Finite Group Databasebeta

← #6435244

Subgroups — #6435244

Order 512.

2478 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2856C211yesmaximal, characteristic
2#34012C211yesmaximal, characteristic, complemented
3#29925C218yesmaximal, complemented
4#29921C214yesmaximal, complemented
5#29920C214yesmaximal, complemented
6#23731C212yesmaximal, complemented
7#23628C212yesmaximal, complemented
8#23480C214yesmaximal, complemented
9#21260C212yesmaximal, complemented
10#21171C212yesmaximal, complemented
11#20626C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×#121—22no—
14C22×#121C2212yes—
15C23×C42C2211yescharacteristic
16C2×#428C2214yes—
17C2×#423C2212yes—
18C2×#418C2214yes—
19C2×#418C2214yes—
20C2×#407C2212yes—
21#2856—22no—
22#2856—22no—
23#2856C2214yes—
24#2856C2214yes—
25C2×#390—22no—
26C2×#390C2212yes—
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29#2285C2212yescomplemented
30#2285C2212yescomplemented
31C2×#384C2211yescharacteristic
32#2283—22no—
33#2283—22no—
34#2283C2214yescomplemented
35#2280—22no—
36#2280C2212yes—
37C4×#117—22no—
38C4×#117C2212yes—
39#2094C2214yescomplemented
40#2093C2214yescomplemented
41#2092C2218yescomplemented
42#2073C2218yescomplemented
43#2071C2218yescomplemented
44#2034C22116yescomplemented
45#2033C22116yescomplemented
46#1982C2218yescomplemented
47#1982C2218yescomplemented
48#1948C2218yescomplemented
49#1947C2218yescomplemented
50#1797C2214yescomplemented
51#1797C2214yescomplemented
52#1736C2214yes—
53#1728C2212yes—
54C23×D4—22no—
55C23×D4—22no—
56C23×D4C2312yes—
57C24×C4D412yes—
58C24×C4C2311yescharacteristic
59C22×M16C2314yes—
60C2×#127—21no—
61C2×#127—21no—
62C2×#127—28no—
63C2×#127C2314yes—
64C2×#121—24no—
65C2×#121—24no—
66C2×#121—24no—
67C2×#121—24no—
68C2×#121—24no—
69C2×#121—48no—
70C2×#121—416no—
71C2×#121C2318yes—
72C2×#117—22no—
73C2×#117—22no—
74C2×#117C2312yes—
75C2×#117C2312yes—
76C22×C4⋊C4—22no—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—24no—
79C22×C22⋊C4—22no—
80C22×C42—28no—
81C22×C42D412yes—
82C22×C42D412yes—
83C22×C42D412yes—
84C22×C42C2312yes—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#428—48no—
89C2×#107C2318yes—
90C2×#106C2318yes—
91#423—28no—
92C2×C4⋊C8C2318yes—
93#418—48no—
94#418—48no—
95C2×#102C23116yes—
96C2×#102C2318yes—
97C2×#102C2318yes—
98#407—44no—
99C2×C22⋊C8C2318yes—
100#390—24no—
101#390—24no—
102#390—24no—
103#390—24no—
104#390—24no—
105#390C2314yes—
106#390C2314yes—
107#384—21no—
108#384—21no—
109#384C2312yes—
110#384C2314yes—
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#383—24no—
114#382—22no—
115#382—22no—
116#382C2312yes—
117#382C2312yes—
118#381—24no—
119#381—22no—
120#380—22no—
121#380—22no—
122#380—22no—
123#380—22no—
124#380C2314yes—
125#380C2314yes—
126#379—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4C2314yes—
132C4×C4⋊C4C2314yes—
133C4×C22⋊C4—24no—
134C4×C22⋊C4—24no—
135C4×C22⋊C4—24no—
136#376—24no—
137#376—22no—
138C43—22no—
139#340C23116yes—
140#336C2314yes—
141#336C2314yes—
142C25—22no—
143C25C2×D411yescharacteristic
144C22×D4—28no—
145C22×D4—22no—
146C22×D4—24no—
147C22×D4—24no—
148C22×D4—28no—
149C22×D4—28no—
150C22×D4—24no—
151C22×D4—24no—
152C22×D4—48no—
153C22×D4—48no—
154C22×D4—416no—
155C22×D4—48no—
156C22×D4—48no—
157C22×D4C2414yes—
158C23×C4—28no—
159C23×C4—28no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—44no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2412yes—
175C23×C4C2411yescharacteristic
176C2×M16—28no—
177C2×M16—28no—
178C2×M16—28no—
179C22×C8C2418yes—
180#127—216no—
181#127—44no—
182#127—44no—
183#121—416no—
184#121—416no—
185#121—416no—
186#121—416no—
187#121—416no—
188#121—416no—
189#117—24no—
190#117—24no—
191#117—24no—
192#117—24no—
193#117—48no—
194#117—48no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—48no—
204C2×C4⋊C4C2414yes—
205C2×C4⋊C4C2414yes—
206C2×C4⋊C4C2412yes—
207C2×C4⋊C4C2412yes—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—48no—
216C2×C22⋊C4—416no—
217C2×C42—28no—
218C2×C42—24no—
219C2×C42—28no—
220C2×C42—28no—
221C2×C42—28no—
222C2×C42—28no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42#4118yes—
235C2×C42C2411yescharacteristic
236C2×C42C2411yescharacteristic
237C2×C42C2412yes—
238#107—48no—
239#106—48no—
240C4⋊C8—216no—
241#102—416no—
242#102—48no—
243#102—48no—
244C22⋊C8—48no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251C24—22no—
252C24—24no—
253C24—22no—
254C24—22no—
255C24—24no—
256C24—42no—
257C24—48no—
258C24—48no—
259C24—44no—
260C24—44no—
261C24—44no—
262C24C22×D412yes—
263C24C22×D411yescharacteristic
264C2×D4—28no—
265C2×D4—24no—
266C2×D4—28no—
267C2×D4—24no—
268C2×D4—48no—
269C2×D4—48no—
270C2×D4—416no—
271C2×D4—416no—
272C2×D4—416no—
273C2×D4—48no—
274C2×D4—48no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—416no—
279C2×D4—416no—
280C2×D4—48no—
281C2×D4—48no—
282C2×D4—832no—
283C2×D4—832no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—432no—
315C22×C4—48no—
316C22×C4—48no—
317C22×C4—48no—
318C22×C4—44no—
319C22×C4#12318yes—
320C22×C4AGL(1,ℤ/8)18yes—
321C22×C4C22×D412yes—
322C22×C4C22×D412yes—
323C22×C4C22×D412yes—
324C22×C4C22×D412yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D412yes—
327C22×C4C2×#4114yes—
328C22×C4H(5,𝔽2)14yes—
329C22×C4H(5,𝔽2)12yes—
330C22×C42−1+412yes—
331C22×C4C2511yescharacteristic
332M16—416no—
333C2×C8—28no—
334C2×C8—28no—
335C2×C8—28no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—28no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—48no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—48no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C22⋊C4—416no—
351C22⋊C4—416no—
352C22⋊C4—48no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—48no—
356C22⋊C4—416no—
357C42—24no—
358C42—24no—
359C42—24no—
360C42—24no—
361C42—24no—
362C42—22no—
363C42—22no—
364C42—44no—
365C42—44no—
366C42—48no—
367C42—48no—
368C42#12314yes—
369C42#12314yes—
370C42C2×#4118yes—
371C23—22no—
372C23—22no—
373C23—28no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—48no—
381C23—48no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—48no—
388C23—48no—
389C23—816no—
390C23—88no—
391C23—88no—
392C23—816no—
393C23—88no—
394C23C2×#12312yes—
395C23#53512yes—
396C23#53712yes—
397C23C2×AGL(1,ℤ/8)14yes—
398C23C23×D411yescharacteristic
399D4—48no—
400D4—48no—
401D4—816no—
402D4—832no—
403D4—816no—
404D4—88no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—216no—
410C2×C4—216no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—48no—
418C2×C4—48no—
419C2×C4—48no—
420C2×C4—44no—
421C2×C4—48no—
422C2×C4—48no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—48no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—88no—
435C2×C4C2×#12314yes—
436C2×C4C2×#12314yes—
437C2×C4C2×#12312yes—
438C2×C4#53512yes—
439C2×C4#53712yes—
440C2×C4#54018yes—
441C2×C4#54218yes—
442C2×C4C2×AGL(1,ℤ/8)14yes—
443C2×C4C2×AGL(1,ℤ/8)14yes—
444C2×C4#57514yes—
445C2×C4C22×#4112yes—
446C2×C4C2×H(5,𝔽2)11yescharacteristic
447C2×C4C2×H(5,𝔽2)12yes—
448C2×C4C2×2−1+411yescharacteristic
449C8—48no—
450C22—24no—
451C22—22no—
452C22—22no—
453C22—24no—
454C22—24no—
455C22—24no—
456C22—24no—
457C22—416no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—84no—
463C22—88no—
464C22—84no—
465C22—84no—
466C22—88no—
467C22—1616no—
468C22#262414yes—
469C22C22×#12311yescharacteristic
470C22C2×#53511yescharacteristic
471C22C2×#53711yescharacteristic
472C22C22×AGL(1,ℤ/8)12yes—
473C22C2×#57512yes—
474C4—24no—
475C4—24no—
476C4—42no—
477C4—42no—
478C4—82no—
479C4—84no—
480C4—82no—
481C4—84no—
482C4#262414yes—
483C4#309418yes—
484C4C2×#54012yes—
485C4C2×#54212yes—
486C2—24no—
487C2—44no—
488C2—164no—
489C2C2×#262411yescharacteristic
490C2C2×#309412yes—
491C2#6079811yescharacteristic
492C2#6080211yescharacteristic
493C2#6091612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2856C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34012C2semidirect
3. #29925C2semidirect
4. #29921C2semidirect
5. #29920C2semidirect
6. #23731C2semidirect
7. #23628C2semidirect
8. #23480C2semidirect
9. #21260C2semidirect
10. #21171C2semidirect
11. #20626C2semidirect
29. #2285C22semidirect
30. #2285C22semidirect
34. #2283C22semidirect
39. #2094C22semidirect
40. #2093C22semidirect
41. #2092C22semidirect
42. #2073C22semidirect
43. #2071C22semidirect
44. #2034C22semidirect
45. #2033C22semidirect
46. #1982C22semidirect
47. #1982C22semidirect
48. #1948C22semidirect
49. #1947C22semidirect
50. #1797C22semidirect
51. #1797C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28562
#340122
#299252
#299212
#299202
#237312
#236282
#234802
#212602
#211712
#206262