Finite Group Databasebeta

← #228522

Subgroups — #228522

Order 512.

603 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758C211yesmaximal, characteristic
2#22990C211yesmaximal, characteristic
3#22640C211yesmaximal, characteristic, complemented
4#9599C211yesmaximal, characteristic, complemented
5#9558C211yesmaximal, characteristic, complemented
6#9546C211yesmaximal, characteristic, complemented
7#9535C211yesmaximal, characteristic, complemented
8#9533C211yesmaximal, characteristic, complemented
9#9074C211yesmaximal, characteristic
10#8849C211yesmaximal, characteristic, complemented
11#8833C211yesmaximal, characteristic, complemented
12#8290C211yesmaximal, characteristic, complemented
13#8212C211yesmaximal, characteristic, complemented
14#8149C211yesmaximal, characteristic, complemented
15#7899C211yesmaximal, characteristic, complemented
16C2×#436—21no—
17C2×#436C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C4×#121C2211yescharacteristic
21#2283C2211yescharacteristic
22#2101C411yescharacteristic
23#2092C411yescharacteristic
24#2085C411yescharacteristic
25#2077C411yescharacteristic
26#2076C411yescharacteristic
27#2073C411yescharacteristic
28#2034C411yescharacteristic
29#2033C411yescharacteristic
30#1954C2211yescharacteristic
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1924—21no—
35#1923—21no—
36#1923—21no—
37#1870—21no—
38#1851—21no—
39#1840—21no—
40#1797C2211yescharacteristic
41C4×#107C2211yescharacteristic
42C4×#106C2211yescharacteristic
43#1770C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46#1768—21no—
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#334C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16—21no—
68C22×M16—21no—
69C2×#121C2×C411yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4C2311yescharacteristic
72C22×C42C2311yescharacteristic
73#436—22no—
74#436—41no—
75#436—44no—
76#431—22no—
77#431—41no—
78C2×#107C2×C411yescharacteristic
79C2×#107C2×C411yescharacteristic
80C2×#106C2×C411yescharacteristic
81C2×#106C2×C411yescharacteristic
82#423—22no—
83#423—41no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8C2×C411yescharacteristic
103C2×C22⋊C8C2×C411yescharacteristic
104C2×C22⋊C8D411yescharacteristic
105C2×C22⋊C8D411yescharacteristic
106C2×#97—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#390C2×C411yescharacteristic
116#389C2×C411yescharacteristic
117#388—21no—
118#386—21no—
119#383—21no—
120#382C2×C411yescharacteristic
121#382C2×C411yescharacteristic
122#381—21no—
123#380C2×C411yescharacteristic
124#380C2×C411yescharacteristic
125#380C2×C411yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131#376—21no—
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336C2×C411yescharacteristic
148#336C2×C411yescharacteristic
149#336D411yescharacteristic
150#336D411yescharacteristic
151#334—22no—
152#334—41no—
153D4⋊C8—22no—
154D4⋊C8—41no—
155#322—22no—
156#322—41no—
157C22×D4—21no—
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4#4111yescharacteristic
163C2×M16—42no—
164C2×M16—42no—
165C2×M16—42no—
166C2×M16—42no—
167C2×M16—81no—
168C2×M16—81no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183#121—22no—
184#121—41no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—42no—
190C4×D4—42no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42#4111yescharacteristic
216C2×C42C2411yescharacteristic
217#107—22no—
218#107—41no—
219#106—22no—
220#106—41no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—42no—
229C4⋊C8—42no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232C4⋊C8—42no—
233C4⋊C8#4112yes—
234#102—41no—
235#102—41no—
236#102—41no—
237#102—41no—
238C22⋊C8—22no—
239C22⋊C8—41no—
240C22⋊C8—41no—
241C22⋊C8—41no—
242C22⋊C8—42no—
243C22⋊C8—42no—
244C22⋊C8—42no—
245C22⋊C8—81no—
246C22⋊C8—81no—
247#97—81no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—42no—
260C4×C8—81no—
261C4×C8#4112yes—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273C24—21no—
274C24C4×D411yescharacteristic
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—81no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C23×C411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C2×#4111yescharacteristic
319M16—84no—
320M16—84no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—22no—
330C2×C8—21no—
331C2×C8—22no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8AGL(1,ℤ/8)12yes—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—42no—
377C4⋊C4#13112yes—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—81no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42C2×#4111yescharacteristic
399C42C2×#4111yescharacteristic
400C42H(5,𝔽2)11yescharacteristic
401C422−1+411yescharacteristic
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23C2×C4×D411yescharacteristic
417D4—41no—
418D4—41no—
419D4—41no—
420D4—41no—
421D4—84no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4C4×#4111yescharacteristic
469C2×C4#52011yescharacteristic
470C2×C4#54611yescharacteristic
471C2×C4#54711yescharacteristic
472C2×C4#55111yescharacteristic
473C2×C4#55211yescharacteristic
474C2×C4C2×#13111yescharacteristic
475C2×C4#56811yescharacteristic
476C2×C4#56811yescharacteristic
477C2×C4#56811yescharacteristic
478C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
479C2×C4#57511yescharacteristic
480C2×C4#57711yescharacteristic
481C2×C4#57711yescharacteristic
482C8—22no—
483C8—42no—
484C8—41no—
485C8—41no—
486C8—41no—
487C8—41no—
488C8—41no—
489C8—41no—
490C8—42no—
491C8—41no—
492C8—81no—
493C8—81no—
494C8—81no—
495C8—81no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22#235911yescharacteristic
511C22#293211yescharacteristic
512C22#293311yescharacteristic
513C22C4×AGL(1,ℤ/8)11yescharacteristic
514C22#295211yescharacteristic
515C22#295511yescharacteristic
516C22#295711yescharacteristic
517C4—21no—
518C4—21no—
519C4—42no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—81no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4#296711yescharacteristic
530C4#296911yescharacteristic
531C4#299311yescharacteristic
532C4#299511yescharacteristic
533C4#332411yescharacteristic
534C4#332611yescharacteristic
535C4#334511yescharacteristic
536C4#335011yescharacteristic
537C2—81no—
538C2—161no—
539C2#1798311yescharacteristic
540C2#1799311yescharacteristic
541C2#1800611yescharacteristic
542C2#1803111yescharacteristic
543C2#2290711yescharacteristic
544C2#2294411yescharacteristic
545C2#2294911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22640C2semidirect
4. #9599C2semidirect
5. #9558C2semidirect
6. #9546C2semidirect
7. #9535C2semidirect
8. #9533C2semidirect
10. #8849C2semidirect
11. #8833C2semidirect
12. #8290C2semidirect
13. #8212C2semidirect
14. #8149C2semidirect
15. #7899C2semidirect

Maximal subgroups

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

subgroupindex
#237582
#229902
#226402
#95992
#95582
#95462
#95352
#95332
#90742
#88492
#88332
#82902
#82122
#81492
#78992