Finite Group Databasebeta

← #403610

Subgroups — #403610

Order 512.

1354 conjugacy classes of subgroups.

The subgroup lattice has 513 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×#1918C211yesmaximal, characteristic
4#15158C211yesmaximal, characteristic
5#11380C211yesmaximal, characteristic
6#11373C211yesmaximal, characteristic
7#11324C212yesmaximal
8#11207C212yesmaximal
9#10907C212yesmaximal
10#10884C212yesmaximal
11#10806C211yesmaximal, characteristic
12C22×C4×Q8—21no—
13C22×C4×Q8C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#107C2212yes—
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#2083—24no—
28#1918—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—44no—
55C2×C4×Q8—44no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4D412yes—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C42—22no—
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73C22×C42C2312yes—
74C2×#107—22no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#107—24no—
78C2×#107—24no—
79C2×#107—42no—
80#395—44no—
81#389—28no—
82#389—216no—
83#389—22no—
84#389—22no—
85#389—24no—
86#389—24no—
87#384—24no—
88#384—24no—
89#384—22no—
90#384D414yes—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4D414yes—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103#340—24no—
104#340—24no—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—42no—
109#340—42no—
110#340Q818yescomplemented
111#336—22no—
112#336—22no—
113#336—24no—
114#336—24no—
115C22×Q8—22no—
116C22×Q8—24no—
117C22×Q8—22no—
118C22×Q8—24no—
119C22×Q8D814yes—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2411yescharacteristic
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8C2×Q814yes—
147C22×C8#4112yes—
148C4×Q8—28no—
149C4×Q8—28no—
150C4×Q8—216no—
151C4×Q8—432no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4—81no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C2×D414yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×Q814yes—
205C2×C4⋊C4C2×Q814yes—
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#107—28no—
224#107—48no—
225#95—28no—
226#95—24no—
227#95—42no—
228#95—44no—
229#95—42no—
230#95—44no—
231#95—82no—
232#95—81no—
233C24C22×D411yescharacteristic
234C2×Q8—28no—
235C2×Q8—216no—
236C2×Q8—28no—
237C2×Q8—28no—
238C2×Q8—216no—
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—24no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4#12012yes—
294C22×C4#12012yes—
295C22×C4#12314yes—
296C22×C4#12314yes—
297C22×C4C2×D812yes—
298C22×C4C2×D814yes—
299C22×C4AGL(1,ℤ/8)12yes—
300C22×C4#13714yes—
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×Q812yes—
304C22×C4C2×#4112yes—
305C22×C4C2×#4112yes—
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4H(5,𝔽2)11yescharacteristic
309C22×C42−1+412yes—
310C22×C42−1+411yescharacteristic
311C2×C8—22no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—41no—
317C2×C8—42no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—42no—
328C4⋊C4—28no—
329C4⋊C4—216no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—416no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—48no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—84no—
354C4⋊C4—88no—
355C4⋊C4—84no—
356C4⋊C4—88no—
357C4⋊C4C4⋊Q814yes—
358C42—22no—
359C42—22no—
360C42—24no—
361C42—24no—
362C42—24no—
363C42—24no—
364C42—24no—
365C42—416no—
366C42—48no—
367C23C2×#12011yescharacteristic
368C23C2×#12312yes—
369C23#54811yescharacteristic
370C23D4×Q812yes—
371C23#55011yescharacteristic
372C23C22×D811yescharacteristic
373C23C2×AGL(1,ℤ/8)11yescharacteristic
374C23C2×#13712yes—
375C23#57611yescharacteristic
376C23#57711yescharacteristic
377C23#57812yes—
378Q8—232no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—81no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—82no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—82no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4#44714yes—
440C2×C4#48018yes—
441C2×C4#52511yescharacteristic
442C2×C4#52511yescharacteristic
443C2×C4#52511yescharacteristic
444C2×C4#52512yes—
445C2×C4C2×C4⋊Q811yescharacteristic
446C2×C4#53312yes—
447C2×C4#53312yes—
448C2×C4#53611yescharacteristic
449C2×C4#53711yescharacteristic
450C2×C4#54112yes—
451C2×C4#54212yes—
452C2×C4#54611yescharacteristic
453C2×C4#54711yescharacteristic
454C2×C4D4×Q812yes—
455C2×C4#55212yes—
456C2×C4#55412yes—
457C2×C4#55612yes—
458C2×C4#55712yes—
459C8—42no—
460C8—44no—
461C8—81no—
462C22#240711yescharacteristic
463C22#242312yes—
464C22#243411yescharacteristic
465C22#244612yes—
466C22#284911yescharacteristic
467C22C2×#44711yescharacteristic
468C22#300411yescharacteristic
469C22#301111yescharacteristic
470C22#301211yescharacteristic
471C22C2×#48012yes—
472C22#309212yes—
473C22#309712yes—
474C22#309812yes—
475C22#335811yescharacteristic
476C22#336611yescharacteristic
477C22#337412yes—
478C22Q8×D812yes—
479C22#338512yes—
480C22#338612yes—
481C22#338712yes—
482C22#339111yescharacteristic
483C22#339711yescharacteristic
484C22#339812yes—
485C4—24no—
486C4—28no—
487C4—44no—
488C4—42no—
489C4—44no—
490C4—82no—
491C4—82no—
492C4—84no—
493C4—84no—
494C4—161no—
495C4#239911yescharacteristic
496C4#246212yes—
497C4#246512yes—
498C4#258411yescharacteristic
499C4#259911yescharacteristic
500C4#263711yescharacteristic
501C2#1487911yescharacteristic
502C2#2418411yescharacteristic
503C2#2420311yescharacteristic
504C2#2505612yes—
505C2#2507012yes—
506C2#2583811yescharacteristic
507C2#2584711yescharacteristic
508C2#2627612yes—
509C2#2628812yes—
510C2#3297811yescharacteristic
511C2#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

Complements

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

NH ≅ G/Ntype
110. #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×#19182
#151582
#113802
#113732
#113242
#112072
#109072
#108842
#108062