Finite Group Databasebeta

← #378865

Subgroups — #378865

Order 512.

1256 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2076C211yesmaximal, characteristic
2C2×C4×#107C211yesmaximal, characteristic, complemented
3C2×C8×C22⋊C4C211yesmaximal, characteristic
4#13993C212yesmaximal
5#11377C211yesmaximal, characteristic, complemented
6#11328C211yesmaximal, characteristic, complemented
7#11096C211yesmaximal, characteristic
8#10880C212yesmaximal, complemented
9#10820C211yesmaximal, characteristic, complemented
10#10807C212yesmaximal, complemented
11#10721C212yesmaximal
12C24×C8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#389C2212yes—
19C2×#380C2212yes—
20C2×#380C2212yes—
21C2×C4×C4⋊C4C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23#2076C418yescomplemented
24C4×#107—24no—
25C8×C22⋊C4—24no—
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1443—22no—
37#1439—22no—
38#1437—22no—
39#1436—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C24×C4C2311yescharacteristic
43C23×C8—22no—
44C23×C8—22no—
45C23×C8—22no—
46C23×C8—24no—
47C23×C8—42no—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4D411yescharacteristic
60C22×C22⋊C4D411yescharacteristic
61C22×C42—22no—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×#107—22no—
65C2×#107—22no—
66C2×#107—22no—
67C2×#107—24no—
68C2×#107C2×C414yes—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8C2×C414yes—
72C2×C4×C8—24no—
73C2×C4×C8—22no—
74C2×C4×C8—42no—
75C2×C4×C8D414yes—
76#389—24no—
77#389—42no—
78#380—24no—
79#380—24no—
80#380—42no—
81#380C2×C418yes—
82C4×C4⋊C4—24no—
83C4×C4⋊C4—24no—
84C4×C22⋊C4—22no—
85C4×C22⋊C4—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95D411yescharacteristic
95C2×#95D411yescharacteristic
96#340—24no—
97#340—24no—
98#340—24no—
99#340—22no—
100#340—42no—
101#340—42no—
102#340C2×C414yes—
103#340C2×C418yes—
104#336—22no—
105#336—22no—
106C25C2×D411yescharacteristic
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4C22×C412yes—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4#4112yes—
126C23×C4C2411yescharacteristic
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—28no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—48no—
142C22×C8—48no—
143C22×C8—44no—
144C22×C8—44no—
145C22×C8C22×C412yes—
146C22×C8C22×C412yes—
147C22×C8C2×D412yes—
148C22×C8C2×D412yes—
149C22×C8#4112yes—
150C22×C8#4112yes—
151C22×C8#4114yes—
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—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4C22×C414yes—
177C2×C4⋊C4C22×C414yes—
178C2×C4⋊C4#4114yes—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—82no—
188C2×C22⋊C4—82no—
189C2×C22⋊C4—82no—
190C2×C22⋊C4Q1614yes—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—44no—
198C2×C42—82no—
199C2×C42—82no—
200C2×C42C2×D412yes—
201C2×C42#4112yes—
202#107—28no—
203#107—28no—
204C22⋊C8—28no—
205C4×C8—28no—
206C4×C8—44no—
207#95—28no—
208#95—22no—
209#95—22no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—82no—
215#95—82no—
216#95—82no—
217C24—21no—
218C24—21no—
219C24—22no—
220C24—22no—
221C24—21no—
222C24—21no—
223C24—22no—
224C24—22no—
225C24C4×D412yes—
226C24C2×Q1612yes—
227C24#13512yes—
228C24C22×D411yescharacteristic
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
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—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4C4×D412yes—
294C22×C4#12112yes—
295C22×C4#12112yes—
296C22×C4C2×Q1612yes—
297C22×C4C2×Q1612yes—
298C22×C4#13512yes—
299C22×C4#13512yes—
300C22×C4C23×C411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C22×C4C2×#4112yes—
304C22×C4C2×#4112yes—
305C22×C4C2×#4111yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4H(5,𝔽2)12yes—
309C22×C42−1+412yes—
310C2×C8—24no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—24no—
315C2×C8—22no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—22no—
319C2×C8—24no—
320C2×C8—22no—
321C2×C8—24no—
322C2×C8—416no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8C4×D414yes—
330C2×C8#12314yes—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—42no—
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—88no—
348C22⋊C4—28no—
349C22⋊C4—28no—
350C22⋊C4—88no—
351C22⋊C4—88no—
352C42—22no—
353C42—22no—
354C42—24no—
355C42—41no—
356C42—41no—
357C42—42no—
358C42—88no—
359C23—24no—
360C23—24no—
361C23—24no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—24no—
366C23—24no—
367C23—24no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—41no—
375C23—42no—
376C23—42no—
377C23—41no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23C4×Q1614yes—
382C23C2×C4×D411yescharacteristic
383C23C2×#12111yescharacteristic
384C23#53211yescharacteristic
385C23#54612yes—
386C23#54712yes—
387C23C22×Q1611yescharacteristic
388C23C2×#13511yescharacteristic
389C23C2×#13511yescharacteristic
390C23C2×#13511yescharacteristic
391C23#57612yes—
392C23#57712yes—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
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—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4#46714yes—
452C2×C4C2×C4×D411yescharacteristic
453C2×C4C4×#4111yescharacteristic
454C2×C4C4×#4112yes—
455C2×C4#52012yes—
456C2×C4C2×#12311yescharacteristic
457C2×C4#52511yescharacteristic
458C2×C4#52511yescharacteristic
459C2×C4#52911yescharacteristic
460C2×C4#52912yes—
461C2×C4#53511yescharacteristic
462C2×C4#53611yescharacteristic
463C2×C4#54212yes—
464C2×C4#55112yes—
465C2×C4#55112yes—
466C2×C4#55212yes—
467C2×C4#55212yes—
468C2×C4#55312yes—
469C2×C4#55612yes—
470C8—24no—
471C8—24no—
472C8—42no—
473C8—42no—
474C22—24no—
475C22—28no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—44no—
480C22—42no—
481C22—44no—
482C22—44no—
483C22—42no—
484C22—42no—
485C22C4×#12111yescharacteristic
486C22#235912yes—
487C22#248611yescharacteristic
488C22#252112yes—
489C22#253511yescharacteristic
490C22C2×C4×Q1611yescharacteristic
491C22C4×#13511yescharacteristic
492C22#294612yes—
493C22C2×#46711yescharacteristic
494C22#305011yescharacteristic
495C22#305811yescharacteristic
496C22#306011yescharacteristic
497C22#316811yescharacteristic
498C22#317011yescharacteristic
499C22#317912yes—
500C22#329712yes—
501C22#329712yes—
502C22#329912yes—
503C22#330412yes—
504C22#333712yes—
505C22#333812yes—
506C22#333912yes—
507C22#334012yes—
508C4—24no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—42no—
513C4—84no—
514C4—84no—
515C4—84no—
516C4—162no—
517C4C4×#12311yescharacteristic
518C4#232611yescharacteristic
519C4#237312yes—
520C4#248211yescharacteristic
521C4#251512yes—
522C4#259911yescharacteristic
523C2—44no—
524C2#1422511yescharacteristic
525C2C4×#46711yescharacteristic
526C2#2153611yescharacteristic
527C2#2292212yes—
528C2#2292312yes—
529C2#2765911yescharacteristic
530C2#2766211yescharacteristic
531C2#2806812yes—
532C2#2806912yes—
533C2#2814211yescharacteristic
534C2#2814311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×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×C4×#107C2semidirect
5. #11377C2semidirect
6. #11328C2semidirect
8. #10880C2semidirect
9. #10820C2semidirect
10. #10807C2semidirect
23. #2076C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20762
C2×C4×#1072
C2×C8×C22⋊C42
#139932
#113772
#113282
#110962
#108802
#108202
#108072
#107212