Finite Group Databasebeta

← #376989

Subgroups — #376989

Order 512.

1436 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#125C211yesmaximal, characteristic, complemented
2C2×#2025C211yesmaximal, characteristic
3C2×#1934C212yesmaximal
4#14700C211yesmaximal, characteristic
5#11385C211yesmaximal, characteristic, complemented
6#11268C211yesmaximal, characteristic, complemented
7#11051C211yesmaximal, characteristic
8#10931C212yesmaximal
9#10885C212yesmaximal, complemented
10#10880C212yesmaximal, complemented
11C4×#336C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×#102C2212yes—
14C22×#102C2212yes—
15C22×C4×C8C2212yes—
16C4×#125—28no—
17C2×#390C2211yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#389C2212yes—
22C2×#388—22no—
23C2×#387—22no—
24C2×#382C2211yescharacteristic
25C2×C4×C4⋊C4—22no—
26C2×C4×C4⋊C4C2212yes—
27C2×C43C2211yescharacteristic
28#2025C418yescomplemented
29#1934—28no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39#1437—22no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C23×D4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#125—24no—
46C2×#125—24no—
47C2×#125—22no—
48C2×#125C2×C414yes—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
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—22no—
60C22×C22⋊C4—21no—
61C22×C42—22no—
62C22×C42D412yes—
63C22×C42D412yes—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×#102—24no—
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—42no—
72C2×#102C2×C418yes—
73C2×C4×C8—24no—
74C2×C4×C8—24no—
75C2×C4×C8—42no—
76#390—24no—
77#390—22no—
78#390—24no—
79#390C2×C414yes—
80#389—24no—
81#389—24no—
82#389D418yes—
83#388—48no—
84#387—216no—
85#382—24no—
86C4×C4⋊C4—216no—
87C4×C4⋊C4—24no—
88C4×C4⋊C4—24no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C43—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—42no—
101#340C2×C418yes—
102#336—22no—
103#336—24no—
104#336C2×C414yes—
105C25—21no—
106C22×D4—22no—
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—21no—
111C22×D4—21no—
112C22×D4—21no—
113C22×D4—21no—
114C22×D4—21no—
115C22×D4C22×C412yes—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
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×C8C22×C414yes—
139C22×C8#4114yes—
140#125—216no—
141#125—216no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—28no—
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—22no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C414yes—
171C2×C4⋊C4C2×D414yes—
172C2×C4⋊C4C2×D414yes—
173C2×C4⋊C4#4114yes—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—28no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—42no—
199C2×C42—44no—
200C2×C42—44no—
201C2×C42C22×C412yes—
202C2×C42C2×D414yes—
203C2×C42#4112yes—
204C2×C42#4114yes—
205#102—216no—
206#102—48no—
207C4×C8—48no—
208#95—28no—
209#95—28no—
210#95—28no—
211#95—44no—
212#95—42no—
213#95—44no—
214C24—22no—
215C24—41no—
216C24—41no—
217C24—41no—
218C24—41no—
219C24—41no—
220C24—41no—
221C24—42no—
222C24—42no—
223C24—42no—
224C24—42no—
225C24C22×D411yescharacteristic
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4C4×D414yes—
289C22×C4#12014yes—
290C22×C4#12412yes—
291C22×C4#12412yes—
292C22×C4#13514yes—
293C22×C4#13514yes—
294C22×C4AGL(1,ℤ/8)14yes—
295C22×C4C23×C411yescharacteristic
296C22×C4C22×D412yes—
297C22×C4C2×#4112yes—
298C22×C4C2×#4112yes—
299C22×C4C2×#4112yes—
300C22×C4C2×#4111yescharacteristic
301C22×C4H(5,𝔽2)11yescharacteristic
302C22×C42−1+411yescharacteristic
303C22×C42−1+411yescharacteristic
304C22×C42−1+411yescharacteristic
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C4⋊C4—232no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—416no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—84no—
335C4⋊C4C4×D418yes—
336C22⋊C4—416no—
337C22⋊C4—84no—
338C22⋊C4—88no—
339C22⋊C4—84no—
340C42—24no—
341C42—24no—
342C42—24no—
343C42—24no—
344C42—24no—
345C42—24no—
346C42—48no—
347C42—416no—
348C42—48no—
349C23—42no—
350C23—44no—
351C23—44no—
352C23—42no—
353C23—42no—
354C23—82no—
355C23—81no—
356C23—82no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—82no—
363C23—82no—
364C23—81no—
365C23—81no—
366C23—82no—
367C23—82no—
368C23—82no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23C2×C4×D411yescharacteristic
373C23C2×#12011yescharacteristic
374C23#52511yescharacteristic
375C23C2×#12411yescharacteristic
376C23#53211yescharacteristic
377C23#53611yescharacteristic
378C23#53711yescharacteristic
379C23C2×#13512yes—
380C23C2×#13512yes—
381C23C2×AGL(1,ℤ/8)12yes—
382C23#57512yes—
383D4—44no—
384D4—44no—
385D4—82no—
386D4—82no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—28no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—81no—
429C2×C4—82no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4#44918yes—
439C2×C4#48814yes—
440C2×C4C2×C4×D412yes—
441C2×C4C4×#4112yes—
442C2×C4#51911yescharacteristic
443C2×C4#52011yescharacteristic
444C2×C4#52512yes—
445C2×C4#52912yes—
446C2×C4#54111yescharacteristic
447C2×C4#54211yescharacteristic
448C2×C4#54612yes—
449C2×C4#54712yes—
450C2×C4#54812yes—
451C2×C4#54812yes—
452C2×C4#55212yes—
453C2×C4#55612yes—
454C2×C4#55612yes—
455C2×C4#55912yes—
456C8—44no—
457C8—44no—
458C22—84no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—84no—
463C22—161no—
464C22—161no—
465C22—162no—
466C22—162no—
467C22—161no—
468C22—161no—
469C22C4×#12011yescharacteristic
470C22C4×#12411yescharacteristic
471C22#232311yescharacteristic
472C22#243011yescharacteristic
473C22#243611yescharacteristic
474C22#259211yescharacteristic
475C22#260411yescharacteristic
476C22C4×#13512yes—
477C22C4×AGL(1,ℤ/8)12yes—
478C22C2×#44912yes—
479C22#300212yes—
480C22#310212yes—
481C22#310612yes—
482C22C2×#48811yescharacteristic
483C22#313111yescharacteristic
484C22#313811yescharacteristic
485C22#314211yescharacteristic
486C22#316812yes—
487C22#318012yes—
488C22#322812yes—
489C22#323212yes—
490C22#324712yes—
491C22#325012yes—
492C4—24no—
493C4—28no—
494C4—42no—
495C4—44no—
496C4—48no—
497C4—82no—
498C4—84no—
499C4—161no—
500C4—161no—
501C4#236212yes—
502C4#236712yes—
503C4#245812yes—
504C4#246912yes—
505C2—162no—
506C2#1418811yescharacteristic
507C2C4×#44912yes—
508C2C4×#48811yescharacteristic
509C2#2165611yescharacteristic
510C2#2211412yes—
511C2#2550012yes—
512C2#2590511yescharacteristic
513C2#2590711yescharacteristic
514C2#2934811yescharacteristic
515C2#2935111yescharacteristic
516C2#2951612yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C4×#125C2semidirect
5. #11385C2semidirect
6. #11268C2semidirect
9. #10885C2semidirect
10. #10880C2semidirect
11. C4×#336C2semidirect
28. #2025C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1252
C2×#20252
C2×#19342
#147002
#113852
#112682
#110512
#109312
#108852
#108802
C4×#3362