Finite Group Databasebeta

← #6434418

Subgroups — #6434418

Order 512.

1438 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35339C211yesmaximal, characteristic, complemented
2#35266C211yesmaximal, characteristic
3#30395C214yesmaximal, complemented
4#30389C214yesmaximal, complemented
5#30386C214yesmaximal, complemented
6#30381C214yesmaximal, complemented
7#23744C211yesmaximal, characteristic, complemented
8#23731C211yesmaximal, characteristic
9#23671C212yesmaximal, complemented
10#23659C211yesmaximal, characteristic, complemented
11C2×#2073C211yesmaximal, characteristic
12#23614C212yesmaximal
13#21160C212yesmaximal, complemented
14#21091C212yesmaximal
15#20430C211yesmaximal, characteristic, complemented
16C2×#428C2212yes—
17C2×#428C2212yes—
18C2×#424C2212yes—
19C2×#418C2212yes—
20C22×#102C2212yes—
21C2×#405C2212yes—
22#2864C2214yescomplemented
23#2860C2214yes—
24#2647C2212yescomplemented
25#2647C2212yescomplemented
26#2644C2214yes—
27#2570—22no—
28#2570—22no—
29#2569—24no—
30#2513—22no—
31#2512—22no—
32#2480—24no—
33#2317C2212yes—
34#2317C2211yescharacteristic
35#2316C2212yes—
36#2316C2211yescharacteristic
37#2309C2211yescharacteristic
38#2309C2211yescharacteristic
39#2296C2211yescharacteristic
40C2×#390C2211yescharacteristic
41#2285C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C4×#117C2211yescharacteristic
44#2101C2212yescomplemented
45#2101C2212yescomplemented
46#2100C2212yescomplemented
47#2099C2212yescomplemented
48#2094C2212yes—
49#2093C2212yes—
50#2092C2214yes—
51#2088C2214yescomplemented
52#2088C2214yescomplemented
53#2087C2214yescomplemented
54#2086C2214yescomplemented
55#2085C2214yescomplemented
56#2085C2214yescomplemented
57#2073C2212yes—
58#2073C2212yes—
59#2073C2214yes—
60#2067C2214yes—
61#2067C2214yes—
62#2065C2214yes—
63#2065C2214yes—
64#1944C2214yescomplemented
65#1943C2214yescomplemented
66#1942C2214yescomplemented
67#1941C2214yescomplemented
68#1936C2214yes—
69#1936C2214yes—
70#1934C2214yes—
71#1934C2214yes—
72#1768C2214yescomplemented
73#1767C2212yescomplemented
74C4×C4⋊C8C2212yescomplemented
75#1736C2212yes—
76#1735C2212yes—
77#1735C2212yes—
78C2×#340C2212yes—
79#1728C2211yescharacteristic
80C2×#336C2211yescharacteristic
81C23×D4C2311yescharacteristic
82C22×M16C2312yes—
83C23×C8C2312yes—
84C2×#126—24no—
85C2×#117C2311yescharacteristic
86C2×#117C2311yescharacteristic
87C2×#117C2312yes—
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4—22no—
93C22×C42C2311yescharacteristic
94#428—28no—
95#428—44no—
96C2×#107C2314yes—
97C2×#107C2314yes—
98C2×#106C2314yes—
99C2×#106C2314yes—
100#424—28no—
101C2×C4⋊C8C2314yes—
102C2×C4⋊C8C2314yes—
103#418—28no—
104C2×#102—44no—
105C2×#102C2314yes—
106C2×#102C2314yes—
107C2×#102C2314yes—
108C2×#102C2314yes—
109#405—44no—
110C2×#97C2314yes—
111C2×C4×C8C2314yes—
112#400—24no—
113#400—24no—
114#398—24no—
115#398—24no—
116#397—24no—
117#397—24no—
118#397—24no—
119#396—24no—
120#390—21no—
121#390—22no—
122#390—21no—
123#390—22no—
124#390C2314yes—
125#390C2314yes—
126#390C2312yes—
127#390C2312yes—
128#389—22no—
129#389—22no—
130#389—22no—
131#389C2314yes—
132#389C2314yes—
133#389C2314yes—
134#389C2314yes—
135#389C2314yes—
136#389C2314yes—
137#389C2312yes—
138#389C2312yes—
139#388—24no—
140#388—24no—
141#388—24no—
142#387—24no—
143#387—24no—
144#387—24no—
145#387—24no—
146#385—22no—
147#385—24no—
148#385—22no—
149#384—22no—
150#384—21no—
151#384C2311yescharacteristic
152#384C2311yescharacteristic
153#382—22no—
154#382—21no—
155#382—22no—
156#382C2312yes—
157#382C2312yes—
158#382C2312yes—
159#380—22no—
160#380—24no—
161#380—22no—
162#379—22no—
163#379—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4D412yes—
167C4×C4⋊C4D412yes—
168C4×C22⋊C4D414yes—
169#376C2312yes—
170C2×#95—22no—
171C43C2312yes—
172#340C2312yes—
173#340C2312yes—
174#340C2312yes—
175#340C2312yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2312yes—
181#340C2312yes—
182#340C2312yes—
183#340C2312yes—
184#336C2312yes—
185#336C2311yescharacteristic
186#336C2311yescharacteristic
187#336C2312yes—
188#336C2312yes—
189C25—21no—
190C22×D4—21no—
191C22×D4—24no—
192C22×D4—22no—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4C2412yes—
196C23×C4—21no—
197C23×C4—22no—
198C23×C4C2×D412yes—
199C23×C4C2411yescharacteristic
200C2×M16—24no—
201C2×M16—24no—
202C2×M16#4118yes—
203C22×C8—24no—
204C22×C8—24no—
205C22×C8—24no—
206C22×C8C2412yes—
207C22×C8C2412yes—
208C22×C8C2414yes—
209#126—88no—
210#117—24no—
211#117—22no—
212#117—22no—
213#117—24no—
214#117—24no—
215#117—22no—
216#117—24no—
217#117—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2414yes—
237C2×C4⋊C4C2412yes—
238C2×C4⋊C4C2412yes—
239C2×C4⋊C4C2412yes—
240C2×C4⋊C4C2412yes—
241C2×C4⋊C4C2412yes—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—84no—
252C2×C22⋊C4C2×D412yes—
253C2×C22⋊C4C2×D412yes—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42C2×D412yes—
259C2×C42C2×D414yes—
260C2×C42C2×D414yes—
261C2×C42C2412yes—
262C2×C42C2412yes—
263C2×C42C2411yescharacteristic
264C2×C42C2411yescharacteristic
265#107—28no—
266#107—44no—
267#106—28no—
268#106—44no—
269C4⋊C8—28no—
270C4⋊C8—28no—
271#102—28no—
272#102—28no—
273#102—44no—
274#102—44no—
275#97—44no—
276C4×C8—44no—
277#95—24no—
278#95—24no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95C2×D414yes—
290#95C2×D414yes—
291C24—22no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—82no—
296C24—82no—
297C24C22×D411yescharacteristic
298C2×D4—24no—
299C2×D4—22no—
300C2×D4—24no—
301C2×D4—22no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—48no—
306C2×D4—48no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4C22×D412yes—
322C22×C4C22×D412yes—
323C22×C4C22×D412yes—
324C22×C4C22×D414yes—
325C22×C4C22×D412yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4114yes—
328C22×C4H(5,𝔽2)12yes—
329C22×C42−1+414yes—
330C22×C42−1+412yes—
331C22×C4C2511yescharacteristic
332M16—216no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—48no—
342C2×C8C2×#4118yes—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—44no—
357C4⋊C4—42no—
358C4⋊C4—44no—
359C4⋊C4—84no—
360C4⋊C4—84no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—84no—
364C4⋊C4—84no—
365C4⋊C4—82no—
366C4⋊C4—84no—
367C4⋊C4—84no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—48no—
371C22⋊C4—44no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—84no—
375C22⋊C4—84no—
376C22⋊C4—84no—
377C22⋊C4—84no—
378C22⋊C4—84no—
379C42—24no—
380C42—24no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—44no—
386C42—44no—
387C42—44no—
388C42—42no—
389C42—42no—
390C23—22no—
391C23—22no—
392C23—44no—
393C23—44no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—88no—
398C23—84no—
399C23—162no—
400C23—162no—
401C23—164no—
402C23#53212yes—
403C23#53612yes—
404C23#53712yes—
405C23C23×D411yescharacteristic
406D4—44no—
407D4—44no—
408D4—84no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—84no—
440C2×C4—162no—
441C2×C4—164no—
442C2×C4#53212yes—
443C2×C4#53612yes—
444C2×C4#53712yes—
445C2×C4#53918yes—
446C2×C4#54118yes—
447C2×C4C22×#4112yes—
448C2×C4C2×H(5,𝔽2)11yescharacteristic
449C2×C4C2×2−1+412yes—
450C2×C4C2×2−1+411yescharacteristic
451C8—28no—
452C8—42no—
453C8—42no—
454C22—22no—
455C22—22no—
456C22—24no—
457C22—24no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—84no—
462C22—84no—
463C22—168no—
464C22#266414yes—
465C22C2×#53211yescharacteristic
466C22C2×#53611yescharacteristic
467C22C2×#53711yescharacteristic
468C22#358412yes—
469C22#358512yes—
470C4—42no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—44no—
475C4—82no—
476C4—82no—
477C4—82no—
478C4—82no—
479C4—82no—
480C4—84no—
481C4#266414yes—
482C4C2×#53912yes—
483C4C2×#54112yes—
484C2—24no—
485C2—162no—
486C2C2×#266411yescharacteristic
487C2#6072411yescharacteristic
488C2#6072511yescharacteristic
489C2#6087512yes—
490C2#6092612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35339C2semidirect
3. #30395C2semidirect
4. #30389C2semidirect
5. #30386C2semidirect
6. #30381C2semidirect
7. #23744C2semidirect
9. #23671C2semidirect
10. #23659C2semidirect
13. #21160C2semidirect
15. #20430C2semidirect
22. #2864C22semidirect
24. #2647C22semidirect
25. #2647C22semidirect
44. #2101C22semidirect
45. #2101C22semidirect
46. #2100C22semidirect
47. #2099C22semidirect
51. #2088C22semidirect
52. #2088C22semidirect
53. #2087C22semidirect
54. #2086C22semidirect
55. #2085C22semidirect
56. #2085C22semidirect
64. #1944C22semidirect
65. #1943C22semidirect
66. #1942C22semidirect
67. #1941C22semidirect
72. #1768C22semidirect
73. #1767C22semidirect
74. C4×C4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#353392
#352662
#303952
#303892
#303862
#303812
#237442
#237312
#236712
#236592
C2×#20732
#236142
#211602
#210912
#204302