Finite Group Databasebeta

← #477704

Subgroups — #477704

Order 512.

1286 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×#2071C212yesmaximal, complemented
2#15134C212yesmaximal, complemented
3#11346C211yesmaximal, characteristic, complemented
4#11328C211yesmaximal, characteristic, complemented
5#11267C212yesmaximal, complemented
6#11123C212yesmaximal, complemented
7#11066C212yesmaximal, complemented
8#11022C212yesmaximal
9#11022C211yesmaximal, characteristic
10C23×D8C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#102C2212yes—
13C22×#102C2212yes—
14C22×#102C2212yes—
15C2×#396—22no—
16C2×#396—22no—
17C2×#390C2212yes—
18C2×#390C2212yes—
19C2×#390C2212yes—
20C2×#389C2212yes—
21C2×#384C2212yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23#2071—28no—
24C2×#340C2212yes—
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33#1439—22no—
34#1439—22no—
35C4⋊#95C2212yes—
36C23×D4C2312yes—
37C22×D8—24no—
38C22×D8—22no—
39C22×D8—41no—
40C22×D8—42no—
41C22×D8—41no—
42C23×C8C2312yes—
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C42C2312yes—
60C22×C42C2311yescharacteristic
61C2×#107—42no—
62C2×#107—41no—
63C2×#102—24no—
64C2×#102—24no—
65C2×#102—24no—
66C2×#102—24no—
67C2×#102—42no—
68C2×#102—42no—
69C2×#102—42no—
70C2×#102—42no—
71C2×#102—42no—
72#396—48no—
73#396—48no—
74#390—24no—
75#390—42no—
76#390—42no—
77#390—42no—
78#390—42no—
79#390—42no—
80#389—42no—
81#389—42no—
82#384—24no—
83#384—42no—
84C4×C4⋊C4—41no—
85C4×C4⋊C4—41no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95D412yes—
90#340—24no—
91#340—42no—
92#340—42no—
93#340—41no—
94#340D414yes—
95#336—24no—
96#336—24no—
97#336—41no—
98#336—41no—
99C25—22no—
100C22×D4—22no—
101C22×D4—22no—
102C22×D4—22no—
103C22×D4—22no—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4—22no—
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4#4114yes—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D412yes—
120C23×C4C2×D412yes—
121C23×C4C2411yescharacteristic
122C2×D8—48no—
123C2×D8—44no—
124C2×D8—44no—
125C2×D8—48no—
126C2×D8—44no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—21no—
133C22×C8—22no—
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×C8C2×D412yes—
143C22×C8#4112yes—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
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—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4D814yescomplemented
175C2×C4⋊C4C2×D414yes—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
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—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—82no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4—82no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204#107—44no—
205#102—48no—
206#102—48no—
207#102—48no—
208#95—42no—
209#95—44no—
210#95—44no—
211#95—44no—
212C24—24no—
213C24—42no—
214C24—42no—
215C24—42no—
216C24—42no—
217C24—42no—
218C24—42no—
219C24—42no—
220C24—42no—
221C24—42no—
222C24—42no—
223C24—42no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24C22×D411yescharacteristic
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
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×C4C2×D814yes—
299C22×C4C2×D812yes—
300C22×C4AGL(1,ℤ/8)14yes—
301C22×C4C22×D411yescharacteristic
302C22×C4C2×#4112yes—
303C22×C4C2×#4111yescharacteristic
304C22×C4H(5,𝔽2)11yescharacteristic
305C22×C4H(5,𝔽2)12yes—
306C22×C4H(5,𝔽2)11yescharacteristic
307C22×C4H(5,𝔽2)11yescharacteristic
308C22×C4H(5,𝔽2)12yes—
309C22×C42−1+411yescharacteristic
310C22×C42−1+412yes—
311D8—416no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—88no—
344C4⋊C4—88no—
345C4⋊C4—88no—
346C22⋊C4—88no—
347C22⋊C4—88no—
348C22⋊C4—88no—
349C22⋊C4—88no—
350C22⋊C4—88no—
351C22⋊C4—88no—
352C42—42no—
353C42—42no—
354C42—42no—
355C42—41no—
356C42—42no—
357C42—41no—
358C42—42no—
359C23—44no—
360C23—44no—
361C23—44no—
362C23—44no—
363C23—44no—
364C23—44no—
365C23—44no—
366C23—82no—
367C23—82no—
368C23—82no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23—82no—
373C23—82no—
374C23—82no—
375C23—82no—
376C23—82no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23—82no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23#53411yescharacteristic
395C23#53512yes—
396C23D4211yescharacteristic
397C23#54711yescharacteristic
398C23#54812yes—
399C23C22×D811yescharacteristic
400C23C2×AGL(1,ℤ/8)12yes—
401C23#57511yescharacteristic
402C23#57611yescharacteristic
403C23#57712yes—
404C23#57811yescharacteristic
405D4—44no—
406D4—44no—
407D4—44no—
408D4—44no—
409D4—82no—
410D4—82no—
411D4—82no—
412D4—82no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—44no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—41no—
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#53711yescharacteristic
459C2×C4#53712yes—
460C2×C4#53812yes—
461C2×C4#53811yescharacteristic
462C2×C4#53912yes—
463C2×C4#53912yes—
464C2×C4#54212yes—
465C2×C4#54612yes—
466C2×C4#55011yescharacteristic
467C2×C4#55111yescharacteristic
468C2×C4#55112yes—
469C2×C4#56011yescharacteristic
470C2×C4#56012yes—
471C2×C4#56011yescharacteristic
472C2×C4#56211yescharacteristic
473C2×C4#56212yes—
474C2×C4#56312yes—
475C2×C4#56311yescharacteristic
476C8—42no—
477C8—42no—
478C8—82no—
479C22—84no—
480C22—84no—
481C22—84no—
482C22—84no—
483C22—84no—
484C22—84no—
485C22—84no—
486C22—162no—
487C22—162no—
488C22—162no—
489C22—162no—
490C22—162no—
491C22—162no—
492C22—162no—
493C22—162no—
494C22#267612yes—
495C22#268211yescharacteristic
496C22#272312yes—
497C22#272811yescharacteristic
498C22#281511yescharacteristic
499C22#318611yescharacteristic
500C22#319211yescharacteristic
501C22#320311yescharacteristic
502C22#320411yescharacteristic
503C22#320512yes—
504C22#321412yes—
505C22#322412yes—
506C22#322512yes—
507C22#326512yes—
508C22#327711yescharacteristic
509C22D4×D811yescharacteristic
510C22#333411yescharacteristic
511C22#334512yes—
512C22#335511yescharacteristic
513C22#335812yes—
514C22#336612yes—
515C22#336712yes—
516C22#337612yes—
517C4—44no—
518C4—44no—
519C4—42no—
520C4—81no—
521C4—84no—
522C4—162no—
523C4—162no—
524C4#269311yescharacteristic
525C4#270512yes—
526C4#273511yescharacteristic
527C4#282311yescharacteristic
528C4#282712yes—
529C4#283011yescharacteristic
530C2—164no—
531C2#1712111yescharacteristic
532C2#3087412yes—
533C2#3090212yes—
534C2#3118811yescharacteristic
535C2#3120411yescharacteristic
536C2#3242612yes—
537C2#3245212yes—
538C2#3254411yescharacteristic
539C2#3255911yescharacteristic
540C2#3275811yescharacteristic
541C2#3276911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2071C2semidirect
2. #15134C2semidirect
3. #11346C2semidirect
4. #11328C2semidirect
5. #11267C2semidirect
6. #11123C2semidirect
7. #11066C2semidirect
174. C2×C4⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#20712
#151342
#113462
#113282
#112672
#111232
#110662
#110222