Finite Group Databasebeta

← #506211

Subgroups — #506211

Order 512.

894 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2232C211yesmaximal, characteristic, complemented
2C2×#2184C211yesmaximal, characteristic
3C4×#466C211yesmaximal, characteristic, complemented
4C2×#1970C211yesmaximal, characteristic, complemented
5#13474C212yesmaximal, complemented
6#13448C212yesmaximal, complemented
7#13130C212yesmaximal, complemented
8#13012C212yesmaximal, complemented
9#12866C211yesmaximal, characteristic, complemented
10#12783C211yesmaximal, characteristic, complemented
11#12731C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×D16C2212yes—
14C2×#466C2211yescharacteristic
15C2×C4×D8C2211yescharacteristic, complemented
16C22×#107C2211yescharacteristic
17C22×#102—21no—
18C22×C4×C8C2211yescharacteristic
19C4×#121—21no—
20C2×#390—21no—
21#2232—24no—
22#2184—24no—
23C2×#366C2212yescomplemented
24C2×C4⋊C16C2212yescomplemented
25C2×#357C2212yes—
26C2×#357C2212yes—
27C2×#357C2212yes—
28C2×#357C2212yescomplemented
29C2×C22⋊C16C2211yescharacteristic
30C2×C22⋊C16C2211yescharacteristic
31#1970—21no—
32#1970—21no—
33#1970C2212yes—
34#1970C2212yes—
35#1970C2211yescharacteristic, complemented
36#1942C2211yescharacteristic, complemented
37#1934—21no—
38#1893—21no—
39#1808C2211yescharacteristic
40C4×#107C2211yescharacteristic, complemented
41C4×#102—21no—
42#1380C2212yescomplemented
43#1380C2212yescomplemented
44#1374C2212yescomplemented
45C23×D4—21no—
46C22×D8—24no—
47C22×D8—21no—
48C22×D8—22no—
49C22×D8—22no—
50C22×D8—42no—
51C22×D8C2311yescharacteristic
52C22×D8C2312yes—
53C23×C8D411yescharacteristic
54C23×C8C2311yescharacteristic
55C2×#121—21no—
56C2×C4×D4—21no—
57C2×C4×D4—41no—
58C22×C4⋊C4—21no—
59C22×C22⋊C4—41no—
60C22×C22⋊C4—41no—
61C22×C42D411yescharacteristic
62C2×D16—24no—
63C2×D16—24no—
64C2×D16—42no—
65C22×C16C2312yes—
66C22×C16C2312yes—
67#466—22no—
68#466D414yes—
69C4×D8—42no—
70C2×#107—22no—
71C2×#107—22no—
72C2×#107—21no—
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2312yes—
76C2×#107C2311yescharacteristic
77C2×#102—21no—
78C2×#102—22no—
79C2×#102—22no—
80C2×#102—21no—
81C2×#102—21no—
82C2×#102—44no—
83C2×C4×C8—42no—
84C2×C4×C8D411yescharacteristic
85C2×C4×C8D411yescharacteristic
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88#390—21no—
89#390—22no—
90#390—22no—
91#390—41no—
92#390—41no—
93#389—21no—
94#388—41no—
95#386—41no—
96#380—21no—
97C4×C4⋊C4—21no—
98C4×C22⋊C4—41no—
99#366—24no—
100C4⋊C16—42no—
101#357—24no—
102#357—24no—
103#357—24no—
104#357—42no—
105C22⋊C16—22no—
106C22⋊C16—22no—
107#340—21no—
108#340—21no—
109#340—21no—
110C25—41no—
111C22×D4—24no—
112C22×D4—21no—
113C22×D4—22no—
114C22×D4—42no—
115C22×D4—42no—
116C22×D4—42no—
117C22×D4—41no—
118C22×D4—42no—
119C22×D4—41no—
120C22×D4—42no—
121C23×C4—41no—
122C23×C4—41no—
123C23×C4D811yescharacteristic
124C23×C4D811yescharacteristic
125C23×C4C2×D411yescharacteristic
126C2×D8—24no—
127C2×D8—22no—
128C2×D8—22no—
129C2×D8—22no—
130C2×D8—22no—
131C2×D8—21no—
132C2×D8—22no—
133C2×D8—22no—
134C2×D8—22no—
135C2×D8—48no—
136C2×D8—48no—
137C2×D8—42no—
138C2×D8—42no—
139C2×D8C2×D412yes—
140C2×D8#4114yes—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—41no—
145C22×C8—41no—
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D412yes—
150C22×C8#4112yes—
151C22×C8C2411yescharacteristic
152#121—24no—
153#121—42no—
154C4×D4—42no—
155C4×D4—42no—
156C4×D4—84no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—84no—
175C2×C22⋊C4—84no—
176C2×C42—41no—
177C2×C42—41no—
178C2×C42—41no—
179C2×C42D811yescharacteristic
180C2×C42D811yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183D16—48no—
184C2×C16—22no—
185C2×C16—22no—
186C2×C16—22no—
187C2×C16—22no—
188C2×C16—22no—
189C2×C16—22no—
190#107—21no—
191#107—22no—
192#107—22no—
193#107—22no—
194#107—22no—
195#107—42no—
196#107C2×D412yes—
197#102—24no—
198#102—42no—
199#102—44no—
200#102—44no—
201#102—42no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—41no—
205C4×C8—41no—
206#95—41no—
207#95—41no—
208#95—41no—
209C24—44no—
210C24—41no—
211C24—42no—
212C24—82no—
213C24—82no—
214C24—82no—
215C24—82no—
216C24—82no—
217C24—82no—
218C24C2×D811yescharacteristic
219C2×D4—22no—
220C2×D4—24no—
221C2×D4—48no—
222C2×D4—48no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—42no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—82no—
236C2×D4—84no—
237C2×D4—84no—
238C2×D4—82no—
239C2×D4—82no—
240C2×D4—82no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4D1614yescomplemented
261C22×C4#12112yes—
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4AGL(1,ℤ/8)12yes—
268C22×C4C22×D411yescharacteristic
269D8—24no—
270D8—24no—
271D8—24no—
272D8—48no—
273D8—42no—
274D8—42no—
275D8—42no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—44no—
287C2×C8#12112yes—
288C2×C8C22×D411yescharacteristic
289C2×C8C2×#4112yes—
290C2×C8C2×#4111yescharacteristic
291C2×C8H(5,𝔽2)12yes—
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C22⋊C4—44no—
303C22⋊C4—82no—
304C22⋊C4—84no—
305C22⋊C4—84no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—84no—
309C22⋊C4—84no—
310C22⋊C4—82no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—41no—
316C42—41no—
317C42—82no—
318C16—42no—
319C16—42no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—44no—
325C23—42no—
326C23—88no—
327C23—88no—
328C23—88no—
329C23—81no—
330C23—82no—
331C23—81no—
332C23—82no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—82no—
337C23—82no—
338C23—81no—
339C23—82no—
340C23—82no—
341C23—164no—
342C23—164no—
343C23—164no—
344C23—164no—
345C23#45912yes—
346C23C2×D1612yes—
347C23#50912yes—
348C23C22×D811yescharacteristic
349D4—44no—
350D4—42no—
351D4—44no—
352D4—42no—
353D4—88no—
354D4—88no—
355D4—81no—
356D4—81no—
357D4—82no—
358D4—82no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—82no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—82no—
389C2×C4—164no—
390C2×C4—164no—
391C2×C4#45912yes—
392C2×C4C2×D1612yes—
393C2×C4C2×D1612yes—
394C2×C4C2×#12111yescharacteristic
395C2×C4#52511yescharacteristic
396C2×C4#53411yescharacteristic
397C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
398C2×C4#57511yescharacteristic
399C2×C4#57511yescharacteristic
400C8—22no—
401C8—41no—
402C8—41no—
403C8#54612yes—
404C8#55112yes—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—82no—
412C22—82no—
413C22—84no—
414C22—84no—
415C22—84no—
416C22—82no—
417C22—1616no—
418C22—162no—
419C22—162no—
420C22—161no—
421C22—161no—
422C22—161no—
423C22—162no—
424C22—161no—
425C22—162no—
426C22#220614yes—
427C22C2×#45911yescharacteristic
428C22#310011yescharacteristic
429C22#318611yescharacteristic
430C22C22×D1611yescharacteristic
431C22C2×#50911yescharacteristic
432C22#341411yescharacteristic
433C22#341411yescharacteristic
434C4—22no—
435C4—22no—
436C4—22no—
437C4—41no—
438C4—41no—
439C4—82no—
440C4—162no—
441C4—161no—
442C4—161no—
443C4#253411yescharacteristic
444C4#304011yescharacteristic
445C4#311411yescharacteristic
446C4#319411yescharacteristic
447C2—24no—
448C2—162no—
449C2—164no—
450C2#2811411yescharacteristic
451C2C2×#220611yescharacteristic
452C2#3358411yescharacteristic
453C2#3366011yescharacteristic
454C2#3367411yescharacteristic
455C2#3375211yescharacteristic
456C2#3376011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#2184C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2232C2semidirect
3. C4×#466C2semidirect
4. C2×#1970C2semidirect
5. #13474C2semidirect
6. #13448C2semidirect
7. #13130C2semidirect
8. #13012C2semidirect
9. #12866C2semidirect
10. #12783C2semidirect
11. #12731C2semidirect
15. C2×C4×D8C22semidirect
23. C2×#366C22semidirect
24. C2×C4⋊C16C22semidirect
28. C2×#357C22semidirect
35. #1970C22semidirect
36. #1942C22semidirect
40. C4×#107C22semidirect
42. #1380C22semidirect
43. #1380C22semidirect
44. #1374C22semidirect
260. C22×C4D16semidirect

Maximal subgroups

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

subgroupindex
C2×#22322
C2×#21842
C4×#4662
C2×#19702
#134742
#134482
#131302
#130122
#128662
#127832
#127312