Finite Group Databasebeta

← #474852

Subgroups — #474852

Order 512.

1222 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1955C211yesmaximal, characteristic
2C2×#1944C211yesmaximal, characteristic, complemented
3#15181C212yesmaximal
4#11227C212yesmaximal, complemented
5#11200C212yesmaximal, complemented
6#11199C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#11095C212yesmaximal, complemented
9#10821C211yesmaximal, characteristic, complemented
10C22×C4×D4C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#106C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×#97C2211yescharacteristic
17C2×#389C2212yes—
18C2×#380C2212yes—
19C2×#380C2212yes—
20C2×#380C2212yes—
21C2×#380C2212yes—
22#1955—24no—
23#1944—24no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32#1442—22no—
33#1439—22no—
34#1439—22no—
35#1438—22no—
36C4⋊#95C2212yes—
37C23×D4D411yescharacteristic
38C24×C4C2312yes—
39C23×C8C2312yes—
40C23×C8C2311yescharacteristic
41C23×C8C2311yescharacteristic
42C2×C4×D4—22no—
43C2×C4×D4—22no—
44C2×C4×D4—41no—
45C2×C4×D4—41no—
46C2×C4×D4—41no—
47C2×C4×D4—41no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4D412yes—
59C22×C42C2311yescharacteristic
60C2×#107—22no—
61C2×#107—41no—
62C2×#107—41no—
63C2×#106—22no—
64C2×#106—24no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—41no—
67C2×C4⋊C8—41no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8—24no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—42no—
73C2×C22⋊C8—42no—
74C2×#97—41no—
75C2×#97—41no—
76C2×#97D414yes—
77#389—24no—
78#389—24no—
79#380—24no—
80#380—24no—
81#380—42no—
82#380—42no—
83#380—42no—
84#380—42no—
85#380—42no—
86#380—42no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94#340—24no—
95#340—24no—
96#340—42no—
97#340—42no—
98#340—41no—
99#340—41no—
100#340—42no—
101#340—42no—
102C25C2×D412yes—
103C22×D4—24no—
104C22×D4—22no—
105C22×D4—41no—
106C22×D4—41no—
107C22×D4—42no—
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×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4C2×D412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2411yescharacteristic
130C22×C8—22no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8C2×D412yes—
145C22×C8C2×D412yes—
146C22×C8#4114yes—
147C4×D4—44no—
148C4×D4—44no—
149C4×D4—44no—
150C4×D4—44no—
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—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4#4114yes—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—82no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4—82no—
195C2×C22⋊C4—82no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42C2×D412yes—
201C2×C42#4112yes—
202#107—44no—
203#106—28no—
204C4⋊C8—44no—
205C22⋊C8—48no—
206C22⋊C8—48no—
207#97—44no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—82no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—22no—
229C24#13714yes—
230C24C22×D411yescharacteristic
231C2×D4—48no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4—48no—
235C2×D4—44no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—42no—
253C22×C4—42no—
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—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—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4#12112yes—
308C22×C4#12112yes—
309C22×C4C22×D411yescharacteristic
310C22×C4C2×#4112yes—
311C22×C4C2×#4112yes—
312C22×C4C2×#4111yescharacteristic
313C22×C4H(5,𝔽2)11yescharacteristic
314C22×C4H(5,𝔽2)11yescharacteristic
315C22×C4H(5,𝔽2)11yescharacteristic
316C22×C4H(5,𝔽2)11yescharacteristic
317C22×C4H(5,𝔽2)12yes—
318C22×C42−1+412yes—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8#12414yes—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—44no—
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—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—88no—
354C22⋊C4—48no—
355C22⋊C4—88no—
356C22⋊C4—88no—
357C22⋊C4—88no—
358C22⋊C4—88no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C23—24no—
364C23—24no—
365C23—24no—
366C23—24no—
367C23—24no—
368C23—24no—
369C23—24no—
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—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23C2×#12111yescharacteristic
399C23#53411yescharacteristic
400C23#53411yescharacteristic
401C23#54612yes—
402C23#54712yes—
403C23C2×#13712yes—
404C23#57512yes—
405C23#57612yes—
406C23#57712yes—
407D4—416no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—82no—
429C2×C4—82no—
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—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—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4#52511yescharacteristic
462C2×C4C2×#12411yescharacteristic
463C2×C4#53411yescharacteristic
464C2×C4#53511yescharacteristic
465C2×C4#53511yescharacteristic
466C2×C4#53611yescharacteristic
467C2×C4#53811yescharacteristic
468C2×C4#53811yescharacteristic
469C2×C4#54012yes—
470C2×C4#54012yes—
471C2×C4#54212yes—
472C2×C4#54212yes—
473C2×C4#55312yes—
474C2×C4#55312yes—
475C2×C4#55312yes—
476C2×C4#55612yes—
477C2×C4#56212yes—
478C2×C4#56212yes—
479C8—24no—
480C8—42no—
481C8—82no—
482C22—44no—
483C22—44no—
484C22—44no—
485C22—44no—
486C22—44no—
487C22—44no—
488C22—44no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22#253511yescharacteristic
498C22#254711yescharacteristic
499C22#261711yescharacteristic
500C22#270112yes—
501C22#270112yes—
502C22#305612yes—
503C22#306212yes—
504C22#319511yescharacteristic
505C22#319612yes—
506C22#319711yescharacteristic
507C22#319812yes—
508C22#319911yescharacteristic
509C22#320411yescharacteristic
510C22#330912yes—
511C22#331112yes—
512C22#331412yes—
513C22#332012yes—
514C22#334412yes—
515C22#334612yes—
516C22#334712yes—
517C22#334812yes—
518C4—42no—
519C4—42no—
520C4—82no—
521C4—84no—
522C4—162no—
523C4—162no—
524C4—162no—
525C4#259111yescharacteristic
526C4#263211yescharacteristic
527C4#264111yescharacteristic
528C4#280012yes—
529C4#280012yes—
530C4#281611yescharacteristic
531C2—84no—
532C2#1701111yescharacteristic
533C2#2815512yes—
534C2#2871512yes—
535C2#2974311yescharacteristic
536C2#2974511yescharacteristic
537C2#3178712yes—
538C2#3178812yes—
539C2#3179412yes—
540C2#3179712yes—
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×#1944C2semidirect
4. #11227C2semidirect
5. #11200C2semidirect
6. #11199C2semidirect
7. #11096C2semidirect
8. #11095C2semidirect
9. #10821C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19552
C2×#19442
#151812
#112272
#112002
#111992
#110962
#110952
#108212