Finite Group Databasebeta

← #415374

Subgroups — #415374

Order 512.

1004 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1984C211yesmaximal, characteristic
2C2×C4×#106C211yesmaximal, characteristic
3#13925C211yesmaximal, characteristic
4#13925C211yesmaximal, characteristic
5#11378C212yesmaximal
6#11316C212yesmaximal
7#11315C212yesmaximal
8#11257C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10850C211yesmaximal, characteristic
11#10842C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#1984—24no—
22C4×#106—24no—
23C2×#340C2211yescharacteristic
24C2×#340C2212yes—
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1446—21no—
34#1446—21no—
35#1445—21no—
36#1445—21no—
37#1436—22no—
38#1436—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C23×Q8D411yescharacteristic
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C42—21no—
62C22×C42—21no—
63C22×C42C2312yes—
64C22×C42C2311yescharacteristic
65C2×#106—22no—
66C2×#106—22no—
67C2×#106—22no—
68C2×#106—22no—
69C2×#106—22no—
70C2×#106Q814yescomplemented
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4×C8—22no—
75C2×C4×C8—22no—
76C2×C4×C8D414yes—
77#382—22no—
78#382—22no—
79#382—22no—
80#382—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99#340—22no—
100#340—22no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—42no—
105#340—42no—
106#340—42no—
107#336—24no—
108#336—24no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—24no—
113C22×Q8—42no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2411yescharacteristic
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8C2×D412yes—
143C22×C8C2×D412yes—
144C22×C8C2×Q812yes—
145C22×C8#4112yes—
146C4×Q8—44no—
147C4×Q8—44no—
148C4×Q8—44no—
149C4×Q8—44no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4C2×Q812yes—
189C2×C4⋊C4C2×Q812yes—
190C2×C4⋊C4#4112yes—
191C2×C4⋊C4#4112yes—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—82no—
201C2×C42—82no—
202C2×C42C2×D412yes—
203C2×C42#4112yes—
204C2×C42#4112yes—
205C2×C42#4112yes—
206C2×C42#4112yes—
207C2×C42#4112yes—
208#106—28no—
209#106—28no—
210C4⋊C8—44no—
211C4×C8—28no—
212#95—44no—
213#95—44no—
214#95—42no—
215#95—42no—
216#95—42no—
217#95—42no—
218#95—42no—
219#95—42no—
220#95—44no—
221#95—44no—
222#95—82no—
223#95—82no—
224#95—82no—
225#95—82no—
226C24C22×D411yescharacteristic
227C2×Q8—28no—
228C2×Q8—48no—
229C2×Q8—48no—
230C2×Q8—44no—
231C2×Q8—44no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4#12112yes—
284C22×C4#12112yes—
285C22×C4#12312yes—
286C22×C4#12312yes—
287C22×C4#12312yes—
288C22×C4#12312yes—
289C22×C4#13514yes—
290C22×C4C22×D411yescharacteristic
291C22×C4C22×Q811yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4111yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4H(5,𝔽2)11yescharacteristic
301C22×C42−1+411yescharacteristic
302C22×C42−1+411yescharacteristic
303C22×C42−1+411yescharacteristic
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8C22⋊Q814yes—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—48no—
342C4⋊C4—44no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—42no—
355C42—84no—
356C42—84no—
357C42#12314yes—
358C42#12414yes—
359C42#12614yes—
360C42#12614yes—
361C23C2×#12111yescharacteristic
362C23C2×#12311yescharacteristic
363C23C2×#12311yescharacteristic
364C23#54811yescharacteristic
365C23#54811yescharacteristic
366C23D4×Q811yescharacteristic
367C23#55011yescharacteristic
368C23C2×#13512yes—
369C23#57512yes—
370C23#57712yes—
371C23#57712yes—
372Q8—416no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4#39714yes—
430C2×C4#39714yes—
431C2×C4C2×C22⋊Q811yescharacteristic
432C2×C4C2×#12311yescharacteristic
433C2×C4#52511yescharacteristic
434C2×C4C2×#12411yescharacteristic
435C2×C4C2×#12611yescharacteristic
436C2×C4C2×#12611yescharacteristic
437C2×C4#52911yescharacteristic
438C2×C4#52911yescharacteristic
439C2×C4#53311yescharacteristic
440C2×C4#53311yescharacteristic
441C2×C4#53611yescharacteristic
442C2×C4#53711yescharacteristic
443C2×C4#54111yescharacteristic
444C2×C4#54111yescharacteristic
445C2×C4#54211yescharacteristic
446C2×C4#54211yescharacteristic
447C2×C4#55211yescharacteristic
448C2×C4#55211yescharacteristic
449C2×C4#55411yescharacteristic
450C2×C4#55611yescharacteristic
451C2×C4#55611yescharacteristic
452C2×C4#55611yescharacteristic
453C2×C4#55611yescharacteristic
454C2×C4#55611yescharacteristic
455C2×C4#55711yescharacteristic
456C2×C4#55711yescharacteristic
457C2×C4#55911yescharacteristic
458C2×C4#55911yescharacteristic
459C8—24no—
460C8—42no—
461C8—82no—
462C22C2×#39711yescharacteristic
463C22#244611yescharacteristic
464C22#244611yescharacteristic
465C22#244811yescharacteristic
466C22#244811yescharacteristic
467C22#253611yescharacteristic
468C22#253811yescharacteristic
469C22#305011yescharacteristic
470C22#305411yescharacteristic
471C22#305711yescharacteristic
472C22#305711yescharacteristic
473C22#309112yes—
474C22#309112yes—
475C22#309912yes—
476C22#309912yes—
477C22#336012yes—
478C22#336512yes—
479C22#337112yes—
480C22#337612yes—
481C22#338112yes—
482C22#338812yes—
483C22#339412yes—
484C22#340112yes—
485C4—24no—
486C4—24no—
487C4—44no—
488C4—82no—
489C4—82no—
490C4—82no—
491C4—82no—
492C4—82no—
493C4—162no—
494C4—162no—
495C4#251111yescharacteristic
496C4#251111yescharacteristic
497C4#251411yescharacteristic
498C4#251411yescharacteristic
499C4#256811yescharacteristic
500C4#259211yescharacteristic
501C4#262511yescharacteristic
502C4#285511yescharacteristic
503C2#1519211yescharacteristic
504C2#2363811yescharacteristic
505C2#2364011yescharacteristic
506C2#2628512yes—
507C2#2628612yes—
508C2#2632412yes—
509C2#2632712yes—
510C2#2818911yescharacteristic
511C2#2819211yescharacteristic
512C2#2826611yescharacteristic
513C2#2827011yescharacteristic
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

Complements

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

NH ≅ G/Ntype
70. C2×#106Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19842
C2×C4×#1062
#139252
#113782
#113162
#113152
#112572
#108502
#108422