Finite Group Databasebeta

← #403611

Subgroups — #403611

Order 512.

1354 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic
3C2×#1919C211yesmaximal, characteristic
4#15158C211yesmaximal, characteristic
5#11380C211yesmaximal, characteristic
6#11371C211yesmaximal, characteristic
7#11323C212yesmaximal
8#11207C212yesmaximal
9#10908C212yesmaximal
10#10884C212yesmaximal
11#10807C211yesmaximal, characteristic
12C22×C4×Q8—21no—
13C22×C4×Q8C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×#106C2212yes—
16C2×#395—21no—
17Q8×C4⋊C4—24no—
18Q8×C4⋊C4—24no—
19C2×#389—22no—
20C2×#389—21no—
21C2×#389C2211yescharacteristic
22C2×#389C2212yes—
23C2×#384C2212yes—
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27#2084—24no—
28#1919—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1441—21no—
42#1441—22no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8D411yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C2×C4×Q8—28no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—24no—
53C2×C4×Q8—24no—
54C2×C4×Q8—48no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C42—22no—
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72C22×C42C2312yes—
73C2×#106—22no—
74C2×#106—22no—
75C2×#106—22no—
76C2×#106—24no—
77C2×#106—24no—
78C2×#106—42no—
79#395—44no—
80#389—28no—
81#389—216no—
82#389—22no—
83#389—22no—
84#389—24no—
85#389—24no—
86#384—24no—
87#384—24no—
88#384—22no—
89#384D414yes—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4D414yes—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—22no—
102#340—24no—
103#340—24no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—42no—
108#340—42no—
109#340Q818yescomplemented
110#336—22no—
111#336—22no—
112#336—24no—
113#336—24no—
114C22×Q8—22no—
115C22×Q8—24no—
116C22×Q8—24no—
117C22×Q8—22no—
118C22×Q8SD1614yes—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2411yescharacteristic
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8C2×Q814yes—
146C22×C8#4112yes—
147C4×Q8—28no—
148C4×Q8—28no—
149C4×Q8—216no—
150C4×Q8—432no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
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—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
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—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×Q814yes—
202C2×C4⋊C4C2×Q814yes—
203C2×C4⋊C4#4112yes—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—28no—
211C2×C42—24no—
212C2×C42—44no—
213C2×C42—44no—
214C2×C42—44no—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42#4114yes—
218C2×C42#4114yes—
219#106—28no—
220#106—48no—
221#95—28no—
222#95—24no—
223#95—42no—
224#95—44no—
225#95—42no—
226#95—44no—
227#95—82no—
228#95—81no—
229C24C22×D411yescharacteristic
230C2×Q8—28no—
231C2×Q8—28no—
232C2×Q8—216no—
233C2×Q8—28no—
234C2×Q8—216no—
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—24no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4#12012yes—
288C22×C4#12012yes—
289C22×C4#12314yes—
290C22×C4#12314yes—
291C22×C4C2×SD1612yes—
292C22×C4C2×SD1614yes—
293C22×C4AGL(1,ℤ/8)14yes—
294C22×C4#13712yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×Q812yes—
298C22×C4C2×#4112yes—
299C22×C4C2×#4112yes—
300C22×C4C2×#4111yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4H(5,𝔽2)11yescharacteristic
303C22×C42−1+412yes—
304C22×C42−1+411yescharacteristic
305C2×C8—22no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—41no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—42no—
321C2×C8—42no—
322C4⋊C4—28no—
323C4⋊C4—216no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—48no—
335C4⋊C4—416no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—42no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—84no—
348C4⋊C4—88no—
349C4⋊C4—84no—
350C4⋊C4—88no—
351C4⋊C4C4⋊Q814yes—
352C42—22no—
353C42—22no—
354C42—24no—
355C42—24no—
356C42—24no—
357C42—24no—
358C42—24no—
359C42—48no—
360C42—416no—
361C23C2×#12011yescharacteristic
362C23C2×#12312yes—
363C23#54811yescharacteristic
364C23D4×Q812yes—
365C23#55011yescharacteristic
366C23C22×SD1611yescharacteristic
367C23C2×AGL(1,ℤ/8)12yes—
368C23C2×#13711yescharacteristic
369C23#57712yes—
370C23#57711yescharacteristic
371C23#57811yescharacteristic
372Q8—232no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
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—44no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—81no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4#44814yes—
432C2×C4#48118yes—
433C2×C4#52511yescharacteristic
434C2×C4#52511yescharacteristic
435C2×C4#52511yescharacteristic
436C2×C4#52512yes—
437C2×C4C2×C4⋊Q811yescharacteristic
438C2×C4#53312yes—
439C2×C4#53312yes—
440C2×C4#53611yescharacteristic
441C2×C4#53711yescharacteristic
442C2×C4#54112yes—
443C2×C4#54212yes—
444C2×C4#54611yescharacteristic
445C2×C4#54711yescharacteristic
446C2×C4D4×Q812yes—
447C2×C4#55212yes—
448C2×C4#55412yes—
449C2×C4#55612yes—
450C2×C4#55712yes—
451C8—42no—
452C8—44no—
453C8—81no—
454C22#240711yescharacteristic
455C22#242312yes—
456C22#243411yescharacteristic
457C22#244612yes—
458C22#284911yescharacteristic
459C22C2×#44811yescharacteristic
460C22#300311yescharacteristic
461C22#300911yescharacteristic
462C22#301311yescharacteristic
463C22C2×#48112yes—
464C22#309212yes—
465C22#309712yes—
466C22#309912yes—
467C22#335911yescharacteristic
468C22#336812yes—
469C22#337211yescharacteristic
470C22Q8×SD1612yes—
471C22#338412yes—
472C22#338612yes—
473C22#338712yes—
474C22#339211yescharacteristic
475C22#339712yes—
476C22#339811yescharacteristic
477C4—24no—
478C4—28no—
479C4—44no—
480C4—42no—
481C4—44no—
482C4—82no—
483C4—82no—
484C4—84no—
485C4—84no—
486C4—161no—
487C4#239911yescharacteristic
488C4#246212yes—
489C4#246512yes—
490C4#258411yescharacteristic
491C4#259911yescharacteristic
492C4#263711yescharacteristic
493C2#1487911yescharacteristic
494C2#2418711yescharacteristic
495C2#2420011yescharacteristic
496C2#2505812yes—
497C2#2507012yes—
498C2#2583711yescharacteristic
499C2#2584411yescharacteristic
500C2#2627812yes—
501C2#2629012yes—
502C2#3297911yescharacteristic
503C2#3298811yescharacteristic
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
109. #340Q8semidirect

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×#20842
C2×#19192
#151582
#113802
#113712
#113232
#112072
#109082
#108842
#108072