Finite Group Databasebeta

← #460499

Subgroups — #460499

Order 512.

1316 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C2×#2101C212yesmaximal, complemented
3#14682C211yesmaximal, characteristic
4#11385C211yesmaximal, characteristic, complemented
5#11385C211yesmaximal, characteristic, complemented
6#11268C211yesmaximal, characteristic, complemented
7#11268C211yesmaximal, characteristic, complemented
8#11208C212yesmaximal, complemented
9#11200C212yesmaximal
10#11199C212yesmaximal
11#10712C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15#2851—28no—
16C2×#389C2212yes—
17C2×#387—22no—
18C2×#385—22no—
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#380C2212yes—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25#2101—28no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39#1434—22no—
40C22⋊#95—22no—
41C24×C4D412yes—
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#125—24no—
46C2×#125—22no—
47C2×#125—22no—
48C2×#125—42no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C42—24no—
61C22×C42—22no—
62C22×C42—22no—
63C22×C42—42no—
64C22×C42D412yes—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×C22⋊C8—24no—
68C2×C22⋊C8—42no—
69C2×C22⋊C8—42no—
70#389—24no—
71#389D418yes—
72#387—216no—
73#385—216no—
74#382—24no—
75#382—24no—
76#382—42no—
77#380—24no—
78#380—24no—
79#380—24no—
80#380—24no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—21no—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—42no—
95#340—42no—
96#340—42no—
97#340—42no—
98#336—24no—
99#336—22no—
100#336—22no—
101C25C2×D411yescharacteristic
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—24no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—24no—
115C23×C4—24no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4#4114yes—
128C23×C4C2411yescharacteristic
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×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143#125—48no—
144#125—48no—
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—22no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—28no—
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—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
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—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4C2×D414yes—
178C2×C4⋊C4C2×D414yes—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—44no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—48no—
194C2×C42—48no—
195C2×C42—42no—
196C2×C42C2×D414yes—
197C2×C42#4112yes—
198C2×C42#4112yes—
199C22⋊C8—48no—
200#95—28no—
201#95—28no—
202#95—28no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—44no—
207#95—42no—
208#95—42no—
209C24—21no—
210C24—21no—
211C24—21no—
212C24—21no—
213C24—21no—
214C24—21no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24#12412yes—
219C24#13514yes—
220C24C22×D411yescharacteristic
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—48no—
248C22×C4—48no—
249C22×C4—48no—
250C22×C4—48no—
251C22×C4—48no—
252C22×C4—48no—
253C22×C4—48no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
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—42no—
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—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4#12014yes—
297C22×C4#12314yes—
298C22×C4#12412yes—
299C22×C4C22×D412yes—
300C22×C4C2×#4112yes—
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C22×C4H(5,𝔽2)11yescharacteristic
304C22×C4H(5,𝔽2)12yes—
305C22×C42−1+411yescharacteristic
306C22×C42−1+412yes—
307C22×C42−1+411yescharacteristic
308C22×C42−1+411yescharacteristic
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—416no—
328C4⋊C4—416no—
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—42no—
341C22⋊C4—416no—
342C22⋊C4—84no—
343C22⋊C4—84no—
344C42—24no—
345C42—24no—
346C42—24no—
347C42—24no—
348C42—48no—
349C23—24no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—24no—
355C23—22no—
356C23—24no—
357C23—24no—
358C23—24no—
359C23—24no—
360C23—22no—
361C23—41no—
362C23—42no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23#48814yes—
372C23C2×#12011yescharacteristic
373C23C2×#12311yescharacteristic
374C23C2×#12411yescharacteristic
375C23#53411yescharacteristic
376C23#53611yescharacteristic
377C23#53611yescharacteristic
378C23#53711yescharacteristic
379C23C2×#13512yes—
380C23#57512yes—
381C23#57512yes—
382C23#57512yes—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—432no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
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—82no—
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—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4#53612yes—
444C2×C4#53712yes—
445C2×C4#53812yes—
446C2×C4#53912yes—
447C2×C4#54111yescharacteristic
448C2×C4#54111yescharacteristic
449C2×C4#54211yescharacteristic
450C2×C4#54211yescharacteristic
451C2×C4#54612yes—
452C2×C4#54612yes—
453C2×C4#54812yes—
454C2×C4#54812yes—
455C2×C4#55212yes—
456C2×C4#55212yes—
457C2×C4#56312yes—
458C2×C4#56312yes—
459C8—82no—
460C8—82no—
461C22—24no—
462C22—24no—
463C22—24no—
464C22—44no—
465C22—42no—
466C22—42no—
467C22—44no—
468C22—42no—
469C22—44no—
470C22—42no—
471C22—44no—
472C22#240511yescharacteristic
473C22#243611yescharacteristic
474C22#250311yescharacteristic
475C22#261611yescharacteristic
476C22#262511yescharacteristic
477C22#263911yescharacteristic
478C22#281711yescharacteristic
479C22#300212yes—
480C22#300512yes—
481C22#309112yes—
482C22#309312yes—
483C22C2×#48811yescharacteristic
484C22#313511yescharacteristic
485C22#314011yescharacteristic
486C22#314211yescharacteristic
487C22#319112yes—
488C22#319712yes—
489C22#323212yes—
490C22#323212yes—
491C22#324012yes—
492C22#324012yes—
493C22#325012yes—
494C22#325612yes—
495C4—44no—
496C4—44no—
497C4—42no—
498C4—84no—
499C4—84no—
500C4—161no—
501C4—161no—
502C4—161no—
503C4—161no—
504C4#269012yes—
505C4#269012yes—
506C4#270212yes—
507C4#270212yes—
508C2—44no—
509C2#1636611yescharacteristic
510C2#2400812yes—
511C2#2590711yescharacteristic
512C2#2590911yescharacteristic
513C2#2796411yescharacteristic
514C2#2796711yescharacteristic
515C2#2969312yes—
516C2#2994012yes—
517C2#3018611yescharacteristic
518C2#3019811yescharacteristic
519C2#3279112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2101C2semidirect
4. #11385C2semidirect
5. #11385C2semidirect
6. #11268C2semidirect
7. #11268C2semidirect
8. #11208C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#21012
#146822
#113852
#112682
#112082
#112002
#111992
#107122