Finite Group Databasebeta

← #400839

Subgroups — #400839

Order 512.

1162 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2075C211yesmaximal, characteristic
2C2×#1942C211yesmaximal, characteristic, complemented
3C2×#1923C211yesmaximal, characteristic
4C2×C4×#107C211yesmaximal, characteristic, complemented
5#13974C212yesmaximal
6#11379C211yesmaximal, characteristic, complemented
7#11256C211yesmaximal, characteristic, complemented
8#11232C212yesmaximal, complemented
9#11096C211yesmaximal, characteristic
10#11096C212yesmaximal
11#10884C212yesmaximal, complemented
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C2×#389C2212yes—
20C2×#388—22no—
21C2×#380C2212yes—
22C2×#380C2212yes—
23C2×C4×C4⋊C4C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#2075—24no—
26#1942—24no—
27#1923—24no—
28C4×#107—24no—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1443—22no—
39#1440—22no—
40#1436—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C24×C4C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C22×C4⋊C4—22no—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4D411yescharacteristic
56C22×C22⋊C4D411yescharacteristic
57C22×C42—22no—
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×#107—22no—
61C2×#107—24no—
62C2×#107—24no—
63C2×#107—24no—
64C2×#107—24no—
65C2×#107—22no—
66C2×#107—22no—
67C2×#107—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—24no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—24no—
75C2×C4×C8—22no—
76C2×C4×C8—22no—
77C2×C4×C8D414yes—
78#389—24no—
79#389—24no—
80#388—48no—
81#380—24no—
82#380—24no—
83#380—24no—
84#380—42no—
85C4×C4⋊C4—24no—
86C4×C4⋊C4—42no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D411yescharacteristic
98C2×#95D411yescharacteristic
99#340—24no—
100#340—22no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—42no—
105#340D414yes—
106#336—22no—
107#336—22no—
108C25C2×D411yescharacteristic
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2411yescharacteristic
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8C2×D412yes—
145C22×C8C2×D412yes—
146C22×C8C2×D412yes—
147C22×C8#4112yes—
148C22×C8#4114yes—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
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—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4C2×D414yes—
174C2×C4⋊C4#4114yes—
175C2×C4⋊C4#4114yes—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4D814yes—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—44no—
196C2×C42—44no—
197C2×C42—82no—
198C2×C42C2×D412yes—
199C2×C42#4112yes—
200#107—216no—
201#107—28no—
202#107—44no—
203C4⋊C8—44no—
204C22⋊C8—44no—
205C22⋊C8—44no—
206C4×C8—28no—
207#95—28no—
208#95—22no—
209#95—22no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—82no—
217C24—21no—
218C24—22no—
219C24—21no—
220C24—21no—
221C24—22no—
222C24—21no—
223C24—22no—
224C24—21no—
225C24—22no—
226C24—21no—
227C24C2×D812yes—
228C24C22×D411yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—21no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
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—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—42no—
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—42no—
296C22×C4—42no—
297C22×C4#12112yes—
298C22×C4#12112yes—
299C22×C4C2×D812yes—
300C22×C4C2×D812yes—
301C22×C4#13512yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C22×C4C2×#4112yes—
306C22×C4C2×#4112yes—
307C22×C4C2×#4112yes—
308C22×C4C2×#4111yescharacteristic
309C22×C4H(5,𝔽2)11yescharacteristic
310C22×C42−1+412yes—
311C22×C42−1+411yescharacteristic
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8#12114yes—
331C2×C8#12414yes—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—88no—
349C22⋊C4—28no—
350C22⋊C4—44no—
351C22⋊C4—416no—
352C22⋊C4—88no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—41no—
358C42—41no—
359C42—42no—
360C42—88no—
361C23—24no—
362C23—22no—
363C23—24no—
364C23—22no—
365C23—22no—
366C23—41no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—41no—
375C23—41no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23C2×#12111yescharacteristic
385C23#52512yes—
386C23D4211yescharacteristic
387C23#54711yescharacteristic
388C23#54712yes—
389C23C22×D811yescharacteristic
390C23C2×#13511yescharacteristic
391C23#57512yes—
392C23#57611yescharacteristic
393C23#57812yes—
394C23#57811yescharacteristic
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4#46614yes—
456C2×C4C2×#12111yescharacteristic
457C2×C4#52511yescharacteristic
458C2×C4#52512yes—
459C2×C4C2×#12411yescharacteristic
460C2×C4#52912yes—
461C2×C4#52912yes—
462C2×C4#53211yescharacteristic
463C2×C4#53611yescharacteristic
464C2×C4#53711yescharacteristic
465C2×C4#53912yes—
466C2×C4#54112yes—
467C2×C4#54712yes—
468C2×C4#54811yescharacteristic
469C2×C4#55212yes—
470C2×C4#55311yescharacteristic
471C2×C4#55612yes—
472C2×C4#55612yes—
473C2×C4#55912yes—
474C8—24no—
475C8—24no—
476C8—81no—
477C8—81no—
478C22—42no—
479C22—44no—
480C22—42no—
481C22—44no—
482C22—42no—
483C22—81no—
484C22—82no—
485C22—81no—
486C22—82no—
487C22—81no—
488C22—81no—
489C22#239211yescharacteristic
490C22#241611yescharacteristic
491C22#245512yes—
492C22#247812yes—
493C22#254711yescharacteristic
494C22C2×#46611yescharacteristic
495C22#305411yescharacteristic
496C22#305911yescharacteristic
497C22#306111yescharacteristic
498C22#310012yes—
499C22#310212yes—
500C22#312412yes—
501C22#312712yes—
502C22#326912yes—
503C22D4×D811yescharacteristic
504C22#328511yescharacteristic
505C22#333412yes—
506C22#333411yescharacteristic
507C22#333911yescharacteristic
508C22#334212yes—
509C22#334912yes—
510C22#335212yes—
511C22#335512yes—
512C4—24no—
513C4—42no—
514C4—42no—
515C4—81no—
516C4—84no—
517C4—84no—
518C4—84no—
519C4—81no—
520C4—162no—
521C4#241311yescharacteristic
522C4#246912yes—
523C4#259211yescharacteristic
524C4#259612yes—
525C4#266411yescharacteristic
526C4#285011yescharacteristic
527C2—82no—
528C2#1480811yescharacteristic
529C2#2369611yescharacteristic
530C2#2370511yescharacteristic
531C2#2478011yescharacteristic
532C2#2478511yescharacteristic
533C2#2654612yes—
534C2#2656112yes—
535C2#2745412yes—
536C2#2746812yes—
537C2#2870411yescharacteristic
538C2#2870911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1942C2semidirect
4. C2×C4×#107C2semidirect
6. #11379C2semidirect
7. #11256C2semidirect
8. #11232C2semidirect
11. #10884C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20752
C2×#19422
C2×#19232
C2×C4×#1072
#139742
#113792
#112562
#112322
#110962
#108842