Finite Group Databasebeta

← #414490

Subgroups — #414490

Order 512.

1096 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1934C211yesmaximal, characteristic
2C2×#1770C211yesmaximal, characteristic, complemented
3#15149C211yesmaximal, characteristic, complemented
4#15147C211yesmaximal, characteristic
5#11316C212yesmaximal, complemented
6#11227C211yesmaximal, characteristic, complemented
7#11076C211yesmaximal, characteristic
8#11066C212yesmaximal
9#10927C212yesmaximal
10#10881C212yesmaximal, complemented
11#10820C211yesmaximal, characteristic, complemented
12C22×C4⋊C8C2211yescharacteristic
13C22×#102C2211yescharacteristic
14C22×#102C2211yescharacteristic
15C22×#102C2212yes—
16C22×#97C2212yes—
17C22×C4×C8C2211yescharacteristic
18C2×#398—21no—
19C2×#397—21no—
20C2×#390C2211yescharacteristic
21C2×#390C2212yes—
22C2×#389C2211yescharacteristic
23C2×#389C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25#1934—24no—
26#1770—24no—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#336C2212yes—
35#1442—22no—
36#1441—21no—
37#1440—22no—
38#1439—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C23×D4C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—21no—
60C22×C42C2312yes—
61C22×C42C2311yescharacteristic
62C2×C4⋊C8—24no—
63C2×C4⋊C8—22no—
64C2×#102—22no—
65C2×#102—22no—
66C2×#102—22no—
67C2×#102—22no—
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—42no—
72C2×#97—24no—
73C2×#97—42no—
74C2×#97—42no—
75C2×C4×C8—24no—
76C2×C4×C8D414yes—
77#398—44no—
78#397—44no—
79#390—24no—
80#390—22no—
81#390—22no—
82#390—42no—
83#389—24no—
84#389—22no—
85#389—22no—
86#389—42no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95D412yes—
100#340—22no—
101#340—22no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—42no—
107#336—24no—
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—22no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2411yescharacteristic
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8C2×D412yes—
143C22×C8C2×D412yes—
144C22×C8#4114yes—
145C22×C8#4114yes—
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—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4#4114yes—
182C2×C4⋊C4#4112yes—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4—82no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42C2×D412yes—
203C2×C42#4112yes—
204C4⋊C8—28no—
205#102—28no—
206#102—48no—
207#102—44no—
208#97—48no—
209C4×C8—28no—
210#95—24no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—42no—
216#95—42no—
217#95—42no—
218#95—81no—
219#95—81no—
220#95—82no—
221C24—22no—
222C24—41no—
223C24—42no—
224C24—41no—
225C24—42no—
226C24—41no—
227C24—41no—
228C24—42no—
229C24—41no—
230C24—42no—
231C24—41no—
232C24C22×D411yescharacteristic
233C2×D4—22no—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—22no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—41no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4#12112yes—
303C22×C4#12112yes—
304C22×C4AGL(1,ℤ/8)12yes—
305C22×C4AGL(1,ℤ/8)12yes—
306C22×C4C22×D411yescharacteristic
307C22×C4C2×#4112yes—
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4112yes—
311C22×C4C2×#4112yes—
312C22×C4C2×#4111yescharacteristic
313C22×C4H(5,𝔽2)11yescharacteristic
314C22×C4H(5,𝔽2)11yescharacteristic
315C22×C42−1+412yes—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—24no—
323C2×C8—22no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8#12314yes—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—44no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—44no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—88no—
354C4⋊C4—84no—
355C22⋊C4—84no—
356C22⋊C4—88no—
357C22⋊C4—88no—
358C22⋊C4—84no—
359C42—22no—
360C42—22no—
361C42—24no—
362C42—42no—
363C42—42no—
364C42—42no—
365C42—42no—
366C23—42no—
367C23—42no—
368C23—44no—
369C23—42no—
370C23—44no—
371C23—82no—
372C23—81no—
373C23—81no—
374C23—82no—
375C23—82no—
376C23—82no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—82no—
389C23C2×#12111yescharacteristic
390C23#53212yes—
391C23#54611yescharacteristic
392C23#54611yescharacteristic
393C23#54812yes—
394C23C2×AGL(1,ℤ/8)11yescharacteristic
395C23C2×AGL(1,ℤ/8)11yescharacteristic
396C23#57512yes—
397C23#57611yescharacteristic
398C23#57712yes—
399C23#57811yescharacteristic
400D4—44no—
401D4—42no—
402D4—42no—
403D4—82no—
404D4—81no—
405D4—81no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
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—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—81no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—82no—
441C2×C4—81no—
442C2×C4—82no—
443C2×C4—81no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4#46914yes—
461C2×C4C2×#12311yescharacteristic
462C2×C4#52511yescharacteristic
463C2×C4#52511yescharacteristic
464C2×C4#52912yes—
465C2×C4#52912yes—
466C2×C4#52912yes—
467C2×C4#53211yescharacteristic
468C2×C4#53511yescharacteristic
469C2×C4#53611yescharacteristic
470C2×C4#53912yes—
471C2×C4#53912yes—
472C2×C4#55112yes—
473C2×C4#55212yes—
474C2×C4#55311yescharacteristic
475C2×C4#55311yescharacteristic
476C2×C4#55312yes—
477C2×C4#55612yes—
478C2×C4#55612yes—
479C8—24no—
480C8—44no—
481C8—42no—
482C22—82no—
483C22—84no—
484C22—84no—
485C22—82no—
486C22—82no—
487C22—161no—
488C22—161no—
489C22—162no—
490C22—162no—
491C22—161no—
492C22—161no—
493C22#253511yescharacteristic
494C22#254511yescharacteristic
495C22#255812yes—
496C22#256012yes—
497C22#285411yescharacteristic
498C22C2×#46911yescharacteristic
499C22#305411yescharacteristic
500C22#305711yescharacteristic
501C22#306111yescharacteristic
502C22#317812yes—
503C22#317912yes—
504C22#318012yes—
505C22#318012yes—
506C22#332312yes—
507C22#332511yescharacteristic
508C22#332511yescharacteristic
509C22#332612yes—
510C22#333011yescharacteristic
511C22#333011yescharacteristic
512C22#336812yes—
513C22#337012yes—
514C22#337312yes—
515C22#337712yes—
516C4—24no—
517C4—42no—
518C4—84no—
519C4—82no—
520C4—82no—
521C4—82no—
522C4—162no—
523C4—161no—
524C4—161no—
525C4#239511yescharacteristic
526C4#257712yes—
527C4#259412yes—
528C4#259911yescharacteristic
529C4#260311yescharacteristic
530C4#262311yescharacteristic
531C2—162no—
532C2#1517411yescharacteristic
533C2#2814711yescharacteristic
534C2#2814911yescharacteristic
535C2#2864311yescharacteristic
536C2#2864811yescharacteristic
537C2#2901612yes—
538C2#2901912yes—
539C2#2907912yes—
540C2#2908612yes—
541C2#3313111yescharacteristic
542C2#3313511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1770C2semidirect
3. #15149C2semidirect
5. #11316C2semidirect
6. #11227C2semidirect
10. #10881C2semidirect
11. #10820C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19342
C2×#17702
#151492
#151472
#113162
#112272
#110762
#110662
#109272
#108812
#108202