Finite Group Databasebeta

← #403500

Subgroups — #403500

Order 512.

1188 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic
3C2×#1918C211yesmaximal, characteristic
4#15156C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11293C212yesmaximal
8#11220C212yesmaximal
9#10908C212yesmaximal
10#10907C212yesmaximal
11#10806C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2212yes—
15C2×#400—21no—
16#2369—24no—
17#2369—24no—
18C2×#391—22no—
19C2×#389—21no—
20C2×#389C2212yes—
21C2×#389C2211yescharacteristic
22C2×C4×C4⋊C4—21no—
23C2×C4×C4⋊C4C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#2058—24no—
27#1918—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40#1443—22no—
41#1441—21no—
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8—21no—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8D414yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C42—21no—
65C22×C42—22no—
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68C2×#107—22no—
69C2×#107—22no—
70C2×#107Q814yescomplemented
71C2×#106—24no—
72C2×#106—24no—
73C2×#106—24no—
74#400—44no—
75#391—216no—
76#389—28no—
77#389—24no—
78#389—24no—
79#389—22no—
80#389—22no—
81C4×C4⋊C4—28no—
82C4×C4⋊C4—24no—
83C4×C4⋊C4—24no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4D414yes—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95D412yes—
99#340—24no—
100#340—24no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—24no—
105#340—24no—
106#340—24no—
107#336—24no—
108#336—24no—
109#336—22no—
110#336—22no—
111C22×Q8—28no—
112C22×Q8—24no—
113C22×Q8—81no—
114C22×Q8—81no—
115C22×Q8—82no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2411yescharacteristic
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8C2×Q812yes—
143C22×C8#4114yes—
144C4⋊Q8—28no—
145C4⋊Q8—28no—
146C4⋊Q8—216no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D414yes—
187C2×C4⋊C4C2×Q812yes—
188C2×C4⋊C4C2×Q812yes—
189C2×C4⋊C4#4112yes—
190C2×C4⋊C4#4114yes—
191C2×C4⋊C4#4114yes—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—24no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—28no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—82no—
205C2×C42—82no—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208#107—28no—
209#106—48no—
210#95—28no—
211#95—28no—
212#95—24no—
213#95—24no—
214#95—44no—
215#95—44no—
216#95—42no—
217#95—42no—
218#95—44no—
219#95—82no—
220#95—81no—
221C24C22×D411yescharacteristic
222C2×Q8—216no—
223C2×Q8—84no—
224C2×Q8—84no—
225C2×Q8—88no—
226C2×Q8—88no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
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—22no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4#12012yes—
285C22×C4#12012yes—
286C22×C4#13512yes—
287C22×C4AGL(1,ℤ/8)12yes—
288C22×C4#13714yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×Q811yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4112yes—
294C22×C4C2×#4112yes—
295C22×C4C2×#4112yes—
296C22×C42−1+411yescharacteristic
297C22×C42−1+412yes—
298C22×C42−1+411yescharacteristic
299C2×C8—22no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—42no—
315C2×C8—42no—
316C4⋊C4—28no—
317C4⋊C4—216no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—44no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—88no—
340C4⋊C4—84no—
341C4⋊C4C22⋊Q814yes—
342C4⋊C4#12318yes—
343C42—24no—
344C42—22no—
345C42—22no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—48no—
351C42—88no—
352C23C2×#12011yescharacteristic
353C23#52512yes—
354C23#54712yes—
355C23#54811yescharacteristic
356C23D4×Q811yescharacteristic
357C23C2×#13511yescharacteristic
358C23C2×AGL(1,ℤ/8)11yescharacteristic
359C23C2×#13712yes—
360C23#57611yescharacteristic
361C23#57712yes—
362C23#57711yescharacteristic
363Q8—816no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—28no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—81no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4#44914yes—
426C2×C4C2×C22⋊Q811yescharacteristic
427C2×C4C2×#12312yes—
428C2×C4#52511yescharacteristic
429C2×C4#52511yescharacteristic
430C2×C4#52912yes—
431C2×C4#53211yescharacteristic
432C2×C4#53312yes—
433C2×C4#53611yescharacteristic
434C2×C4#53611yescharacteristic
435C2×C4#54112yes—
436C2×C4#54112yes—
437C2×C4#54811yescharacteristic
438C2×C4#54812yes—
439C2×C4D4×Q811yescharacteristic
440C2×C4#55212yes—
441C2×C4#55612yes—
442C2×C4#55612yes—
443C2×C4#55712yes—
444C8—42no—
445C8—44no—
446C8—81no—
447C22#240911yescharacteristic
448C22#243211yescharacteristic
449C22#246212yes—
450C22#246912yes—
451C22#284911yescharacteristic
452C22C2×#44911yescharacteristic
453C22#300411yescharacteristic
454C22#300911yescharacteristic
455C22#301211yescharacteristic
456C22#310812yes—
457C22#311212yes—
458C22#311812yes—
459C22#312312yes—
460C22#334012yes—
461C22#334412yes—
462C22#334612yes—
463C22#335412yes—
464C22#336011yescharacteristic
465C22#336611yescharacteristic
466C22#336912yes—
467C22#338011yescharacteristic
468C22#338411yescharacteristic
469C22#338712yes—
470C4—24no—
471C4—44no—
472C4—44no—
473C4—42no—
474C4—48no—
475C4—82no—
476C4—82no—
477C4—82no—
478C4—162no—
479C4—161no—
480C4#239611yescharacteristic
481C4#242112yes—
482C4#244612yes—
483C4#258111yescharacteristic
484C4#260411yescharacteristic
485C4#263411yescharacteristic
486C2#1487611yescharacteristic
487C2#2432711yescharacteristic
488C2#2434511yescharacteristic
489C2#2570011yescharacteristic
490C2#2571211yescharacteristic
491C2#2686312yes—
492C2#2687712yes—
493C2#2713112yes—
494C2#2714712yes—
495C2#3300311yescharacteristic
496C2#3301811yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
70. C2×#107Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
C2×#20582
C2×#19182
#151562
#113842
#113462
#112932
#112202
#109082
#109072
#108062