Finite Group Databasebeta

← #403615

Subgroups — #403615

Order 512.

1344 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic
3C2×#1921C211yesmaximal, characteristic
4#15202C211yesmaximal, characteristic
5#11380C211yesmaximal, characteristic
6#11371C211yesmaximal, characteristic
7#11324C212yesmaximal
8#11207C212yesmaximal
9#10907C212yesmaximal
10#10885C212yesmaximal
11#10807C211yesmaximal, characteristic
12C22×C4×Q8—21no—
13C22×C4×Q8C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#107C2212yes—
16Q8×C4⋊C4—24no—
17Q8×C4⋊C4—24no—
18C2×#389—22no—
19C2×#389—21no—
20C2×#389C2211yescharacteristic
21C2×#389C2212yes—
22C2×#384C2212yes—
23C2×#384C2211yescharacteristic
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C4⋊C4C2211yescharacteristic
26#2083—24no—
27#1921—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1446—21no—
41#1444—22no—
42#1444—21no—
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×#107—22no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#107—24no—
77C2×#107—24no—
78C2×#107—42no—
79#389—28no—
80#389—216no—
81#389—22no—
82#389—22no—
83#389—24no—
84#389—24no—
85#384—24no—
86#384—24no—
87#384—22no—
88#384D414yes—
89C4×C4⋊C4—24no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4D414yes—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101#340—24no—
102#340—24no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—42no—
107#340—42no—
108#340Q818yescomplemented
109#336—22no—
110#336—22no—
111#336—24no—
112#336—24no—
113C22×Q8—22no—
114C22×Q8—24no—
115C22×Q8—24no—
116C22×Q8—22no—
117C22×Q8Q1614yes—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2411yescharacteristic
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—21no—
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×C8C2×Q814yes—
145C22×C8#4112yes—
146C4×Q8—28no—
147C4×Q8—28no—
148C4×Q8—216no—
149C4×Q8—432no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4C2×D414yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×Q814yes—
200C2×C4⋊C4C2×Q814yes—
201C2×C4⋊C4#4112yes—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—28no—
209C2×C42—24no—
210C2×C42—44no—
211C2×C42—44no—
212C2×C42—44no—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42#4114yes—
216C2×C42#4114yes—
217#107—28no—
218#107—48no—
219#95—28no—
220#95—24no—
221#95—42no—
222#95—44no—
223#95—44no—
224#95—42no—
225#95—82no—
226#95—81no—
227C24C22×D411yescharacteristic
228C2×Q8—28no—
229C2×Q8—28no—
230C2×Q8—216no—
231C2×Q8—28no—
232C2×Q8—216no—
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—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
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—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4#12012yes—
284C22×C4#12012yes—
285C22×C4#12314yes—
286C22×C4#12314yes—
287C22×C4C2×Q1612yes—
288C22×C4C2×Q1614yes—
289C22×C4AGL(1,ℤ/8)14yes—
290C22×C4#13712yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×Q812yes—
294C22×C4C2×#4112yes—
295C22×C4C2×#4112yes—
296C22×C4C2×#4111yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4H(5,𝔽2)11yescharacteristic
299C22×C42−1+412yes—
300C22×C42−1+411yescharacteristic
301C2×C8—22no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—41no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—42no—
318C4⋊C4—28no—
319C4⋊C4—216no—
320C4⋊C4—22no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—48no—
331C4⋊C4—416no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—84no—
344C4⋊C4—88no—
345C4⋊C4—84no—
346C4⋊C4—88no—
347C4⋊C4C4⋊Q814yes—
348C42—22no—
349C42—22no—
350C42—24no—
351C42—24no—
352C42—24no—
353C42—24no—
354C42—24no—
355C42—48no—
356C42—416no—
357C23C2×#12011yescharacteristic
358C23C2×#12312yes—
359C23#54811yescharacteristic
360C23D4×Q812yes—
361C23#55011yescharacteristic
362C23C22×Q1611yescharacteristic
363C23C2×AGL(1,ℤ/8)12yes—
364C23C2×#13711yescharacteristic
365C23#57612yes—
366C23#57711yescharacteristic
367C23#57811yescharacteristic
368Q8—232no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4#45114yes—
426C2×C4#48418yes—
427C2×C4#52511yescharacteristic
428C2×C4#52511yescharacteristic
429C2×C4#52511yescharacteristic
430C2×C4#52512yes—
431C2×C4C2×C4⋊Q811yescharacteristic
432C2×C4#53312yes—
433C2×C4#53312yes—
434C2×C4#53611yescharacteristic
435C2×C4#53711yescharacteristic
436C2×C4#54112yes—
437C2×C4#54212yes—
438C2×C4#54611yescharacteristic
439C2×C4#54711yescharacteristic
440C2×C4D4×Q812yes—
441C2×C4#55212yes—
442C2×C4#55412yes—
443C2×C4#55612yes—
444C2×C4#55712yes—
445C8—42no—
446C8—44no—
447C8—81no—
448C22#240711yescharacteristic
449C22#242312yes—
450C22#243411yescharacteristic
451C22#244612yes—
452C22#284911yescharacteristic
453C22C2×#45111yescharacteristic
454C22#300311yescharacteristic
455C22#300811yescharacteristic
456C22#301311yescharacteristic
457C22C2×#48412yes—
458C22#309212yes—
459C22#309612yes—
460C22#309812yes—
461C22#336311yescharacteristic
462C22#336612yes—
463C22#337411yescharacteristic
464C22Q8×Q1612yes—
465C22#338412yes—
466C22#338512yes—
467C22#338612yes—
468C22#339511yescharacteristic
469C22#339912yes—
470C22#340011yescharacteristic
471C4—24no—
472C4—28no—
473C4—44no—
474C4—42no—
475C4—44no—
476C4—82no—
477C4—82no—
478C4—84no—
479C4—84no—
480C4—161no—
481C4#239911yescharacteristic
482C4#246212yes—
483C4#246512yes—
484C4#258411yescharacteristic
485C4#259911yescharacteristic
486C4#263711yescharacteristic
487C2#1487911yescharacteristic
488C2#2419111yescharacteristic
489C2#2419611yescharacteristic
490C2#2506212yes—
491C2#2506812yes—
492C2#2584111yescharacteristic
493C2#2584411yescharacteristic
494C2#2628212yes—
495C2#2628812yes—
496C2#3298311yescharacteristic
497C2#3298611yescharacteristic
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
108. #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×#20832
C2×#19212
#152022
#113802
#113712
#113242
#112072
#109072
#108852
#108072