Finite Group Databasebeta

← #460472

Subgroups — #460472

Order 512.

1230 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#2085C212yesmaximal, complemented
3#14340C211yesmaximal, characteristic
4#11385C211yesmaximal, characteristic, complemented
5#11385C211yesmaximal, characteristic, complemented
6#11324C211yesmaximal, characteristic, complemented
7#11323C211yesmaximal, characteristic, complemented
8#11207C212yesmaximal, complemented
9#11096C212yesmaximal
10#11095C212yesmaximal
11#10713C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×C22⋊C8C2212yes—
17#2629—28no—
18C2×#398—22no—
19C2×#396—22no—
20C2×#389C2212yes—
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#380C2212yes—
25C2×#380C2211yescharacteristic
26C2×#380C2211yescharacteristic
27#2085—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37#1441—22no—
38#1439—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C24×C4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×#125—24no—
45C2×#125—22no—
46C2×#125—22no—
47C2×#125—42no—
48C2×#117—24no—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—42no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4D412yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4D412yes—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×#107—24no—
70C2×#107—42no—
71C2×#107—42no—
72C2×#106—24no—
73C2×#106—42no—
74C2×#106—42no—
75C2×C22⋊C8—42no—
76C2×C22⋊C8—42no—
77C2×C22⋊C8Q818yes—
78#398—216no—
79#396—216no—
80#389—24no—
81#389D418yes—
82#384—24no—
83#384—42no—
84#382—24no—
85#380—24no—
86#380—24no—
87#380—42no—
88#380—42no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96#340—24no—
97#340—24no—
98#340—42no—
99#340—42no—
100#340—42no—
101#340—42no—
102#336—24no—
103#336—22no—
104#336—22no—
105C25C2×D411yescharacteristic
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×Q814yes—
125C23×C4C2411yescharacteristic
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8C2×Q814yes—
139C22×C8C2×Q814yes—
140#125—48no—
141#125—48no—
142#117—48no—
143#117—48no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
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—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4C2×D414yes—
186C2×C4⋊C4C2×D414yes—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4—82no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4—82no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42C2×D414yes—
203C2×C42#4112yes—
204C2×C42#4112yes—
205#107—48no—
206#106—48no—
207C22⋊C8—48no—
208#95—28no—
209#95—28no—
210#95—42no—
211#95—42no—
212#95—44no—
213#95—82no—
214#95—82no—
215C24—21no—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—21no—
220C24—21no—
221C24—22no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24C4⋊Q812yes—
226C24C22×D411yescharacteristic
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
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—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
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—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4#12014yes—
296C22×C4#12314yes—
297C22×C4C4⋊Q812yes—
298C22×C4C22×D412yes—
299C22×C4C22×Q812yes—
300C22×C4C2×#4111yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4H(5,𝔽2)11yescharacteristic
303C22×C4H(5,𝔽2)12yes—
304C22×C4H(5,𝔽2)12yes—
305C22×C42−1+411yescharacteristic
306C22×C42−1+411yescharacteristic
307C22×C42−1+411yescharacteristic
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—48no—
327C4⋊C4—416no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—88no—
341C4⋊C4—88no—
342C4⋊C4—88no—
343C22⋊C4—48no—
344C22⋊C4—88no—
345C22⋊C4—84no—
346C22⋊C4—84no—
347C42—24no—
348C42—24no—
349C42—24no—
350C42—24no—
351C23—24no—
352C23—24no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—41no—
357C23—42no—
358C23—42no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23C2×#12011yescharacteristic
375C23C2×#12311yescharacteristic
376C23C2×C4⋊Q811yescharacteristic
377C23#53411yescharacteristic
378C23#53511yescharacteristic
379C23#53611yescharacteristic
380C23#53711yescharacteristic
381C23#57612yes—
382C23#57712yes—
383C23#57712yes—
384C23#57812yes—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4#53512yes—
445C2×C4#53712yes—
446C2×C4#54111yescharacteristic
447C2×C4#54111yescharacteristic
448C2×C4#54211yescharacteristic
449C2×C4#54211yescharacteristic
450C2×C4#54312yes—
451C2×C4#54412yes—
452C2×C4#54612yes—
453C2×C4#54612yes—
454C2×C4D4×Q812yes—
455C2×C4D4×Q812yes—
456C2×C4#55412yes—
457C2×C4#55412yes—
458C2×C4#56212yes—
459C2×C4#56212yes—
460C8—44no—
461C8—44no—
462C22—42no—
463C22—42no—
464C22—44no—
465C22—44no—
466C22—42no—
467C22—81no—
468C22—81no—
469C22—82no—
470C22—81no—
471C22—81no—
472C22—82no—
473C22#240511yescharacteristic
474C22#243411yescharacteristic
475C22#250211yescharacteristic
476C22#261511yescharacteristic
477C22#262511yescharacteristic
478C22#265911yescharacteristic
479C22#281611yescharacteristic
480C22#301212yes—
481C22#301312yes—
482C22#309812yes—
483C22#309912yes—
484C22#316312yes—
485C22#316412yes—
486C22#319812yes—
487C22#320112yes—
488C22#322011yescharacteristic
489C22#322111yescharacteristic
490C22#322211yescharacteristic
491C22#322311yescharacteristic
492C22#324112yes—
493C22#324212yes—
494C22#325712yes—
495C22#325812yes—
496C4—44no—
497C4—44no—
498C4—42no—
499C4—84no—
500C4—84no—
501C4—161no—
502C4—161no—
503C4—161no—
504C4—161no—
505C4#269212yes—
506C4#269212yes—
507C4#270412yes—
508C4#270412yes—
509C2—82no—
510C2#1636511yescharacteristic
511C2#2401712yes—
512C2#2582212yes—
513C2#2795711yescharacteristic
514C2#2795811yescharacteristic
515C2#2967511yescharacteristic
516C2#2967611yescharacteristic
517C2#2994612yes—
518C2#3029912yes—
519C2#3278211yescharacteristic
520C2#3278411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2085C2semidirect
4. #11385C2semidirect
5. #11385C2semidirect
6. #11324C2semidirect
7. #11323C2semidirect
8. #11207C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#20852
#143402
#113852
#113242
#113232
#112072
#110962
#110952
#107132