Finite Group Databasebeta

← #449611

Subgroups — #449611

Order 512.

1620 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2485C212yesmaximal
2C2×#1854C212yesmaximal, complemented
3C2×#1854C212yesmaximal, complemented
4#11328C211yesmaximal, characteristic, complemented
5#11328C211yesmaximal, characteristic, complemented
6#11327C212yesmaximal, complemented
7#11200C212yesmaximal, complemented
8#11199C212yesmaximal, complemented
9#10964C211yesmaximal, characteristic
10C22×C4×D4C2211yescharacteristic
11C22×#117C2212yes—
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17#2485—28no—
18#2485—28no—
19C2×#397—22no—
20C2×#394—22no—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#388—22no—
24C2×#387—22no—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28#1854—28no—
29#1854—28no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C23×D4D411yescharacteristic
40C24×C4C2312yes—
41C23×C8C2312yes—
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C2×C4×D4—24no—
45C2×C4×D4—24no—
46C2×C4×D4—41no—
47C2×C4×D4—41no—
48C2×#117—24no—
49C2×#117—24no—
50C2×#117—24no—
51C2×#117—24no—
52C2×#117—42no—
53C2×#117—42no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4D412yes—
66C22×C42—22no—
67C22×C42C2311yescharacteristic
68C2×#107—24no—
69C2×#107—24no—
70C2×#107—41no—
71C2×#107—41no—
72C2×#106—24no—
73C2×#106—24no—
74C2×#106—42no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81#397—216no—
82#394—216no—
83#389—24no—
84#389—24no—
85#389—24no—
86#389—24no—
87#388—216no—
88#387—216no—
89#380—24no—
90#380—24no—
91#380—24no—
92#380—24no—
93#380D418yes—
94#380D418yes—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100#340—24no—
101#340—22no—
102#340—22no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—24no—
107#336—41no—
108#336—41no—
109C25C2×D412yes—
110C22×D4—24no—
111C22×D4—22no—
112C22×D4—41no—
113C22×D4—42no—
114C22×D4—41no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D414yes—
134C23×C4C2×D414yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8#4114yes—
152C22×C8#4112yes—
153C22×C8#4112yes—
154C4×D4—48no—
155C4×D4—44no—
156C4×D4—44no—
157#117—48no—
158#117—48no—
159#117—48no—
160#117—48no—
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—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4C2×D414yes—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2×D414yes—
192C2×C4⋊C4C2×D414yes—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—82no—
205C2×C22⋊C4—82no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4—82no—
208C2×C42—28no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—44no—
214C2×C42—44no—
215#107—44no—
216#107—44no—
217#106—48no—
218C22⋊C8—216no—
219C22⋊C8—216no—
220#95—28no—
221#95—28no—
222#95—28no—
223#95—28no—
224#95—28no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—22no—
229C24—22no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24#12014yes—
240C24C22×D411yescharacteristic
241C2×D4—48no—
242C2×D4—44no—
243C2×D4—44no—
244C2×D4—48no—
245C2×D4—44no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—21no—
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×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4#12014yes—
318C22×C4#13512yes—
319C22×C4C22×D412yes—
320C22×C4C22×D412yes—
321C22×C4C2×#4112yes—
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4H(5,𝔽2)11yescharacteristic
325C22×C4H(5,𝔽2)11yescharacteristic
326C22×C42−1+412yes—
327C22×C42−1+411yescharacteristic
328C22×C42−1+411yescharacteristic
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—41no—
345C2×C8—41no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—44no—
355C4⋊C4—416no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—88no—
365C22⋊C4—48no—
366C22⋊C4—416no—
367C22⋊C4—416no—
368C22⋊C4—88no—
369C22⋊C4—88no—
370C42—416no—
371C42—42no—
372C42—41no—
373C42—41no—
374C23—24no—
375C23—24no—
376C23—24no—
377C23—24no—
378C23—24no—
379C23—24no—
380C23—24no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23C2×#12012yes—
410C23#53411yescharacteristic
411C23#54712yes—
412C23#54711yescharacteristic
413C23#54711yescharacteristic
414C23C2×#13511yescharacteristic
415C23#57512yes—
416C23#57511yescharacteristic
417C23#57611yescharacteristic
418C23#57712yes—
419C23#57811yescharacteristic
420D4—416no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—41no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4#52512yes—
475C2×C4#52512yes—
476C2×C4#53612yes—
477C2×C4#53612yes—
478C2×C4#53912yes—
479C2×C4#53911yescharacteristic
480C2×C4#53911yescharacteristic
481C2×C4D4212yes—
482C2×C4D4211yescharacteristic
483C2×C4D4211yescharacteristic
484C2×C4#54712yes—
485C2×C4#54712yes—
486C2×C4#54712yes—
487C2×C4#54712yes—
488C2×C4#55911yescharacteristic
489C2×C4#55911yescharacteristic
490C2×C4#56311yescharacteristic
491C2×C4#56311yescharacteristic
492C8—44no—
493C8—42no—
494C8—42no—
495C22—44no—
496C22—44no—
497C22—44no—
498C22—44no—
499C22—44no—
500C22—44no—
501C22—44no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22#240311yescharacteristic
511C22#240612yes—
512C22#240612yes—
513C22#272411yescharacteristic
514C22#272411yescharacteristic
515C22#300212yes—
516C22#300512yes—
517C22#301212yes—
518C22#301312yes—
519C22#319111yescharacteristic
520C22#319711yescharacteristic
521C22#320311yescharacteristic
522C22#320411yescharacteristic
523C22#334012yes—
524C22#334111yescharacteristic
525C22#334211yescharacteristic
526C22#334712yes—
527C22#334812yes—
528C22#334912yes—
529C22#335012yes—
530C22#335112yes—
531C22#335211yescharacteristic
532C22#335511yescharacteristic
533C4—44no—
534C4—42no—
535C4—84no—
536C4—84no—
537C4—84no—
538C4—81no—
539C4—84no—
540C4#245512yes—
541C4#245511yescharacteristic
542C4#245511yescharacteristic
543C4#268011yescharacteristic
544C4#268012yes—
545C4#268011yescharacteristic
546C2—84no—
547C2#1594911yescharacteristic
548C2#2392311yescharacteristic
549C2#2392911yescharacteristic
550C2#2409312yes—
551C2#2409612yes—
552C2#2410312yes—
553C2#2410612yes—
554C2#3246111yescharacteristic
555C2#3246211yescharacteristic
556C2#3247011yescharacteristic
557C2#3247111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1854C2semidirect
3. C2×#1854C2semidirect
4. #11328C2semidirect
5. #11328C2semidirect
6. #11327C2semidirect
7. #11200C2semidirect
8. #11199C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24852
C2×#18542
#113282
#113272
#112002
#111992
#109642