Finite Group Databasebeta

← #450367

Subgroups — #450367

Order 512.

1380 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1854C212yesmaximal, complemented
2C2×#1854C212yesmaximal, complemented
3#13871C212yesmaximal
4#11378C212yesmaximal, complemented
5#11256C211yesmaximal, characteristic, complemented
6#11256C211yesmaximal, characteristic, 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—
17C2×#380C2212yes—
18C2×#380C2212yes—
19C2×#380C2212yes—
20#1854—28no—
21#1854—28no—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31#1440—22no—
32#1438—22no—
33#1437—22no—
34#1436—22no—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C23×D4D411yescharacteristic
38C24×C4C2312yes—
39C23×C8C2312yes—
40C23×C8C2311yescharacteristic
41C23×C8C2311yescharacteristic
42C2×C4×D4—24no—
43C2×C4×D4—24no—
44C2×C4×D4—41no—
45C2×C4×D4—41no—
46C2×#117—24no—
47C2×#117—24no—
48C2×#117—42no—
49C2×#117—42no—
50C2×#117—42no—
51C2×#117—42no—
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4D412yes—
62C22×C42—22no—
63C22×C42C2311yescharacteristic
64C2×#107—24no—
65C2×#107—24no—
66C2×#107—41no—
67C2×#107—41no—
68C2×#106—42no—
69C2×#106—42no—
70C2×#106D418yescomplemented
71C2×C22⋊C8—24no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77#380—24no—
78#380—24no—
79#380—24no—
80#380—24no—
81#380—42no—
82#380—42no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90#340—24no—
91#340—24no—
92#340—22no—
93#340—22no—
94#340—24no—
95#340—24no—
96#340—42no—
97#336—41no—
98#336—41no—
99C25C2×D412yes—
100C22×D4—24no—
101C22×D4—22no—
102C22×D4—41no—
103C22×D4—42no—
104C22×D4—41no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4#4114yes—
125C23×C4#4114yes—
126C23×C4C2411yescharacteristic
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—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8C2×D414yes—
142C22×C8#4112yes—
143C22×C8#4112yes—
144C4×D4—48no—
145C4×D4—44no—
146C4×D4—44no—
147#117—48no—
148#117—48no—
149#117—48no—
150#117—48no—
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—24no—
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—22no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4C2×D414yes—
176C2×C4⋊C4C2×D414yes—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—82no—
189C2×C22⋊C4—82no—
190C2×C22⋊C4—82no—
191C2×C22⋊C4—82no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—44no—
197C2×C42—44no—
198C2×C42—44no—
199#107—44no—
200#107—44no—
201#106—48no—
202C22⋊C8—216no—
203C22⋊C8—216no—
204#95—44no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—82no—
210#95—82no—
211C24—22no—
212C24—22no—
213C24—22no—
214C24—22no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24#12314yes—
226C24C22×D411yescharacteristic
227C2×D4—48no—
228C2×D4—44no—
229C2×D4—44no—
230C2×D4—48no—
231C2×D4—44no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—21no—
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—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×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#12314yes—
304C22×C4#13512yes—
305C22×C4C22×D412yes—
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4112yes—
309C22×C4C2×#4112yes—
310C22×C4H(5,𝔽2)11yescharacteristic
311C22×C4H(5,𝔽2)12yes—
312C22×C4H(5,𝔽2)11yescharacteristic
313C22×C42−1+411yescharacteristic
314C22×C42−1+411yescharacteristic
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—44no—
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—42no—
348C4⋊C4—42no—
349C22⋊C4—48no—
350C22⋊C4—416no—
351C22⋊C4—88no—
352C22⋊C4—88no—
353C22⋊C4—88no—
354C42—416no—
355C42—42no—
356C42—41no—
357C42—41no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23—24no—
362C23—24no—
363C23—24no—
364C23—24no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23C2×#12312yes—
394C23#53411yescharacteristic
395C23D4212yes—
396C23#54711yescharacteristic
397C23#54711yescharacteristic
398C23C2×#13511yescharacteristic
399C23#57512yes—
400C23#57511yescharacteristic
401C23#57611yescharacteristic
402C23#57712yes—
403C23#57811yescharacteristic
404D4—416no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
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—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4#52512yes—
459C2×C4#52512yes—
460C2×C4#53412yes—
461C2×C4#53911yescharacteristic
462C2×C4#53911yescharacteristic
463C2×C4#53912yes—
464C2×C4#53912yes—
465C2×C4D4211yescharacteristic
466C2×C4#54812yes—
467C2×C4#54812yes—
468C2×C4#55112yes—
469C2×C4#55111yescharacteristic
470C2×C4#55111yescharacteristic
471C2×C4#55312yes—
472C2×C4#55312yes—
473C2×C4#55911yescharacteristic
474C2×C4H(3,𝔽4)11yescharacteristic
475C2×C4#56311yescharacteristic
476C8—44no—
477C8—42no—
478C8—42no—
479C22—44no—
480C22—44no—
481C22—44no—
482C22—44no—
483C22—44no—
484C22—44no—
485C22—44no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22#244312yes—
495C22#244312yes—
496C22#248311yescharacteristic
497C22#267911yescharacteristic
498C22#272411yescharacteristic
499C22#309112yes—
500C22#309312yes—
501C22#309612yes—
502C22#309712yes—
503C22#319111yescharacteristic
504C22#319711yescharacteristic
505C22#320311yescharacteristic
506C22#320411yescharacteristic
507C22#326812yes—
508C22#326912yes—
509C22#327512yes—
510C22#328412yes—
511C22#334111yescharacteristic
512C22#334211yescharacteristic
513C22#334812yes—
514C22#335012yes—
515C22#335211yescharacteristic
516C22#335511yescharacteristic
517C4—44no—
518C4—42no—
519C4—84no—
520C4—84no—
521C4—81no—
522C4—162no—
523C4—162no—
524C4#246311yescharacteristic
525C4#246311yescharacteristic
526C4#260012yes—
527C4#272812yes—
528C4#278211yescharacteristic
529C4#278211yescharacteristic
530C2—84no—
531C2#1596611yescharacteristic
532C2#2613512yes—
533C2#2614112yes—
534C2#2615212yes—
535C2#2615412yes—
536C2#2760511yescharacteristic
537C2#2760711yescharacteristic
538C2#3105911yescharacteristic
539C2#3106611yescharacteristic
540C2#3246211yescharacteristic
541C2#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
1. C2×#1854C2semidirect
2. C2×#1854C2semidirect
4. #11378C2semidirect
5. #11256C2semidirect
6. #11256C2semidirect
7. #11200C2semidirect
8. #11199C2semidirect
70. C2×#106D4semidirect

Maximal subgroups

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

subgroupindex
C2×#18542
#138712
#113782
#112562
#112002
#111992
#109642