Finite Group Databasebeta

← #403750

Subgroups — #403750

Order 512.

1360 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2C2×#2072C211yesmaximal, characteristic
3C2×#1927C211yesmaximal, characteristic
4#15201C211yesmaximal, characteristic
5#11380C211yesmaximal, characteristic
6#11373C211yesmaximal, characteristic
7#11267C212yesmaximal
8#11050C212yesmaximal
9#10932C212yesmaximal
10#10908C212yesmaximal
11#10806C211yesmaximal, characteristic
12C22×C4×Q8—21no—
13C22×C4×Q8C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18Q8×C4⋊C4—24no—
19Q8×C4⋊C4—24no—
20C2×#391C2212yes—
21C2×#391C2211yescharacteristic
22C2×#389—22no—
23C2×#389—21no—
24C2×#384C2212yes—
25C2×#384C2211yescharacteristic
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28#2072—24no—
29#1927—24no—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40#1446—21no—
41#1442—22no—
42#1442—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×Q8D411yescharacteristic
46C23×Q8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4×Q8—28no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—24no—
54C2×C4×Q8—24no—
55C2×C4×Q8—44no—
56C2×C4×Q8—44no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C42—22no—
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71C22×C42C2312yes—
72C2×#106—22no—
73C2×#106—24no—
74C2×#103—22no—
75C2×#103—22no—
76C2×#103—22no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—22no—
80C2×#103—24no—
81C2×#103—42no—
82#391—22no—
83#391—22no—
84#391—24no—
85#391—24no—
86#389—28no—
87#389—216no—
88#384—24no—
89#384—24no—
90#384—22no—
91#384D414yes—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4D414yes—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—22no—
106#340—24no—
107#340—24no—
108#340—22no—
109#340—24no—
110#340—42no—
111#340—42no—
112#336—22no—
113#336—22no—
114#336—24no—
115#336—24no—
116C22×Q8—22no—
117C22×Q8—24no—
118C22×Q8—22no—
119C22×Q8—24no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8SD1614yes—
130C22×Q8#4112yes—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2411yescharacteristic
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8#4112yes—
159C4×Q8—28no—
160C4×Q8—28no—
161C4×Q8—216no—
162C4×Q8—432no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—81no—
199C2×C4⋊C4—81no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C2×D414yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4#4114yes—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—28no—
213C2×C42—24no—
214C2×C42—44no—
215C2×C42—44no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42#4114yes—
221C2×C42#4114yes—
222#106—44no—
223#103—28no—
224#103—48no—
225#103—44no—
226#95—28no—
227#95—24no—
228#95—42no—
229#95—42no—
230#95—44no—
231#95—44no—
232#95—42no—
233#95—44no—
234#95—82no—
235#95—81no—
236C24C22×D411yescharacteristic
237C2×Q8—28no—
238C2×Q8—216no—
239C2×Q8—28no—
240C2×Q8—28no—
241C2×Q8—216no—
242C2×Q8—22no—
243C2×Q8—24no—
244C2×Q8—22no—
245C2×Q8—24no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—41no—
249C2×Q8—42no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—41no—
254C2×Q8—42no—
255C2×Q8—41no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C22×C4—22no—
261C22×C4—22no—
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—24no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4#12012yes—
313C22×C4#12012yes—
314C22×C4#12314yes—
315C22×C4#12314yes—
316C22×C4C2×SD1612yes—
317C22×C4C2×SD1614yes—
318C22×C4AGL(1,ℤ/8)12yes—
319C22×C4#13714yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C2×#4112yes—
323C22×C4C2×#4112yes—
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4112yes—
327C22×C4H(5,𝔽2)11yescharacteristic
328C22×C42−1+412yes—
329C22×C42−1+411yescharacteristic
330C2×C8—22no—
331C2×C8—24no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C4⋊C4—28no—
348C4⋊C4—216no—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—416no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—48no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—84no—
369C4⋊C4—88no—
370C4⋊C4#12414yes—
371C42—22no—
372C42—22no—
373C42—24no—
374C42—24no—
375C42—24no—
376C42—24no—
377C42—24no—
378C42—416no—
379C42—48no—
380C23C2×#12011yescharacteristic
381C23C2×#12312yes—
382C23#54812yes—
383C23#54811yescharacteristic
384C23#55011yescharacteristic
385C23C22×SD1611yescharacteristic
386C23C2×AGL(1,ℤ/8)11yescharacteristic
387C23C2×#13712yes—
388C23#57611yescharacteristic
389C23#57711yescharacteristic
390C23#57812yes—
391Q8—232no—
392Q8—42no—
393Q8—42no—
394Q8—44no—
395Q8—81no—
396Q8—81no—
397Q8—82no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—42no—
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—44no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—81no—
440C2×C4—82no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—82no—
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—82no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4#45014yes—
457C2×C4#48318yes—
458C2×C4#52511yescharacteristic
459C2×C4#52511yescharacteristic
460C2×C4#52511yescharacteristic
461C2×C4#52512yes—
462C2×C4C2×#12411yescharacteristic
463C2×C4#52912yes—
464C2×C4#52912yes—
465C2×C4#53611yescharacteristic
466C2×C4#53711yescharacteristic
467C2×C4#54112yes—
468C2×C4#54212yes—
469C2×C4#54611yescharacteristic
470C2×C4#54711yescharacteristic
471C2×C4#54812yes—
472C2×C4#55212yes—
473C2×C4#55612yes—
474C2×C4#55612yes—
475C2×C4#55912yes—
476C8—42no—
477C8—81no—
478C8—82no—
479C22#240711yescharacteristic
480C22#242112yes—
481C22#243611yescharacteristic
482C22#244812yes—
483C22#284911yescharacteristic
484C22C2×#45011yescharacteristic
485C22#300411yescharacteristic
486C22#301111yescharacteristic
487C22#301211yescharacteristic
488C22C2×#48312yes—
489C22#309212yes—
490C22#309712yes—
491C22#309812yes—
492C22#335912yes—
493C22#336211yescharacteristic
494C22#336712yes—
495C22#336811yescharacteristic
496C22#336912yes—
497C22#337412yes—
498C22#337512yes—
499C22#339211yescharacteristic
500C22#339812yes—
501C22#339911yescharacteristic
502C4—24no—
503C4—28no—
504C4—44no—
505C4—44no—
506C4—42no—
507C4—82no—
508C4—82no—
509C4—84no—
510C4—161no—
511C4—162no—
512C4#239811yescharacteristic
513C4#245812yes—
514C4#246912yes—
515C4#259211yescharacteristic
516C4#259911yescharacteristic
517C4#263711yescharacteristic
518C2#1488211yescharacteristic
519C2#2418811yescharacteristic
520C2#2420311yescharacteristic
521C2#2499412yes—
522C2#2500412yes—
523C2#2592911yescharacteristic
524C2#2594211yescharacteristic
525C2#2632012yes—
526C2#2633012yes—
527C2#3298211yescharacteristic
528C2#3298911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×Q8×C4⋊C42
C2×#20722
C2×#19272
#152012
#113802
#113732
#112672
#110502
#109322
#109082
#108062