Finite Group Databasebeta

← #230486

Subgroups — #230486

Order 512.

627 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23253C211yesmaximal, characteristic, complemented
2#22795C211yesmaximal, characteristic
3C4×#478C211yesmaximal, characteristic
4#9872C211yesmaximal, characteristic, complemented
5#9602C211yesmaximal, characteristic, complemented
6#9556C211yesmaximal, characteristic
7#9554C211yesmaximal, characteristic
8#9542C211yesmaximal, characteristic
9#9542C211yesmaximal, characteristic
10#9534C211yesmaximal, characteristic, complemented
11#9533C211yesmaximal, characteristic, complemented
12#8284C211yesmaximal, characteristic, complemented
13C8×#106C211yesmaximal, characteristic, complemented
14#8264C211yesmaximal, characteristic, complemented
15#8170C211yesmaximal, characteristic
16C2×#478C411yescharacteristic
17C2×#431—21no—
18C4×#125C2211yescharacteristic
19C42×D4C2211yescharacteristic
20#1989C411yescharacteristic
21#1954C2211yescharacteristic
22#1954C2211yescharacteristic
23#1949C411yescharacteristic
24#1939—21no—
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1923—21no—
28#1923—21no—
29#1921C411yescharacteristic
30#1919C411yescharacteristic
31#1916—21no—
32#1916—21no—
33#1864C411yescharacteristic
34#1862C411yescharacteristic
35#1842C2211yescharacteristic
36#1804C411yescharacteristic
37#1797—21no—
38C4×#107C2211yescharacteristic
39C4×#106C2211yescharacteristic
40C8×C4⋊C4—21no—
41C8×C4⋊C4C2211yescharacteristic
42C8×C4⋊C4C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×#102C2211yescharacteristic
50#1744C2211yescharacteristic
51C2×#334C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C2×#125C2×C411yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C22×C42—21no—
72#478C818yescomplemented
73#431—44no—
74C2×#107C2×C411yescharacteristic
75C2×#107C2×C411yescharacteristic
76C2×#106C2×C411yescharacteristic
77C2×#106C2×C411yescharacteristic
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2×C411yescharacteristic
83C2×C4⋊C8C2×C411yescharacteristic
84C2×C4⋊C8D411yescharacteristic
85C2×C4⋊C8D411yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#102C2×C411yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×#102C2×C411yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109#389C2×C411yescharacteristic
110#389C2×C411yescharacteristic
111#387—21no—
112#387—21no—
113#382C2×C411yescharacteristic
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C43C2311yescharacteristic
125#340C2×C411yescharacteristic
126#340C2×C411yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#336—21no—
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336C2×C411yescharacteristic
143#336C2×C411yescharacteristic
144#336D411yescharacteristic
145#336D411yescharacteristic
146#334—22no—
147#334—22no—
148D4⋊C8—22no—
149D4⋊C8—22no—
150#322—22no—
151C82—22no—
152C22×D4C22×C411yescharacteristic
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8C22×C411yescharacteristic
163C22×C8C22×C411yescharacteristic
164C22×C8C22×C411yescharacteristic
165C22×C8C22×C411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170#125—22no—
171#125C2×C814yes—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—41no—
179C4×D4C2×C814yes—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42C2411yescharacteristic
208#107—22no—
209#107C2×C814yes—
210#106—22no—
211#106C2×C814yes—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—41no—
222C4⋊C8—42no—
223C4⋊C8—42no—
224C4⋊C8—81no—
225C4⋊C8—81no—
226C4⋊C8C2×C814yes—
227C4⋊C8#4112yes—
228#102—41no—
229#102—41no—
230#102C2×C814yes—
231#102C2×C814yes—
232C22⋊C8—42no—
233C22⋊C8—42no—
234C22⋊C8—81no—
235C22⋊C8—81no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—42no—
248C4×C8—42no—
249C4×C8—42no—
250C4×C8—81no—
251C4×C8—81no—
252C4×C8#4112yes—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262C24—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4C22×C812yes—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—24no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—81no—
288C22×C4C4×D411yescharacteristic
289C22×C4C4×D411yescharacteristic
290C22×C4C4×D411yescharacteristic
291C22×C4C4×D411yescharacteristic
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8C22×C812yes—
335C2×C8C22×C812yes—
336C2×C8AGL(1,ℤ/8)12yes—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—42no—
350C4⋊C4—81no—
351C4⋊C4C22×C812yes—
352C4⋊C4C22×C812yes—
353C4⋊C4C22×C812yes—
354C4⋊C4#13112yes—
355C22⋊C4—24no—
356C22⋊C4—42no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—41no—
366C42—41no—
367C42—42no—
368C42—42no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42C22×C812yes—
376C42#13112yes—
377C42C2×#4111yescharacteristic
378C42C2×#4111yescharacteristic
379C422−1+411yescharacteristic
380C422−1+411yescharacteristic
381C23—22no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23C2×C4×D411yescharacteristic
389D4—22no—
390D4—22no—
391D4—41no—
392D4—41no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—48no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—84no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4C8×D412yes—
433C2×C4C8×D412yes—
434C2×C4C4×#4111yescharacteristic
435C2×C4#51911yescharacteristic
436C2×C4#54711yescharacteristic
437C2×C4#54811yescharacteristic
438C2×C4#55211yescharacteristic
439C2×C4#55611yescharacteristic
440C2×C4C23×C811yescharacteristic
441C2×C4C2×#13111yescharacteristic
442C2×C4C2×#13111yescharacteristic
443C2×C4#56811yescharacteristic
444C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
445C2×C4#57511yescharacteristic
446C2×C4#57711yescharacteristic
447C2×C4#57711yescharacteristic
448C8—22no—
449C8—22no—
450C8—22no—
451C8—42no—
452C8—41no—
453C8—41no—
454C8—41no—
455C8—42no—
456C8—41no—
457C8—41no—
458C8—81no—
459C8—81no—
460C8—81no—
461C8—81no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22#236211yescharacteristic
470C22C2×C8×D411yescharacteristic
471C22#293311yescharacteristic
472C22C4×AGL(1,ℤ/8)11yescharacteristic
473C22#295211yescharacteristic
474C22#295511yescharacteristic
475C22#295711yescharacteristic
476C4—21no—
477C4—21no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—42no—
482C4—41no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—81no—
487C4—81no—
488C4—81no—
489C4C8×#4111yescharacteristic
490C4#296911yescharacteristic
491C4#298211yescharacteristic
492C4#298611yescharacteristic
493C4#334511yescharacteristic
494C4#334811yescharacteristic
495C4#336811yescharacteristic
496C4#337611yescharacteristic
497C2—82no—
498C2C8×AGL(1,ℤ/8)11yescharacteristic
499C2#1785811yescharacteristic
500C2#1800811yescharacteristic
501C2#1803111yescharacteristic
502C2#2305011yescharacteristic
503C2#2307211yescharacteristic
504C2#2308211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23253C2semidirect
4. #9872C2semidirect
5. #9602C2semidirect
10. #9534C2semidirect
11. #9533C2semidirect
12. #8284C2semidirect
13. C8×#106C2semidirect
14. #8264C2semidirect
72. #478C8semidirect

Maximal subgroups

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

subgroupindex
#232532
#227952
C4×#4782
#98722
#96022
#95562
#95542
#95422
#95342
#95332
#82842
C8×#1062
#82642
#81702