Finite Group Databasebeta

← #403413

Subgroups — #403413

Order 512.

1382 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2C2×#2096C211yesmaximal, characteristic
3C2×#1927C211yesmaximal, characteristic
4#15201C211yesmaximal, characteristic
5#11380C211yesmaximal, characteristic
6#11357C211yesmaximal, characteristic
7#11323C212yesmaximal
8#11050C212yesmaximal
9#10932C212yesmaximal
10#10908C212yesmaximal
11#10806C211yesmaximal, characteristic
12C22×C4×Q8—21no—
13C22×C4×Q8C2211yescharacteristic
14C22×#106C2212yes—
15C22×#106C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2211yescharacteristic
19Q8×C4⋊C4—24no—
20Q8×C4⋊C4—24no—
21C2×#391C2212yes—
22C2×#391C2211yescharacteristic
23C2×#389—22no—
24C2×#389—21no—
25C2×#384C2212yes—
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2096—24no—
30#1927—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40#1446—21no—
41#1442—22no—
42#1442—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8D411yescharacteristic
46C23×Q8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
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⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C42—22no—
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71C22×C42C2312yes—
72C2×#106—24no—
73C2×#106—24no—
74C2×#106—22no—
75C2×#106—42no—
76C2×#106—42no—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—22no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103Q814yescomplemented
85#391—22no—
86#391—22no—
87#391—24no—
88#391—24no—
89#389—28no—
90#389—216no—
91#384—24no—
92#384—24no—
93#384—22no—
94#384D414yes—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4D414yes—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—22no—
109#340—24no—
110#340—22no—
111#340—24no—
112#340—42no—
113#340—42no—
114#336—22no—
115#336—22no—
116#336—24no—
117#336—24no—
118C22×Q8—22no—
119C22×Q8—24no—
120C22×Q8—22no—
121C22×Q8—24no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8SD1614yes—
132C22×Q8C2×Q812yes—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2411yescharacteristic
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8C2×Q812yes—
160C22×C8#4114yes—
161C4×Q8—28no—
162C4×Q8—28no—
163C4×Q8—216no—
164C4×Q8—432no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
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—22no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C2×D414yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×Q812yes—
206C2×C4⋊C4#4112yes—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—28no—
214C2×C42—24no—
215C2×C42—44no—
216C2×C42—44no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42#4114yes—
222C2×C42#4114yes—
223#106—48no—
224#106—44no—
225#103—28no—
226#103—48no—
227#103—44no—
228#95—28no—
229#95—24no—
230#95—42no—
231#95—42no—
232#95—44no—
233#95—44no—
234#95—42no—
235#95—44no—
236#95—82no—
237#95—81no—
238C24C22×D411yescharacteristic
239C2×Q8—28no—
240C2×Q8—216no—
241C2×Q8—28no—
242C2×Q8—28no—
243C2×Q8—216no—
244C2×Q8—22no—
245C2×Q8—24no—
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—41no—
249C2×Q8—42no—
250C2×Q8—41no—
251C2×Q8—42no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
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—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4#12012yes—
315C22×C4#12012yes—
316C22×C4#12314yes—
317C22×C4#12314yes—
318C22×C4C2×SD1612yes—
319C22×C4C2×SD1614yes—
320C22×C4AGL(1,ℤ/8)12yes—
321C22×C4#13714yes—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×Q811yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4112yes—
327C22×C4C2×#4112yes—
328C22×C4C2×#4112yes—
329C22×C4H(5,𝔽2)12yes—
330C22×C42−1+411yescharacteristic
331C22×C42−1+411yescharacteristic
332C2×C8—22no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—42no—
349C4⋊C4—28no—
350C4⋊C4—216no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—416no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—48no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—84no—
371C4⋊C4—88no—
372C4⋊C4C22⋊Q814yes—
373C42—22no—
374C42—22no—
375C42—24no—
376C42—24no—
377C42—24no—
378C42—24no—
379C42—24no—
380C42—416no—
381C42—48no—
382C23C2×#12011yescharacteristic
383C23C2×#12312yes—
384C23#54811yescharacteristic
385C23D4×Q811yescharacteristic
386C23#55012yes—
387C23C22×SD1611yescharacteristic
388C23C2×AGL(1,ℤ/8)11yescharacteristic
389C23C2×#13712yes—
390C23#57712yes—
391C23#57711yescharacteristic
392C23#57811yescharacteristic
393Q8—232no—
394Q8—42no—
395Q8—42no—
396Q8—44no—
397Q8—81no—
398Q8—81no—
399Q8—82no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
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—44no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4#45014yes—
459C2×C4#48318yes—
460C2×C4C2×C22⋊Q811yescharacteristic
461C2×C4#52511yescharacteristic
462C2×C4#52511yescharacteristic
463C2×C4#52512yes—
464C2×C4#52912yes—
465C2×C4#53211yescharacteristic
466C2×C4#53312yes—
467C2×C4#53511yescharacteristic
468C2×C4#53611yescharacteristic
469C2×C4#54212yes—
470C2×C4#54212yes—
471C2×C4#54612yes—
472C2×C4#54811yescharacteristic
473C2×C4D4×Q811yescharacteristic
474C2×C4#55212yes—
475C2×C4#55412yes—
476C2×C4#55612yes—
477C2×C4#55912yes—
478C8—42no—
479C8—44no—
480C8—81no—
481C22#240911yescharacteristic
482C22#242312yes—
483C22#243111yescharacteristic
484C22#244812yes—
485C22#284911yescharacteristic
486C22C2×#45011yescharacteristic
487C22#300411yescharacteristic
488C22#300911yescharacteristic
489C22#301311yescharacteristic
490C22C2×#48312yes—
491C22#309212yes—
492C22#309712yes—
493C22#309912yes—
494C22#335911yescharacteristic
495C22#336912yes—
496C22#337311yescharacteristic
497C22Q8×SD1611yescharacteristic
498C22#338611yescharacteristic
499C22#338712yes—
500C22#339212yes—
501C22#339712yes—
502C22#339812yes—
503C22#340012yes—
504C4—24no—
505C4—28no—
506C4—44no—
507C4—44no—
508C4—42no—
509C4—82no—
510C4—82no—
511C4—84no—
512C4—161no—
513C4—162no—
514C4#239611yescharacteristic
515C4#245812yes—
516C4#246512yes—
517C4#258011yescharacteristic
518C4#260411yescharacteristic
519C4#263211yescharacteristic
520C2#1487511yescharacteristic
521C2#2433011yescharacteristic
522C2#2434511yescharacteristic
523C2#2506012yes—
524C2#2507012yes—
525C2#2562711yescharacteristic
526C2#2564011yescharacteristic
527C2#2632012yes—
528C2#2632812yes—
529C2#3298211yescharacteristic
530C2#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
84. C2×#103Q8semidirect

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×#20962
C2×#19272
#152012
#113802
#113572
#113232
#110502
#109322
#109082
#108062