Finite Group Databasebeta

← #376988

Subgroups — #376988

Order 512.

1310 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#125C211yesmaximal, characteristic
2C2×#2026C211yesmaximal, characteristic
3C2×#1933C212yesmaximal
4#14703C211yesmaximal, characteristic
5#11385C211yesmaximal, characteristic
6#11268C211yesmaximal, characteristic
7#11052C211yesmaximal, characteristic
8#10932C212yesmaximal
9#10884C212yesmaximal
10#10880C212yesmaximal
11C4×#336C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×C4×C8C2212yes—
16C4×#125—28no—
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#389C2212yes—
22C2×#387—22no—
23C2×#382C2211yescharacteristic
24C2×C4×C4⋊C4—22no—
25C2×C4×C4⋊C4C2212yes—
26C2×C43C2211yescharacteristic
27#2026C418yescomplemented
28#1933—28no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38#1436—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41#1434—22no—
42C23×Q8C2311yescharacteristic
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×C42—22no—
58C22×C42D412yes—
59C22×C42D412yes—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×#103—24no—
64C2×#103—24no—
65C2×#103—24no—
66C2×#103—24no—
67C2×#103—42no—
68C2×#103C2×C418yes—
69C2×C4×C8—24no—
70C2×C4×C8—24no—
71C2×C4×C8—42no—
72#391—22no—
73#391—24no—
74#391—24no—
75#391C2×C414yes—
76#389—24no—
77#389—24no—
78#389D418yes—
79#387—216no—
80#382—24no—
81C4×C4⋊C4—216no—
82C4×C4⋊C4—24no—
83C4×C4⋊C4—24no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—21no—
94C43—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—42no—
100#340C2×C418yes—
101#336—22no—
102#336—24no—
103#336C2×C414yes—
104C22×Q8—21no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8C22×C412yes—
114C23×C4—22no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2411yescharacteristic
125C22×C8—22no—
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×C8C22×C414yes—
138C22×C8#4114yes—
139#125—216no—
140#125—216no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—28no—
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—22no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C414yes—
170C2×C4⋊C4C2×D414yes—
171C2×C4⋊C4C2×D414yes—
172C2×C4⋊C4#4114yes—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—28no—
182C2×C42—24no—
183C2×C42—24no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—42no—
188C2×C42—44no—
189C2×C42—44no—
190C2×C42C22×C412yes—
191C2×C42C2×D414yes—
192C2×C42#4112yes—
193C2×C42#4114yes—
194#103—216no—
195#103—48no—
196C4×C8—48no—
197#95—28no—
198#95—28no—
199#95—28no—
200#95—28no—
201#95—44no—
202#95—44no—
203#95—42no—
204#95—42no—
205#95—42no—
206#95—44no—
207C24C22×D411yescharacteristic
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4C4×D414yes—
278C22×C4#12014yes—
279C22×C4#12412yes—
280C22×C4#12412yes—
281C22×C4#13514yes—
282C22×C4#13514yes—
283C22×C4#13714yes—
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D412yes—
286C22×C4C2×#4112yes—
287C22×C4C2×#4112yes—
288C22×C4C2×#4112yes—
289C22×C4C2×#4111yescharacteristic
290C22×C4H(5,𝔽2)11yescharacteristic
291C22×C42−1+411yescharacteristic
292C22×C42−1+411yescharacteristic
293C22×C42−1+411yescharacteristic
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C4⋊C4—232no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—416no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—84no—
324C4⋊C4C4×D418yes—
325C42—24no—
326C42—24no—
327C42—24no—
328C42—24no—
329C42—24no—
330C42—24no—
331C42—48no—
332C42—416no—
333C42—48no—
334C23C2×C4×D411yescharacteristic
335C23C2×#12011yescharacteristic
336C23#52511yescharacteristic
337C23C2×#12411yescharacteristic
338C23#53211yescharacteristic
339C23#53611yescharacteristic
340C23#53711yescharacteristic
341C23C2×#13512yes—
342C23C2×#13512yes—
343C23C2×#13712yes—
344C23#57512yes—
345Q8—44no—
346Q8—44no—
347Q8—82no—
348Q8—82no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—28no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4#45218yes—
408C2×C4#48814yes—
409C2×C4C2×C4×D412yes—
410C2×C4C4×#4112yes—
411C2×C4#51911yescharacteristic
412C2×C4#52011yescharacteristic
413C2×C4#52512yes—
414C2×C4#52912yes—
415C2×C4#54111yescharacteristic
416C2×C4#54211yescharacteristic
417C2×C4#54612yes—
418C2×C4#54712yes—
419C2×C4#54812yes—
420C2×C4#54812yes—
421C2×C4#55212yes—
422C2×C4#55612yes—
423C2×C4#55612yes—
424C2×C4#55912yes—
425C8—44no—
426C8—44no—
427C22C4×#12011yescharacteristic
428C22C4×#12411yescharacteristic
429C22#232311yescharacteristic
430C22#243011yescharacteristic
431C22#243611yescharacteristic
432C22#259211yescharacteristic
433C22#260411yescharacteristic
434C22C4×#13512yes—
435C22C4×#13712yes—
436C22C2×#45212yes—
437C22#300212yes—
438C22#310212yes—
439C22#310712yes—
440C22C2×#48811yescharacteristic
441C22#313111yescharacteristic
442C22#313911yescharacteristic
443C22#314211yescharacteristic
444C22#316812yes—
445C22#318112yes—
446C22#323112yes—
447C22#323212yes—
448C22#324812yes—
449C22#325012yes—
450C4—24no—
451C4—28no—
452C4—42no—
453C4—44no—
454C4—48no—
455C4—82no—
456C4—82no—
457C4—84no—
458C4—161no—
459C4—161no—
460C4#236212yes—
461C4#236712yes—
462C4#245812yes—
463C4#246912yes—
464C2#1418811yescharacteristic
465C2C4×#45212yes—
466C2C4×#48811yescharacteristic
467C2#2165711yescharacteristic
468C2#2211512yes—
469C2#2549912yes—
470C2#2590411yescharacteristic
471C2#2590711yescharacteristic
472C2#2934911yescharacteristic
473C2#2935111yescharacteristic
474C2#2951712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
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
27. #2026C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1252
C2×#20262
C2×#19332
#147032
#113852
#112682
#110522
#109322
#108842
#108802
C4×#3362