×#384C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1948—24no—
26#1926—24no—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1443—22no—
35#1439—22no—
36C4⋊#95C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40#1434—21no—
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⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C42—21no—
59C22×C42C2312yes—
60C22×C42C2311yescharacteristic
61C2×#106—22no—
62C2×#106—24no—
63C2×#106—42no—
64C2×#106—42no—
65C2×#106—42no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—42no—
70C2×C4⋊C8—42no—
71C2×#102—22no—
72C2×#102—22no—
73C2×#102—22no—
74C2×#102—24no—
75C2×#102—22no—
76C2×#102—42no—
77C2×#102—42no—
78C2×#102—42no—
79#397—44no—
80#397—44no—
81#390—24no—
82#390—22no—
83#390—22no—
84#390—42no—
85#384—22no—
86#384—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4D414yes—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95D411yescharacteristic
99C2×#95D412yes—
100C2×#95D411yescharacteristic
101#340—24no—
102#340—22no—
103#340—24no—
104#340—42no—
105#340—42no—
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—21no—
115C22×D4—21no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4#4112yes—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2411yescharacteristic
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
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—22no—
143C22×C8—21no—
144C22×C8#4112yes—
145C22×C8#4114yes—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—81no—
175C2×C4⋊C4—81no—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—82no—
188C2×C22⋊C4—82no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—24no—
198C2×C42—82no—
199C2×C42C2×D412yes—
200C2×C42#4112yes—
201#106—44no—
202#106—48no—
203C4⋊C8—28no—
204C4⋊C8—48no—
205#102—28no—
206#102—44no—
207#102—48no—
208#95—24no—
209#95—22no—
210#95—42no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—42no—
215#95—82no—
216#95—81no—
217#95—82no—
218#95—82no—
219#95SD1614yes—
220C24—22no—
221C24—41no—
222C24—42no—
223C24—42no—
224C24—41no—
225C24—41no—
226C24—41no—
227C24—42no—
228C24—41no—
229C24—41no—
230C24—42no—
231C24C22×D411yescharacteristic
232C2×D4—22no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—41no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4#12112yes—
299C22×C4#12112yes—
300C22×C4C2×SD1614yes—
301C22×C4C2×SD1612yes—
302C22×C4AGL(1,ℤ/8)12yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C2×#4112yes—
305C22×C4C2×#4111yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4H(5,𝔽2)12yes—
309C22×C4H(5,𝔽2)11yescharacteristic
310C22×C4H(5,𝔽2)11yescharacteristic
311C22×C42−1+412yes—
312C22×C42−1+412yes—
313C2×C8—22no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—42no—
319C2×C8—41no—
320C2×C8—42no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—42no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—84no—
349C4⋊C4—84no—
350C22⋊C4—88no—
351C22⋊C4—84no—
352C22⋊C4—88no—
353C22⋊C4—84no—
354C42—22no—
355C42—22no—
356C42—24no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—84no—
362C23—42no—
363C23—44no—
364C23—42no—
365C23—44no—
366C23—42no—
367C23—82no—
368C23—82no—
369C23—82no—
370C23—82no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—82no—
375C23—81no—
376C23—82no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23C2×#12111yescharacteristic
386C23#53712yes—
387C23#54611yescharacteristic
388C23#54712yes—
389C23#55011yescharacteristic
390C23C22×SD1611yescharacteristic
391C23C2×AGL(1,ℤ/8)11yescharacteristic
392C23#57512yes—
393C23#57611yescharacteristic
394C23#57712yes—
395C23#57711yescharacteristic
396D4—42no—
397D4—42no—
398D4—44no—
399D4—81no—
400D4—81no—
401D4—82no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4#46014yes—
455C2×C4#52511yescharacteristic
456C2×C4#52511yescharacteristic
457C2×C4#53511yescharacteristic
458C2×C4#53511yescharacteristic
459C2×C4#53611yescharacteristic
460C2×C4#53611yescharacteristic
461C2×C4#53812yes—
462C2×C4#53812yes—
463C2×C4#53912yes—
464C2×C4#53912yes—
465C2×C4#54212yes—
466C2×C4#55212yes—
467C2×C4#55212yes—
468C2×C4#55311yescharacteristic
469C2×C4#55511yescharacteristic
470C2×C4#55612yes—
471C2×C4#56312yes—
472C2×C4#56312yes—
473C8—44no—
474C8—42no—
475C8—81no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22—84no—
480C22—84no—
481C22—161no—
482C22—162no—
483C22—161no—
484C22—162no—
485C22—161no—
486C22—161no—
487C22#241311yescharacteristic
488C22#241411yescharacteristic
489C22#262011yescharacteristic
490C22#269512yes—
491C22#272712yes—
492C22C2×#46011yescharacteristic
493C22#304011yescharacteristic
494C22#304211yescharacteristic
495C22#304511yescharacteristic
496C22#324612yes—
497C22#325112yes—
498C22#325712yes—
499C22#326312yes—
500C22#329511yescharacteristic
501C22#330712yes—
502C22#332411yescharacteristic
503C22#333512yes—
504C22#334512yes—
505C22#334712yes—
506C22#335112yes—
507C22#339211yescharacteristic
508C22#339911yescharacteristic
509C22#340112yes—
510C4—24no—
511C4—42no—
512C4—44no—
513C4—82no—
514C4—82no—
515C4—82no—
516C4—162no—
517C4—162no—
518C4—161no—
519C4#259911yescharacteristic
520C4#263111yescharacteristic
521C4#263311yescharacteristic
522C4#277012yes—
523C4#277512yes—
524C4#282111yescharacteristic
525C2—162no—
526C2#1667911yescharacteristic
527C2#2457811yescharacteristic
528C2#2459911yescharacteristic
529C2#2462811yescharacteristic
530C2#2464311yescharacteristic
531C2#2982611yescharacteristic
532C2#2983811yescharacteristic
533C2#3160612yes—
534C2#3162112yes—
535C2#3252512yes—
536C2#3254112yes—
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. #13936C2semidirect
5. #11373C2semidirect
6. #11322C2semidirect
7. #11257C2semidirect
8. #11231C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19482
C2×#19262
#151362
#139362
#113732
#113222
#112572
#112312
#111212
#110762